Lech-Mumford constant and stability of local rings

TL;DR

This paper explores the concept of local semistability in singularities, demonstrating that semistable singularities are log canonical and introducing the Lech-Mumford constant as a key invariant in this context.

Contribution

It advances the understanding of Mumford's local semistability, establishes new links with log canonical singularities, and develops the theory of the Lech-Mumford constant with new examples.

Findings

Semistable singularities are log canonical under mild conditions.

Introduced and studied the Lech-Mumford constant as an invariant.

Provided new examples of semistable and unstable singularities.

Abstract

We study further Mumford's notion of local semistability and, in particular, show that semistable singularities are log canonical under mild assumptions. We provide many new examples of semistable and unstable singularities. More generally, we develop the theory of the Lech-Mumford constant, an invariant defined as an optimal constant in the Lech inequality.

Equations

$$e(I)\le d!e(R)\ell (R/I),$$

$${\mathrm{c}}_{\mathrm{LM}}(R):=\underset{\sqrt{I}=\mathfrak{m}}{\sup }\left\{\frac{e(I)}{d!\ell (R/I)}\right\}.$$

$${\mathrm{c}}_{\mathrm{LM}}(R)\ge {\mathrm{c}}_{\mathrm{LM}}(R[[T]])\text{ and strict inequality is possible (and often).}$$

$$\underset{n\to \infty }{\lim }{\mathrm{c}}_{\mathrm{LM}}(R[[T_1,\dots ,T_n]])$$

$$\text{lim-stable}.$$

$$h_M(t):=\sum _{n=-\infty }^{\infty }{\ell }_{G_0}(M_n)t^n\in \mathbb{Z}[[t]].$$

$$h_M(t)=\frac{f(t)}{{\prod }_{i=1}^s(1-t^{d_i})},$$

$${\text{gr}}_I(R):=R/I\oplus I/I^2\oplus I^2/I^3\oplus \cdots$$

$$p_I(n)=e_0(I)\left(\genfrac{}{}{0ex}{}{n+d-1}{d}\right)-e_1(I)\left(\genfrac{}{}{0ex}{}{n+d-2}{d-1}\right)+\cdots +(-1{)}^d{e}_d(I),$$

$$\Delta p_I(n):=p_I(n+1)-p_I(n)=e_0(I)\left(\genfrac{}{}{0ex}{}{n+d-1}{d-1}\right)-e_1(I)\left(\genfrac{}{}{0ex}{}{n+d-2}{d-2}\right)+\cdots +(-1{)}^{d-1}{e}_{d-1}(I)$$

$$e(I):=\underset{n\to \infty }{\lim }\frac{d!\ell (R/I^n)}{n^d}.$$

$$e(I,M):=\underset{n\to \infty }{\lim }\frac{d!\ell (M/I^{n}M)}{n^d}.$$

$$e(I,M)=\sum _{\mathfrak{p}}\ell (M_{\mathfrak{p}})e(I,R/\mathfrak{p}),$$

$$x^n+a_1{x}^{n-1}+\cdots +a_n=0,$$

$$e(I,R)=e(IR/(x_1,\dots ,x_k),R/(x_1,\dots ,x_k)).$$

$$\frac{1}{d!}\sum _{k=0}^d\left(\genfrac{}{}{0ex}{}{d}{d-k}\right)e(I^{[k]}\mid J^{[d-k]})n^k{m}^{d-k}.$$

$$e(I^{[k]}\mid J^{[d-k]})=e(IR/(x_1,\dots ,x_{d-k}))$$

$$e(I)=-(-E{)}^d.$$

$$I^{\frac{b}{a}}:=\{x\in R\mid x^a\in \overline{I^b}\}.$$

$$\rho (I):=\mathrm{lcm}\{v_i(I)\mid v_i\text{ are the Rees valuations of }I\}.$$

$$\ell (R/I^{\frac{n\rho +r}{\rho }})=\frac{e(I)}{d!}{n}^d+O(n^{d-1}).$$

$$0\to I^{\frac{w}{\rho }}/I^n{I}^{\frac{w}{\rho }}\to R/I^{\frac{w+n\rho }{\rho }}\to R/I^{\frac{w}{\rho }}\to 0$$

$$I^{\frac{b}{a}}=\Gamma (Z,{\mathcal{O}}_Z(-\lceil \frac{b}{a}E\rceil ))\cap R.$$

$$Y:=\mathrm{Proj}(R\oplus \overline{I}T\oplus \overline{I^2}{T}^2\oplus \overline{I^3}{T}^3\oplus \cdots )\to \mathrm{Spec}(R).$$

$$(I,T^a{)}^{\frac{n}{a}}=\sum _{k=0}^n{I}^{\frac{k}{a}}{T}^{n-k}.$$

$$(I^{\frac{k}{a}}{T}^{n-k}{)}^a\subseteq \overline{I^k}(T^a{)}^{n-k}\subseteq \overline{(I,T^a{)}^n}.$$

$$(x^a{T}^{a(n-s)}{)}^m+f_1(x^a{T}^{a(n-s)}{)}^{m-1}+\cdots +f_k(x^a{T}^{a(n-s)}{)}^{m-k}+\cdots +f_m=0$$

$$x^{am}+y_1{x}^{a(m-1)}+\cdots +y_m=0,$$

$$\ell (R[[T]]/(I,T^m{)}^{\frac{nm+r}{m}})=\sum _{h=0}^{s-1}\sum _{k=0}^n\frac{m}{\rho }\ell (R/I^{\frac{k\rho +h}{\rho }})+\sum _{h=s}^{\rho -1}\sum _{k=0}^{n-1}\frac{m}{\rho }\ell (R/I^{\frac{k\rho +h}{\rho }})+\left(\frac{m}{\rho }-t\right)\ell (R/I^{\frac{n\rho +s}{\rho }}).$$

$$(I,T^m{)}^{\frac{nm+r}{m}}=\sum _{k=0}^{nm+r}{I}^{\frac{k}{m}}{T}^{nm+r-k}=\sum _{k=0}^{nm+r}{I}^{\frac{\lceil k\rho /m\rceil }{\rho }}{T}^{nm+r-k}.$$

Full text

1. 1 Preliminaries

2. 1.1 Hilbert functions, multiplicities, and integral closure

3. 1.2 Rational powers

4. 1.3 Stable ideals and pseudo-rational rings

5. 2 Definition and basic properties

6. 2.1 Lech–Mumford constant

7. 2.2 Stability of local rings

8. 2.3 Localization and faithfully flat extension

9. 2.4 Weak semicontinuity

10. 3 Examples: surface singularities

11. 3.1 Lech-stable singularities

12. 3.2 Minimally elliptic singularities

13. 3.3 Asymptotic Lech’s inequality

14. 4 Estimates on higher Lech–Mumford constants

15. 4.1 Lower bound on $\operatorname{c^{\infty}_{LM}}(R)$ via graded families of ideals

16. 4.2 Semistability in dimension one

17. 4.3 Strict complete intersections and rings of large multiplicity

18. 5 Connections with singularities in MMP

19. 5.1 Singularities in the minimal model program

20. 5.2 Asymptotic Riemann-Roch formula and its consequence

21. 5.3 Lim-stable and log canonical singularities

22. 5.4 Lech-stable and canonical singularities

23. 5.5 Semi-log canonical singularities

24. 6 Examples: simple normal crossings and determinantal rings

25. 6.1 Inclusion-exclusion formulas

26. 6.2 Simple normal crossings and determinantal rings

27. 6.3 Rational triple points

28. 7 Lech’s inequality revisited

29. 7.1 Optimal Lech’s inequality in dimension three

30. 7.2 Elliptic and Rational polygonal cones

31. 7.3 A more general conjecture

32. 7.4 Three points in $\mathbb{P}^{2}$

33. 7.5 Semi-log canonical hypersurfaces in dimension two

Introduction

Overview

Let $(R,{\mathfrak{m}})$ be a Noetherian local ring. In [Lec60], Lech proved a simple inequality relating two basic invariants of an ${\mathfrak{m}}$-primary ideal $I\subseteq R$ – the Hilbert–Samuel multiplicity (denoted $\operatorname{e}(I)$) and the colength – as follows:

Theorem I** (Lech’s inequality [Lec60, Theorem 3]).**

Let $(R,{\mathfrak{m}})$ be a Noetherian local ring of dimension $d$, and let $I\subseteq R$ be an ${\mathfrak{m}}$-primary ideal. Then

[TABLE]

where $\operatorname{e}(R)\coloneqq\operatorname{e}({\mathfrak{m}})$.

Lech also observed that his inequality is never sharp when $d\geq 2$ (see [Lec60, Page 74, after (4.1)]); that is, the inequality in Theorem I is always strict in this case. This naturally raises the question of whether the constant $\operatorname{e}(R)$ in Lech’s inequality can be improved. As far as the authors are aware, this problem was first considered by Mumford [Mum77]. Hence, for a Noetherian local ring $(R,{\mathfrak{m}})$ of dimension $d$, it seems proper to introduce the following invariant:

[TABLE]

We call $\operatorname{c_{LM}}(R)$ the Lech–Mumford constant of $R$. Note that Mumford introduced the same invariant but called and denoted it differently, see Remark 2.1.2.

Mumford’s motivation for studying this invariant came from understanding singularities on the compactification of the moduli spaces of smooth projective varieties constructed via Geometric Invariant Theory (GIT). More precisely, he proved the following result.

Theorem II** ([Mum77, Proposition 3.12], see also [Sha81, Proposition 1.3]).**

Let $X$ be a projective scheme over a field $k$ and $L$ be an ample line bundle on $X$. Set $N_{n}\coloneqq\dim_{k}\operatorname{H}^{0}(X,L^{n})$ and let $\Phi_{n}\colon X\to\mathbb{P}_{k}^{N_{n}-1}$ be the map defined by $L^{n}$ for sufficiently large $n$. Suppose $(X,L)$ is asymptotically Chow (or Hilbert) semistable,111Abusing notation, we will abbreviate both of them as “asymptotically GIT semistable”. that is, for all $n\gg 0$, the Chow (or the Hilbert) point corresponding to $\Phi_{n}(X)$ is semistable under the natural action of $SL(N_{n})$. Then for every closed point $x\in X$, the local ring $\mathcal{O}_{X,x}$ satisfies $\operatorname{c_{LM}}(\mathcal{O}_{X,x}[[T]])=1$.

In accordance with Theorem II, Mumford called a Noetherian local ring $(R,{\mathfrak{m}})$ semistable if $\operatorname{c_{LM}}(R[[T]])=1$ and initiated the study of such singularities, primarily in dimensions one and two. He also introduced a notion of stable singularities, whose definition is more technical but is still coming from and inspired by GIT (see Definition 2.2.1). Continuing Mumford’s work, Shah [Sha81, Sha86] attempted to classify semistable singularities in dimension two and obtained necessary conditions for such surface singularities. Ultimately, he obtained a list of candidates for semistable singularities of small multiplicity (but did not prove that all singularities on the list are semistable).

In this paper, we improve and generalize Mumford’s and Shah’s classification results by providing necessary conditions for semistable singularities in all dimensions. Our results are based on the Minimal Model Program (MMP), which is natural given that the recent progress in the MMP has brought restrictions on singularities appearing on asymptotically GIT semistable varieties. Inspired by Shah’s classification, Odaka has shown in [Oda13] that asymptotically GIT semistable varieties have at worst semi-log canonical singularities – a class of singularities arising from the MMP introduced by Kollár and Shepherd-Barron to present an MMP approach to the compactification of the moduli space of smooth surfaces [KSB88]. The connection between GIT semistability and MMP singularities is mediated by K-semistability: it is well-known that asymptotic GIT semistability implies K-semistability (see [RT07, Theorem 3.9]), and Odaka proved that singularities on K-semistable varieties are semi-log canonical.

Our first theorem established a direct relation between semistable singularities and semi-log canonical singularities:

Theorem A** (Theorem 5.5.3).**

Let $(R,{\mathfrak{m}})$ be a local ring essentially of finite type over a field of characteristic zero that satisfies $(\textnormal{S}_{2})$ and $(\textnormal{G}_{1})$. Suppose $R$ is ${\mathbb{Q}}$-Gorenstein and semistable. Then $R$ is semi-log canonical.

We note and highlight that Theorem A is a local result without assuming global geometry behind, and it naturally fits into the following diagram:

[TABLE]

We also point out that K-semistable varieties may not have semistable singularities in general, see [WX14, Section 4.2]. Thus, although there have been tremendous breakthroughs in the study of K-stability [Xu25] and we hope to relate them with our study of stability of local rings, we have not been able to see a precise connection or application at the moment.

Our approach to Theorem A is inspired by Odaka’s method, and the key step is similar in spirit: using (semi-)log canonical modification to construct a “destabilizing” graded family of ideals. A crucial difference, however, is that we must reduce to a situation in which the graded family consists of ${\mathfrak{m}}$-primary ideals. This recruitment naturally led us to introduce a weaker notion of lim-stable singularities, which localizes well by Theorem 2.3.2. (By contrast, it remains unclear whether semistability itself localizes.) A further technical difficulty arises while proving that the graded family constructed via the (semi-)log canonical modification actually violates lim-stability. This hinges on a delicate analysis of the Hilbert series of certain rational powers of ideals, see Theorem 5.5.1.

In fact, Theorem 5.5.3 is stronger than Theorem A in many ways: we only need to assume lim-stability, and in dimension two, the result holds in all characteristics $\neq 2$. Moreover, when $R$ is normal, we obtain even stronger results in Theorem 5.3.6. In what follows, we will give a more detailed explanation of our main results and our main computations.

Detailed results

The definition of semistability is based on the following important feature of the Lech–Mumford constant:

[TABLE]

Moreover, it is easy to see that $\operatorname{c_{LM}}(R)\geq 1$ always holds. Thus, the limit

[TABLE]

exists and we will denote it by $\operatorname{c^{\infty}_{LM}}(R)$. This prompted us to introduce two new classes of singularities: Lech-stable (when $\operatorname{c_{LM}}(R)=1$) and lim-stable (when $\operatorname{c^{\infty}_{LM}}(R)=1$). These two classes as well as Mumford’s notions of local stability are related as follows:

[TABLE]

The following is our main result on the connection between stability of local rings and MMP singularities when $R$ is normal:

Theorem B** (Theorem 5.3.6, Theorem 5.4.1).**

Let $(R,{\mathfrak{m}})$ be an excellent normal local domain admitting a dualizing complex. Suppose that either

1. (1)

$\dim(R)\leq 2$, or 2. (2)

$R$* is essentially of finite type over a field of characteristic zero that is numerically ${\mathbb{Q}}$-Gorenstein.*

Then we have that

1. (a)

if $R$ is lim-stable, then $R$ is ${\mathbb{Q}}$-Gorenstein and log canonical; 2. (b)

if $R$ is Lech-stable and $R$ has canonical singularities on the punctured spectrum, then $R$ is ${\mathbb{Q}}$-Gorenstein and canonical.

We also obtain similar results by relaxing the normality condition, as already mentioned in Theorem A. We note that in Theorem B (b), there is an additional assumption on the singularities of the punctured spectrum of $R$. This assumption cannot be removed, see Remark 5.4.2 (which relies on a weak semicontinuity result for the Lech–Mumford constant, Proposition 2.4.10). We also point out that lim-stability (or even Lech-stability) alone does not imply ${\mathbb{Q}}$-Gorenstein in general (Theorem 6.2.5).

The proof of Theorem B (a) relies on a variation of log canonical modification essentially due to Hashizume [Has20], see Theorem 5.1.7, as well as a lower bound on $\operatorname{c^{\infty}_{LM}}(R)$ in terms of the Hilbert–Poincare series of graded families of ${\mathfrak{m}}$-primary ideals, see Proposition 4.1.3. The proof then proceeds by examining the “second coefficient” in the asymptotic colengths of certain rational powers, see Theorem 5.3.2 and Theorem 5.3.4. The proof of part (b) is similar and easier (we use canonical modifications [BCHM10]). The key estimate on $\operatorname{c^{\infty}_{LM}}(R)$, Proposition 4.1.3, applied to powers of maximal ideals, already yields some consequences:

Theorem C** (Corollary 4.3.1, Corollary 4.3.3, Remark 4.3.4).**

Let $(R,{\mathfrak{m}})$ be a Noetherian Cohen–Macaulay local ring of dimension $d$. Then

• •

if $R$ is a strict complete intersection and lim-stable (resp., Lech-stable), then $\operatorname{e}(R)\leq\operatorname{edim}(R)$ (resp., $\operatorname{e}(R)<\operatorname{edim}(R)$);

• •

there exists a constant $C(d)$ depending only on $d$ so that if $R$ is lim-stable, then $\operatorname{e}(R)<C(d)$. Moreover, one can take $C(1)=3$ and $C(2)=17$.

The definition of Lech–Mumford constant involves a supremum, and this creates a major difficulty in computing (or even bounding) the invariant. We have managed to compute this invariant for some surface singularities, which include pseudo-rational singularities ($\operatorname{c_{LM}}(R)=\operatorname{e}(R)/2$, Corollary 3.1.2); standard graded normal domains ($\operatorname{c_{LM}}(R)=\operatorname{e}(R)/2$, Corollary 3.2.5); simple elliptic singularities and minimally elliptic singularities of degree one (Theorem 3.2.3 and Theorem 3.2.8).

We next summarize our main results on classification of (Lech-, semi-, lim-)stable singularities and computations in examples.

Theorem D**.**

Let $(R,{\mathfrak{m}})$ be a Noetherian local ring.

1. (1)

If $\dim(R)=0$, then Lech-stability, semistability, and lim-stability of $R$ are all equivalent to $R$ being a field. Moreover, $R$ is never stable. (Proposition 2.2.5) 2. (2)

If $\dim(R)=1$, then

1. (a)

$R$* is Lech-stable if and only if $R$ is stable if and only if the unmixed part of $\widehat{R}$, ${\widehat{R}}^{\textup{un}}$, is regular. (Proposition 2.2.6)* 2. (b)

$R$* is semistable if and only if $R$ is lim-stable if and only if ${\widehat{R}}^{\textup{un}}$ is either regular or a node. (Theorem 4.2.1)* 3. (3)

If $\dim(R)=2$, then Cohen–Macaulay Lech-stable singularities are essentially rational double points, given explicitly via the ADE classification. (Theorem 3.1.5) 4. (4)

If $\dim(R)=2$, then the following singularities are semistable:

1. (a)

cone of the twisted cubic curve (Proposition 6.3.1); 2. (b)

certain rational triple points (Proposition 6.3.2); 3. (c)

elliptic and rational polygonal cones in Mumford’s terminology (Proposition 7.2.1 and Proposition 7.2.3); 4. (d)

Veronese subrings $k[x,y]^{(n)}$ for $n\leq 6$ (Example 7.2.4); 5. (e)

most of semi-log canonical hypersurfaces (Theorem 7.5.1). 5. (5)

In higher dimensions, simple normal crossings are semistable (Theorem 6.2.1), and generic determinantal rings of maximal minors are Lech-stable (Theorem 6.2.5).

Establishing Theorem D relies on a variety of tools used in computing and bounding the Lech–Mumford constant, including the analysis of its behavior under Gröbner deformations (Proposition 2.1.6, Corollary 2.4.7 and Corollary 2.4.8) and the development of sharper, often optimal, versions of Lech’s inequality (Theorem 6.1.3 and Theorem 7.1.2).

Lastly, we note that while the main motivation of this paper comes from understanding Mumford’s notion of semistability [Mum77], which has geometric origin, our interest on the algebraic side lies in finding refinements, generalizations, and variations of the classical Lech’s inequality (Theorem I). We have developed a series of works in this direction – see [KMQ*+*19, MQS20a, HMQS20, MS23, MQS23]. This paper builds on these works by employing several of their techniques and unifying various aspects.

Acknowledgement

During the (long) preparation of this manuscript, we were inspired by and benefited from discussions and conversations with many colleagues.

We would like to thank Craig Huneke for encouraging us initiating this project and for many fruitful discussions and suggestions, in particular for his suggestion of using rational powers of ideals. We would also like to thank him and Pham Hung Quy for collaborating with us on Lech’s inequality which motivates a large portion of this manuscript.

We would like to thank Joe Waldron and Ziquan Zhuang for their numerous help on the Minimal Model Program and for their patience in answering our questions on the materials presented in Section 5. Among other things, we learned the proof of Theorem 5.1.7 from Ziquan, and we learned Remark 5.3.7 from Joe.

We would like to thank Kei-ichi Watanabe for discussions on two-dimensional singularities related to the techniques in Section 3. In particular, Lemma 3.2.4 is due to Kei-ichi. We would also like to thank Hailong Dao for many communications on low-dimensional singularities. In particular, Lemma 3.1.1 and Corollary 3.1.2 were generalized from the Gorenstein case in discussion with him.

We would like to thank Yuchen Liu for explaining to us several arguments in Mumford’s paper [Mum77] and teaching us the degeneration techniques. We would like to thank Chi Li for inspiring us to consider the second coefficient of the colengths of certain graded family of ideals (Proposition 5.2.2).

Finally, we would like to thank Harold Blum, Aldo Conca, Igor Dolgachev, David Eisenbud, Hang Huang, Shihoko Ishii, Mircea Mustaţă, Yuji Odaka, Suchitra Pande, Quentin Posva, Marilina Rossi, Karl Schwede, Shunsuke Takagi, Kevin Tucker, Bernd Ulrich, Jakub Witaszek, and Chenyang Xu for their expertise and comments on various parts of this paper.

This material is based upon work supported by the National Science Foundation under Grant No. DMS-1928930 and by the Alfred P. Sloan Foundation under grant G-2021-16778, while the author was in residence at the Simons Laufer Mathematical Sciences Institute (formerly MSRI) in Berkeley, California, during the Spring 2024 semester.

The first author was supported in part by NSF Grant DMS #2302430.

The second author was supported by the State Research Agency of Spain through the Ramon y Cajal fellowship RYC2020-028976-I funded by MCIN/AEI/10.13039/501100011033 and by FSE “invest in your future” and grants PID2021-125052NA-I00 and EUR2023-143443 funded by MCIN/AEI/10.13039/501100011033 and the European Union NextGenerationEU/PRTR and by BCAM Severo-Ochoa accreditation CEX2021-001142-S funded by MCIN/AEI/10.13039/501100011033. For a period he was supported through a fellowship from “la Caixa” Foundation (ID 100010434), and the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 847648 (fellowship code “LCF/BQ/PI21/11830033”).

Notations and Conventions

Throughout the rest of this article, all rings are commutative, Noetherian, with multiplicative identity $1$. We will use $(R,{\mathfrak{m}})$ to denote a (Noetherian) local ring with unique maximal ideal ${\mathfrak{m}}$. All schemes are Noetherian and separated.

1. Preliminaries

1.1. Hilbert functions, multiplicities, and integral closure

Let $G=\oplus_{n=0}^{\infty}G_{n}$ be an $\mathbb{N}$-graded ring such that $G_{0}$ is an Artinian local ring and $G$ is finitely generated over $G_{0}$. For any finitely generated $\mathbb{Z}$-graded $G$-module $M$ one defines the Hilbert function $H_{M}(n)\coloneqq\ell_{G_{0}}(M_{n})$ and the Hilbert–Poincare series

[TABLE]

Since $G$ is a finitely generated $G_{0}$-algebra, we may choose homogeneous generators $x_{1},\ldots,x_{s}$ and let $d_{1},\ldots,d_{s}>0$ be their degrees. By the Hilbert–Serre theorem [AM69, Theorem 11.1], $h_{M}(t)$ is a rational function of the form

[TABLE]

where $f(t)\in\mathbb{Z}[t,t^{-1}]$ (and if $M$ is generated in non-negative degrees, then we have $f(t)\in\mathbb{Z}[t]$) and the order of the pole of $h_{M}(t)$ at $t=1$ is $\dim(M)-1$. It follows that $H_{M}(n)$ is eventually, for $n\gg 0$, given by a quasi-polynomial of degree $\dim(M)-1$. In the case where $d_{1}=\cdots=d_{s}=1$, the function $H_{M}(n)$ is eventually a polynomial. We will refer to this (quasi-)polynomial as the Hilbert (quasi-)polynomial of $M$.

Let $I$ be an ${\mathfrak{m}}$-primary ideal of a local ring $(R,{\mathfrak{m}})$. The associated graded ring of $I$,

[TABLE]

is an $\mathbb{N}$-graded ring generated in degree one over $R/I$. If $M$ is a finitely generated $R$-module then $\text{gr}_{I}(M)\coloneqq M/I\oplus IM/I^{2}M\oplus\cdots$ is a finitely generated graded $\text{gr}_{I}(R)$-module. Hence its Hilbert function $H_{\text{gr}_{I}(M)}(n)$ is given by a polynomial of degree $\dim(M)-1$ for $n\gg 0$, which is called the Hilbert polynomial of $M$ with respect to $I$.

We will be mostly working with the Hilbert–Samuel polynomial $p_{I}(t)$ of $I$ which computes $n\mapsto\ell(R/I^{n})$ for $n\gg 0$. The Hilbert and Hilbert–Samuel polynomials are related by the sum/difference transform: if we write the Hilbert–Samuel polynomial as

[TABLE]

then

[TABLE]

is the Hilbert polynomial of $I$. Since the Hilbert–Samuel polynomial is integer-valued, the coefficients $\operatorname{e}_{i}(I)$ are integers and are called the Hilbert coefficients of $I$.

Definition 1.1.1**.**

Let $(R,{\mathfrak{m}})$ be a local ring of dimension $d$ and let $I\subseteq R$ be an ${\mathfrak{m}}$-primary ideal. The integer $\operatorname{e}(I)\coloneqq\operatorname{e}_{0}(I)$ is called the Hilbert–Samuel multiplicity of $I$ and can be directly defined as

[TABLE]

The multiplicity of a finitely generated $R$-module $M$ with respect to $I$ is defined similarly as

[TABLE]

We note that, under the definition above, $\operatorname{e}(I,M)=0$ when $\dim(M)<d$, and if $\dim(M)=d$, then $\operatorname{e}(I,M)$ is the normalized leading coefficient (i.e., $(d-1)!$ times the leading coefficient) of the Hilbert polynomial of $M$ with respect to $I$.

The Hilbert–Samuel multiplicity is additive in short exact sequences of $R$-modules, this property yields what is often called the additivity or associativity formula for multiplicities [HS06, Theorem 11.2.4] which we will use throughout the article:

[TABLE]

where the sum varies through all primes ${\mathfrak{p}}$ such that $\dim(R/{\mathfrak{p}})=\dim(R)$.

The Hilbert–Samuel multiplicity is intrinsically connected to integral closure of ideals. We summarize the main ideas and refer to [HS06] for more background information.

Definition 1.1.2**.**

An element $x\in R$ is integral over an ideal $I\subseteq R$ if it satisfies a monic polynomial equation

[TABLE]

where $a_{k}\in I^{k}$ for $k=1,\ldots,n$. The set of all elements $x$ integral over $I$ forms an ideal and we call this ideal the integral closure of $I$ and denoted it by $\overline{I}$. An ideal $I$ is integrally closed if $I=\overline{I}$ and is normal if $I^{m}=\overline{I^{m}}$ for all $m\in\mathbb{N}$.

We say that an ideal $Q\subseteq J$ is a reduction of $J$ if there is an integer $k$ such that $QJ^{k}=J^{k+1}$. It is well-known that $Q$ is a reduction of $J$ if and only if $J\subseteq\overline{Q}$ (see [HS06, Corollary 1.2.5], note that we always work with Noetherian rings). It follows easily that $\operatorname{e}(I)=\operatorname{e}(\overline{I})$ for all ${\mathfrak{m}}$-primary ideals $I\subseteq R$. Conversely, a celebrated theorem of Rees shows that if $(R,{\mathfrak{m}})$ is formally equidimensional (i.e., the ${\mathfrak{m}}$-adic completion $\widehat{R}$ is equidimensional), then for two ${\mathfrak{m}}$-primary ideals $I\subseteq J$, we have $\operatorname{e}(I)=\operatorname{e}(J)$ if and only if $J\subseteq\overline{I}$ [HS06, Theorem 11.3.1]. Thus the study of multiplicities can often be reduced to integrally closed ideals.

In the opposite way, the study of multiplicities can be often reduced to parameter ideals, i.e., ideals generated by a system of parameters. It is well-known that for a local ring $(R,{\mathfrak{m}})$ with an infinite residue field, any ${\mathfrak{m}}$-primary ideal $I\subseteq R$ has a reduction generated by a system of parameters, and any such reduction is called a minimal reduction of $I$. In fact, it turns out that any $d=\dim(R)$ general elements $x_{1},\ldots,x_{d}\in I$ form a minimal reduction of $I$, see [HS06, Theorem 8.6.6]. Here, we define general elements of $I$ by equipping the vector space $I/{\mathfrak{m}}I$ with the Zariski topology.

More generally, the theory of superficial elements extends the formula $\operatorname{e}(I)=\operatorname{e}(J)$ to shorter sequences, thus providing a suitable tool for inductive proofs, see [HS06, Section 8.6]. We record the following fact which is a consequence of [HS06, Proposition 11.1.9]. If $(R,{\mathfrak{m}})$ is a local ring of dimension $d$ with an infinite residue field, then for any ${\mathfrak{m}}$-primary ideal $I\subseteq R$ and any general elements $x_{1},\ldots,x_{k}\in I$ with $k<d$, we have

[TABLE]

We next recall some basics on mixed multiplicities that will be used in this paper. Mixed multiplicities originate from the work of Risler and Teissier (see [Tei73, Section 2]) which uses the work of Bhattacharya [Bha57]. Given two ${\mathfrak{m}}$-primary ideals $I,J$ in a local ring $(R,{\mathfrak{m}})$ of dimension $d$, the function $(n,m)\mapsto\ell(R/I^{n}J^{m})$ is eventually, i.e., for $n,m\gg 0$, given by a polynomial function of degree $d$. One then defines the mixed multiplicities $\operatorname{e}(I^{[k]}\mid J^{[d-k]})$ by writing the degree $d$ bihomogeneous part of that polynomial as

[TABLE]

All mixed multiplicities are positive integers, and we have $\operatorname{e}(I)=\operatorname{e}(I^{[d]}\mid J^{[0]})$ and $\operatorname{e}(J)=\operatorname{e}(I^{[0]}\mid J^{[d]})$. In [Tei73], the theory of superficial sequences was extended to prove that

[TABLE]

for $k\geq 1$, where $x_{1},\ldots,x_{d-k}\in J$ are general elements (assuming $R$ has an infinite residue field). We will mostly use mixed multiplicities when $\dim(R)=2$. In this case, we will clean up the notation by using $\operatorname{e}(I\mid J)$ instead of $\operatorname{e}(I^{[1]}\mid J^{[1]})$.

We will often express multiplicities in terms of intersection numbers on certain blowups. For this we will use intersection theory for proper schemes over an Artinian local ring, see [CM25] or [Kol96, Chapter VI.2]. This theory was originated from [Sna59], and was further developed in [Mum66, Kle66]. We refer to [CM25, Theorem A] for the generality stated here. Let $(R,{\mathfrak{m}})$ be a local ring of dimension $d$ and let $I\subseteq R$ be an ${\mathfrak{m}}$-primary ideal. Let $Y\to\operatorname{Spec}(R)$ be a projective birational map such that $I\cdot\mathcal{O}_{Y}\cong\mathcal{O}_{Y}(-E)$ is an invertible sheaf corresponding to the (non-effective) Cartier divisor $-E$. Then we have

[TABLE]

One way to interpret the intersection product $-(-E)^{d}$ is to restrict the line bundle $\mathcal{O}_{Y}(-E)$ to $E$ and then compute the intersection number $(\mathcal{O}_{Y}(-E)|_{E})^{d-1}$, here $E$ is projective over an Artinian local ring $R/I$ and the intersection product is defined in [Kle66, Definition I.2.1], see also [Kol96, Chapter VI.2] and [Ful98, Chapter 20] for intersection theory in more general setups. Similarly, we can also express mixed multiplicities as intersection numbers on certain blowups [CM25, Theorem E].

1.2. Rational powers

We will use the theory of rational powers of ideals. We begin by collecting some definitions.

Definition 1.2.1**.**

Let $I\subseteq R$ be an ideal and let $a,b\in\mathbb{N}$. We define the $(b/a)$-th rational power of $I$ as

[TABLE]

Remark 1.2.2*.*

It is easy to see that rational powers are integrally closed and that $I^{\frac{n}{1}}=\overline{I^{n}}$. By the valuation criterion of integral closure [HS06, Theorem 6.8.3], we have $x\in I^{\frac{b}{a}}$ if and only if $\nu(x)\geq(b/a)\nu(I)$ for all valuations $\nu$ supported at the minimal primes of $R$. It follows easily that rational powers are well-defined, i.e., $I^{\frac{b}{a}}=I^{\frac{b^{\prime}}{a^{\prime}}}$ whenever $b/a=b^{\prime}/a^{\prime}$.

Let $I\subseteq R$ be an ideal. There exists finitely many discrete valuations $v_{i}$ supported at minimal primes of $R$ so that for each $n$, $x\in\overline{I^{n}}$ if and only if $v_{i}(x)\geq n$ for all $i$. The minimal collection of such $v_{i}$’s are called the Rees valuations of $I$, see [HS06, Definition 10.1.1] and Proposition 1.2.5 for the construction. It follows that $x\in I^{\frac{b}{a}}$ if and only if $\nu(x)\geq(b/a)\nu(I)$ for all Rees valuations $\nu$ of $I$.

Definition 1.2.3**.**

We define the Rees period of $I\subseteq R$ as

[TABLE]

The Rees period is a common denominator for all rational powers, i.e., if $\rho$ is the Rees period of $I$ and $\alpha\in\mathbb{Q}_{\geq 0}$, then $I^{\alpha}=I^{\frac{\lceil\rho\alpha\rceil}{\rho}}$, see [HS06, Proposition 10.5.5]. The common denominator allows one to show that the Hilbert–Samuel function of rational powers of $I$ is eventually a quasi-polynomial. This fact is well-known to experts and is closely related to [BDHM24, Proposition 2.13] (and alternatively can be deduced from Corollary 5.2.5 in Section 5). We record it here for completeness.

Proposition 1.2.4**.**

Let $(R,{\mathfrak{m}})$ be an analytically unramified local ring (i.e., the ${\mathfrak{m}}$-adic completion $\widehat{R}$ is reduced) of dimension $d\geq 1$ and let $I\subseteq R$ be an ${\mathfrak{m}}$-primary ideal. If $\rho$ is the Rees period of $I$ then for any integer $0\leq r<\rho$ the function $n\mapsto\ell(R/I^{\frac{n\rho+r}{\rho}})$ is given by a polynomial for $n\gg 0$. Moreover,

[TABLE]

Proof.

It was shown in [BDHM24, Proposition 2.13] (note that its proof is still valid when $R$ is not necessarily a domain) that there exists $a\in\mathbb{Z}$ such that $I^{\frac{w+\rho n}{\rho}}=\overline{I^{n}}I^{\frac{w}{\rho}}$ for all integers $w\geq a$ and $n\geq 1$. Since $R$ is analytically unramified, there exists $b\in\mathbb{Z}$ so that $\overline{I^{n+b}}=I^{n}\overline{I^{b}}$ for all $n\geq 1$ [HS06, Corollary 9.2.1]. Hence, after enlarging $a$ if needed, we may assume that $I^{\frac{w+\rho n}{\rho}}=I^{n}I^{\frac{w}{\rho}}$. Thus we have an exact sequence

[TABLE]

and the conclusion follows after noting that the function $n\mapsto\ell(I^{\frac{w}{\rho}}/I^{n}I^{\frac{w}{\rho}})$ is eventually a polynomial for $n\gg 0$ with the prescribed leading coefficient. ∎

We will need the following geometric description of rational powers of ideals, which is also well-known to experts.

Proposition 1.2.5**.**

Let $R$ be a domain and $I\subseteq R$ be an ideal. Suppose that $f\colon Z\to\operatorname{Spec}(R)$ is a projective birational map such that $I\mathcal{O}_{Z}=\mathcal{O}_{Z}(-E)$ for a Cartier divisor $E$ on $Z$ and the generic point of each irreducible component of ${E}_{\textup{red}}$ is regular. Then the Rees valuations of $I$ are the divisorial valuations attached to the irreducible components of ${E}_{\textup{red}}$ and we may compute

[TABLE]

Furthermore, if $R$ is excellent then the result holds for the natural map

[TABLE]

Proof.

Let $X\to\operatorname{Spec}(R)$ be the blow-up of $I$ and $\overline{X}$ be its normalization. The construction given in [HS06, Proposition 10.2.5, Exercises 10.5,6] identifies the Rees valuations as the valuations arising from the minimal primes of $I\mathcal{O}_{\overline{X}}$. Since localization and normalization commute, we could equivalently normalize the local rings of the minimal primes of $I\mathcal{O}_{X}$. Now, by construction $Z\to\operatorname{Spec}(R)$ factors through $X$, so the DVRs at the minimal primes of $I\mathcal{O}_{Z}$ (i.e., the generic points of the irreducible components of ${E}_{\textup{red}}$) factor through the normalizations of the minimal primes of $I\mathcal{O}_{X}$. Therefore, those are precisely the Rees valuations. The formula for rational powers now follows directly from their description in terms of Rees valuations.

We now verify that $Y\to\operatorname{Spec}(R)$ satisfies the necessary assumptions. In the extended Rees algebra $S=R[T^{-1},\overline{I}T,\overline{I^{2}}T^{2},\ldots]$ the principal ideal $(T^{-1})$ is normal by [HS06, Proposition 5.2.1 and Proposition 5.3.1]. Hence, for every associated prime $Q$ of $T^{-1}S$ the ring $S_{Q}$ is a DVR [Ree56, Lemma 4.2]. Since $R$ is excellent, we know that $S$ is Noetherian (e.g., by [HS06, Corollary 9.2.1]). Since $S_{Q}$ is a DVR, it follows that the homogeneous localization $S_{(Q)}$ (which is a $\mathbb{Z}$-graded ring) is regular by [GW78, Proposition 1.2.5]. Thus $[S_{(Q)}]_{0}$ is a DVR: it is normal since it is a direct summand of $S_{(Q)}$ and it is one-dimensional. It is easy to see that $[S_{(Q)}]_{0}$ are exactly the local rings of the minimal primes of $I\mathcal{O}_{Y}$, and thus they are regular. ∎

We end this subsection by providing a formula for the colength of integral closures of powers of $(I,T^{m})$ in the ring $R[[T]]$ using rational powers of $I$.

Lemma 1.2.6**.**

Let $I\subseteq R$ be an ideal. Then for all positive integers $a,n$ we have

[TABLE]

Proof.

It is clear that the right hand side is contained in $(I,T^{a})^{\frac{n}{a}}$. Namely,

[TABLE]

For the opposite inclusion, we note that $T^{n}\in(I,T^{a})^{\frac{n}{a}}$ and so it suffices to characterize when $xT^{n-s}\in(I,T^{a})^{\frac{n}{a}}$. Given that $x^{a}T^{a(n-s)}\in\overline{(I,T^{a})^{n}}$, it satisfies an equation of integral dependence

[TABLE]

where $f_{k}\in(I,T^{a})^{nk}$. We now group the terms of the coefficients of $T^{a(n-s)m}$ to obtain

[TABLE]

where $y_{k}T^{ak(n-s)}\in(I,T^{a})^{nk}$ and so $y_{k}\in I^{ks}$. This shows that $x^{a}\in\overline{I^{s}}$, thus $x\in I^{\frac{s}{a}}$ and the conclusion follows. ∎

We next express the colength of rational powers of $(I,T^{m})$ in $R[[T]]$ when $m$ is a multiple of the Rees period of $I$.

Proposition 1.2.7**.**

Let $(R,{\mathfrak{m}})$ be a local ring and $I\subseteq R$ be an ${\mathfrak{m}}$-primary ideal with the Rees period $\rho$. Let $m$ be a positive integer divisible by $\rho$. For positive integers $n$ and $0\leq r<m$ we write $s=\lceil\frac{r\rho}{m}\rceil$ and $t=s\frac{m}{\rho}-r$. Then we have

[TABLE]

Proof.

Using Lemma 1.2.6, we first rewrite

[TABLE]

It is left to count which of the appearing fractions in the exponent of $I$ are distinct. Note that we have

[TABLE]

We see that:

1. (1)

For each $0\leq h\leq s$, all of $\frac{h}{\rho},\ldots,\frac{n\rho+h}{\rho}$ appear among the exponents $\frac{\lceil k\rho/m\rceil}{\rho}$, and

• •

if $0\leq h<s$, then each $I^{\frac{i\rho+h}{\rho}}$ above appears $\frac{m}{\rho}$ times in the sum;

• •

if $h=s$, then each $I^{\frac{i\rho+h}{\rho}}$ above appears $\frac{m}{\rho}$ times in the sum when $0\leq i\leq n-1$, while $I^{\frac{n\rho+h}{\rho}}$ appears $\frac{m}{\rho}-s\cdot\frac{m}{\rho}+r=\frac{m}{\rho}-t$ times in the sum. 2. (2)

For $s<h\leq\rho-1$, only $\frac{h}{\rho},\ldots,\frac{(n-1)\rho+h}{\rho}$ appear among the exponents $\frac{\lceil k\rho/m\rceil}{\rho}$, and in this case each $I^{\frac{i\rho+h}{\rho}}$ appears $\frac{m}{\rho}$ times for all $h$ and all $0\leq i\leq n-1$.

Putting these together we obtain the formula in the proposition. ∎

Corollary 1.2.8**.**

Let $(R,{\mathfrak{m}})$ be a local ring and $I\subseteq R$ be an ${\mathfrak{m}}$-primary ideal with Rees period $\rho$. Let $m$ be a positive integer divisible by $\rho$ and $n$ be an arbitrary positive integer. Then we have

[TABLE]

Proof.

This follows from Proposition 1.2.7: in this case we have $r=s=t=0$, the first term in Proposition 1.2.7 vanish and we obtain

[TABLE]

which is easily checked to agree with both expressions in the corollary. ∎

1.3. Stable ideals and pseudo-rational rings

In this subsection, we briefly recall the concept of stable ideals and its connections with rings of minimal multiplicity and pseudo-rational rings.

Definition 1.3.1**.**

Let $(R,{\mathfrak{m}})$ be a local ring and $I\subseteq R$ be an ${\mathfrak{m}}$-primary ideal. We say that $I$ is stable if there is a parameter ideal $J\subseteq I$ such that $I^{2}=JI$.

Remark 1.3.2*.*

We note that if $I$ is stable, then the parameter ideal $J$ in the definition is necessarily a minimal reduction of $I$ and clearly $I^{2}\subseteq J$. Furthermore, when $(R,{\mathfrak{m}})$ is a Cohen–Macaulay local ring with an infinite residue field and $I$ is integrally closed, then $I^{2}\subseteq J$ for some minimal reduction $J$ implies $I$ is stable. For if $I^{2}\subseteq J$, then we have

[TABLE]

where the second equality above follows from [Ito88, Theorem 1]. Thus we have equalities throughout and $I$ is stable.

As far as we know, the notion of stable ideals first appeared in the work of Lipman for one-dimensional Cohen–Macaulay local rings, see [Lip71, Lemma 1.11]. The definition in higher dimensions appeared in [Kub85], the definition there is equivalent to ours when the residue field is infinite due to the following result that was independently observed by Huneke and Ooishi.

Theorem 1.3.3** ([Hun87, Theorem 2.1] or [Ooi87, Theorem 3.3]).**

Let $(R,\mathfrak{m})$ be a Cohen–Macaulay local ring of dimension $d$ with an infinite residue field. Then the following conditions are equivalent:

1. (1)

$I$* is stable,* 2. (2)

$I^{2}=JI$* for any minimal reduction $J$ of $I$,* 3. (3)

$\ell(R/I)=\operatorname{e}_{0}(I)-\operatorname{e}_{1}(I)$.

If these conditions hold, then $\operatorname{e}_{2}(I),\ldots,\operatorname{e}_{d}(I)=0$ and

[TABLE]

for all $n\geq 1$.

The notion of minimal multiplicity is defined through the Abhyankar’s inequality.

Theorem 1.3.4** (Abhyankar’s inequality, [Abh67]).**

Let $(R,{\mathfrak{m}})$ be a Cohen–Macaulay local ring of dimension $d$. Then $\operatorname{e}(R)\geq\operatorname{edim}(R)-d+1$. Moreover, if the residue field of $R$ is infinite, then equality holds if and only if ${\mathfrak{m}}$ is stable.

A Cohen–Macaulay local ring $(R,{\mathfrak{m}})$ has minimal multiplicity if Abhyankar’s inequality is an equality, that is, $\operatorname{e}(R)=\operatorname{edim}(R)-\dim(R)+1$. This property is stable on extending residue field (since multiplicity and embedding dimension are preserved under such extensions). Hence by Theorem 1.3.3, if $(R,{\mathfrak{m}})$ is a $d$-dimensional Cohen–Macaulay local ring of minimal multiplicity, then the Hilbert–Samuel polynomial of ${\mathfrak{m}}$ is

[TABLE]

We next recall that a local ring $(R,{\mathfrak{m}})$ is pseudo-rational in the sense of Lipman–Teissier [LT81], if it is Cohen–Macaulay, normal, analytically unramified (i.e., $\widehat{R}$ is reduced) and for every proper birational map $\pi\colon Y\to X\coloneqq\operatorname{Spec}(R)$, the canonical map $H_{\mathfrak{m}}^{d}(R)\to H^{d}(R\Gamma_{\mathfrak{m}}(R\pi_{*}\mathcal{O}_{Y}))$ is injective where $d=\dim(R)$. When $X$ admits a dualizing complex $\omega_{X}^{\bullet}$ (e.g., when $R$ is a homomorphic image of a Gorenstein local ring), then $\omega_{Y}^{\bullet}\coloneqq\pi^{!}\omega_{X}^{\bullet}$ is a dualizing complex of $Y$, and in this case, the last condition in the definition is equivalent to that $\pi_{*}\omega_{Y}=\omega_{X}$.

It is well-known that if $(R,{\mathfrak{m}})$ is essentially of finite type over a field of characteristic zero or if $\dim(R)\leq 2$, then pseudo-rational singularities are exactly rational singularities: that is, $Rf_{*}\mathcal{O}_{Y}=\mathcal{O}_{X}$ for some (equivalently, any) resolution of singularities $f$: $Y\to X=\operatorname{Spec}(R)$. This is essentially a consequence of the Grauert–Riemenschneider vanishing theorem [GR70, Lip78]. The following theorem characterizes pseudo-rational surface singularities in terms of integral closure of ideals.

Theorem 1.3.5** ([LT81, Proposition 5.5]).**

Let $(R,\mathfrak{m})$ be a normal analytically unramified local domain of dimension two. Then $R$ is pseudo-rational if and only if $I\overline{I}=\overline{I^{2}}$ for every ${\mathfrak{m}}$-primary ideal $I$. In particular, if $R$ is pseudo-rational with an infinite residue field, then every ${\mathfrak{m}}$-primary integrally closed ideal is stable.

2. Definition and basic properties

2.1. Lech–Mumford constant

We start by introducing the key concept of this paper.

Definition 2.1.1**.**

Let $(R,\mathfrak{m})$ be a local ring of dimension $d$. We call the quantity

[TABLE]

the Lech–Mumford constant of $R$.

Remark 2.1.2*.*

In [Mum77], $\operatorname{c_{LM}}(R)$ is called the [math]th flat multiplicity of $R$ and is denoted by $\operatorname{e}_{0}(R)$. Furthermore, $\operatorname{c_{LM}}(R[[T_{1},\dots,T_{n}]])$ is called the $n$th flat multiplicity of $R$ and is denoted by $\operatorname{e}_{n}(R)$. We decided not to follow Mumford’s notation because $\operatorname{e}_{i}$ is widely used in commutative algebra for the $i$th Hilbert coefficient.

We collect some basic properties of $\operatorname{c_{LM}}(R)$.

Proposition 2.1.3**.**

Let $(R,{\mathfrak{m}})$ be a local ring. Then we have the following:

1. (1)

the supremum in the definition of $\operatorname{c_{LM}}(R)$ can be restricted to integrally closed ideals; 2. (2)

$\operatorname{c_{LM}}(R)=\operatorname{c_{LM}}(\widehat{R})$; 3. (3)

$\operatorname{e}(R)\geq\operatorname{c_{LM}}(R)\geq\operatorname{e}(R)/\operatorname{e}({R}_{\textup{red}})$, in particular $\operatorname{c_{LM}}(R)\geq 1$; 4. (4)

$\operatorname{c_{LM}}(R)=\operatorname{e}(R)$* if $\dim(R)\leq 1$;* 5. (5)

if $S$ is a quotient of $R$ such that $\dim(S)=\dim(R)$, then $\operatorname{c_{LM}}(S)\leq\operatorname{c_{LM}}(R)$; 6. (6)

if ${R}^{\textup{un}}$ is the unmixed part of $R$,222Recall that ${R}^{\textup{un}}=R/j(R)$ where $j(R)$ is the largest $R$-submodule of $R$ such that $\dim(j(R))<\dim(R)$, see [HH94]. then $\operatorname{c_{LM}}(R)=\operatorname{c_{LM}}({R}^{\textup{un}})$.

Proof.

First of all, (1) is obvious: replacing $I$ by its integral closure will keep the multiplicity and will not increase the colength. Second, there is a one-to-one correspondence between (integrally closed) ${\mathfrak{m}}$-primary ideals of $R$ and $\widehat{R}$ [HS06, Lemma 9.1.1], and multiplicity and colength are preserved by passing to completion, (2) follows.

Next, in (3) we have $\operatorname{c_{LM}}(R)\leq\operatorname{e}(R)$ by Theorem I. Now let $d\coloneqq\dim(R)$ and consider the sequence of ideals $I_{n}=\mathfrak{m}^{n}+\sqrt{0}$. Note that $I_{n}$ is integral over $\mathfrak{m}^{n}$. It follows that

[TABLE]

This proves that $\operatorname{c_{LM}}(R)\geq\operatorname{e}(R)/\operatorname{e}({R}_{\textup{red}})\geq 1$.

To see (4), note that if $\dim(R)\leq 1$, then $\operatorname{e}({\mathfrak{m}})/d!\ell(R/{\mathfrak{m}})=\operatorname{e}(R)$ and thus $\operatorname{c_{LM}}(R)\geq\operatorname{e}(R)$, this together with (2) shows that $\operatorname{c_{LM}}(R)=\operatorname{e}(R)$.

To see (5), let $I$ be an $\mathfrak{m}$-primary ideal of $S$. We may consider its preimage $I^{\prime}$ in $R$. Then $\ell(R/I^{\prime})=\ell(S/I)$ and, since $S$ is a quotient of $R$ with $\dim(S)=\dim(R)$, we have $\operatorname{e}(I^{\prime})\geq\operatorname{e}(I)$. Thus $\operatorname{c_{LM}}(S)\leq\operatorname{c_{LM}}(R)$.

To see (2), note that ${R}^{\textup{un}}$ is a quotient of $R$ with $\dim(\ker(R\to{R}^{\textup{un}}))<\dim(R)$. Thus for any ${\mathfrak{m}}$-primary ideal $I$ of $R$, we have $\operatorname{e}(I)=\operatorname{e}(I{R}^{\textup{un}})$ and $\ell(R/I)\geq\ell({R}^{\textup{un}}/I{R}^{\textup{un}})$. It follows that $\operatorname{c_{LM}}(R)\leq\operatorname{c_{LM}}({R}^{\textup{un}})$. Since $\operatorname{c_{LM}}(R)\geq\operatorname{c_{LM}}({R}^{\textup{un}})$ by (4), we have $\operatorname{c_{LM}}(R)=\operatorname{c_{LM}}({R}^{\textup{un}})$. ∎

We can completely characterize when $\operatorname{c_{LM}}(R)=\operatorname{e}(R)$. This is essentially a reformulation of the main result of [MS23] (see also [HMQS20, Theorem A (2)]).

Theorem 2.1.4**.**

Let $(R,{\mathfrak{m}})$ be a local ring. Then $\operatorname{c_{LM}}(R)=\operatorname{e}(R)$ if and only if one of the following holds.

1. (a)

$\dim(R)\leq 1$. 2. (b)

$\dim(R)\geq 2$* and $\operatorname{e}({\widehat{R}}_{\textup{red}})=1$.*

Proof.

First of all, whenever (a) or (b) holds, we have $\operatorname{c_{LM}}(R)=\operatorname{e}(R)$ by (4) and (3) of Proposition 2.1.3. Conversely, suppose $\dim(R)\geq 2$ and $\operatorname{e}({\widehat{R}}_{\textup{red}})>1$, by [MS23, Theorem 3], there exists $\epsilon>0$ such that $\operatorname{e}(I)\leq d!(\operatorname{e}(R)-\epsilon)\ell(R/I)$ for all ${\mathfrak{m}}$-primary ideals $I\subseteq R$. This implies $\operatorname{c_{LM}}(R)\leq\operatorname{e}(R)-\epsilon<\operatorname{e}(R)$. ∎

It will be useful to be able to restrict in the definition of $\operatorname{c_{LM}}(R)$ to special classes of ideals. The discussion and results below extend some ideas in [Sha81, Proposition 3.4]. Recall that a graded family of ideals in a ring $R$ is a sequence of ideals $\{J_{n}\}_{n}$ such that $J_{n+1}\subseteq J_{n}$ and $J_{n}J_{m}\subseteq J_{n+m}$. For convenience, we shall also set $J_{0}=R$. Examples of such families are powers of an ideal, or integral closures of powers of an ideal, or valuation ideals. For each graded family $\{J_{n}\}_{n}$ we can form

• •

the Rees algebra $R[J_{\bullet}T^{\bullet}]\coloneqq\oplus_{n=0}^{\infty}J_{n}T^{n}$,

• •

the extended Rees algebra $R[J_{\bullet}T^{\bullet},T^{-1}]$,

• •

the associated graded ring $\text{gr}_{J_{\bullet}}(R)\coloneqq R[J_{\bullet}T^{\bullet},T^{-1}]/(T^{-1})=\oplus_{n=0}^{\infty}J_{n}/J_{n+1}$.

The graded family $\{J_{n}\}_{n}$ is called Noetherian if $R[J_{\bullet}T^{\bullet}]$ is a Noetherian ring. Note that this implies $R[J_{\bullet}T^{\bullet},T^{-1}]$ and $\text{gr}_{J_{\bullet}}(R)$ are both Noetherian. Moreover, for a Noetherian graded family, by [Rat79, Remark 2.4.3] there exists $m\in\mathbb{N}$ such that

[TABLE]

This implies (when $R$ is local) that $\cap_{n}J_{n}\subseteq\cap_{n}J_{m}^{n}\subseteq\cap_{n}{\mathfrak{m}}^{n}=0$. Furthermore, we have that $R[J_{\bullet}T^{\bullet},T^{-1}]$ is an integral extension of the usual extended Rees algebra $R[J_{m}^{n}T^{n},T^{-1}]$, and thus by [HS06, Theorem 2.2.5 and Theorem 5.1.4], we have

[TABLE]

For the ease of the presentation of the next few results, we introduce a graded version of Lech–Mumford constant for graded rings. We will show that, in our cases of interest, it agrees with the usual Lech–Mumford constant of the localization at the unique homogeneous maximal ideal so it will not cause any ambiguity later. Let $R$ be an $\mathbb{N}^{n}$-graded algebra over a local ring $(R_{0},{\mathfrak{m}}_{0})$. We define

[TABLE]

where the supremum runs over all $\mathbb{N}^{n}$-graded ideals of finite colength, i.e., those $\mathbb{N}^{n}$-graded ideals $I$ such that $\sqrt{I}={\mathfrak{m}}\coloneqq{\mathfrak{m}}_{0}+R_{>0}$ (${\mathfrak{m}}$ is the unique $\mathbb{N}^{n}$-graded maximal ideal).

Proposition 2.1.5**.**

Let $(R,\mathfrak{m})$ be a local ring and $\{J_{n}\}_{n}$ be a Noetherian graded family. Then $\operatorname{c_{LM}}(R)\leq\operatorname{c^{gr}_{LM}}(\operatorname{gr}_{J_{\bullet}}(R))$. In particular, $\operatorname{c_{LM}}(R)\leq\operatorname{c^{gr}_{LM}}(\operatorname{gr}_{J}(R))$ for any ideal $J\subseteq R$.

Proof.

Let $I$ be an $\mathfrak{m}$-primary ideal of $R$ and let $\operatorname{in}(I)$ be its form ideal in $\operatorname{gr}_{J_{\bullet}}(R)$, i.e.,

[TABLE]

Since the graded family is Noetherian, we have $\cap_{k}J_{k}=0$. It follows that

[TABLE]

Moreover, it is easy to see that $\operatorname{in}(I_{1})\operatorname{in}(I_{2})\subseteq\operatorname{in}(I_{1}I_{2})$ for any two ideals $I_{1},I_{2}$ of $R$, thus

[TABLE]

Since $\dim(R)=\dim(\text{gr}_{J_{\bullet}}(R))$, the above discussion shows that $\operatorname{e}(I)\leq\operatorname{e}(\operatorname{in}(I))$. Since $\operatorname{in}(I)$ is clearly a homogeneous ideal, we have $\operatorname{c_{LM}}(R)\leq\operatorname{c^{gr}_{LM}}(\text{gr}_{J_{\bullet}}(R))$ as wanted. ∎

Proposition 2.1.6**.**

If $R$ is an $\mathbb{N}^{n}$-graded ring over a local ring $(R_{0},{\mathfrak{m}}_{0})$ and ${\mathfrak{m}}={\mathfrak{m}}_{0}+R_{>0}$ is its unique homogeneous maximal ideal, then $\operatorname{c_{LM}}(R_{\mathfrak{m}})=\operatorname{c^{gr}_{LM}}(R)$ and we may restrict to multi-homogeneous ${\mathfrak{m}}$-primary ideals in the supremum computing $\operatorname{c_{LM}}(R_{\mathfrak{m}})$.

Proof.

We consider the degree lexicographic order filtration on $\mathbb{N}^{n}$: for two multi-degrees, $(a_{1},...,a_{n})\prec(b_{1},...,b_{n})$ if either

• •

$\sum^{n}_{i=1}a_{i}<\sum^{n}_{i=1}b_{i}$, or

• •

$\sum^{n}_{i=1}a_{i}=\sum^{n}_{i=1}b_{i}$, and there is $j$ such that $a_{j}<b_{j}$ and $a_{k}=b_{k}$ for all $k<j$.

Given an element $f\in R$, we define its initial form $\operatorname{in}_{\prec}(f)$ to be the smallest graded piece with respect to $\prec$. For any ideal $J\subseteq R$, we define $\operatorname{in}_{\prec}(J)$ to be the ideal generated by all $\{\operatorname{in}_{\prec}(f)\mid f\in J\}$. Furthermore, for any vector $w\in\mathbb{N}^{n}$, we define the weight filtration as follows: if $g$ is any multi-homogeneous element then define $w(g)=w\cdot\deg(g)$. For an arbitrary element $f\in R$, we define its initial form $\operatorname{in}_{w}(f)$ to be the smallest graded piece with respect to $w$, and define $\operatorname{in}_{w}(J)$ similarly.

Now we fix an ${\mathfrak{m}}$-primary ideal $I\subseteq R$ and consider $\operatorname{in}_{\prec}(I)$. By Noetherianness, $\operatorname{in}_{\prec}(I)$ is generated by $\operatorname{in}_{\prec}(f_{i})$ for finitely many $f_{1},\ldots,f_{r}\in I$. It follows that we can find a weight vector $w\in\mathbb{N}^{n}$ so that $\operatorname{in}_{\prec}(f_{i})=\operatorname{in}_{w}(f_{i})$ for all $f_{i}$. Thus $\operatorname{in}_{\prec}(I)=\operatorname{in}_{w}(I)$ (since the two ideals have the same colengths).

Next, we consider a graded family $J_{\bullet}\coloneqq\{J_{n}\}_{n}$, where $J_{n}$ is the ideal in $R$ generated by all multi-homogeneous elements $g$ such that $w(g)\geq n$, and set $\text{gr}_{w}(R)\coloneqq\text{gr}_{J_{\bullet}}(R)$ to be the associated graded ring with respect to $J_{\bullet}$. Note that $\text{gr}_{w}(R)\cong R$ since the grading induced by the weight is coarser than the original $\mathbb{N}^{n}$-grading on $R$. The form ideal of $I$ in $\text{gr}_{w}(R)$ is $\operatorname{in}_{w}(I)=\operatorname{in}_{\prec}(I)$ by construction. By the argument in Proposition 2.1.5, we know that

[TABLE]

Finally, by varying $I$ it follows that $\operatorname{c_{LM}}(R_{\mathfrak{m}})\leq\operatorname{c^{gr}_{LM}}(R)$. But the other direction $\operatorname{c^{gr}_{LM}}(R)\leq\operatorname{c_{LM}}(R_{\mathfrak{m}})$ is clear since each multi-homogeneous ideal of finite colength corresponds to an ${\mathfrak{m}}$-primary ideal of $R_{\mathfrak{m}}$ (and $R_{\mathfrak{m}}$ has possibly more ${\mathfrak{m}}$-primary ideals). Thus $\operatorname{c_{LM}}(R_{\mathfrak{m}})=\operatorname{c^{gr}_{LM}}(R)$ and the latter can be computed by multi-homogeneous ideals. ∎

As a special case of Proposition 2.1.6, we recover the following fundamental result of Mumford.

Corollary 2.1.7** ([Mum77, Lemma 3.6]).**

Let $(R,\mathfrak{m})$ be a local ring. Then $\operatorname{c_{LM}}(R[[T]])$ can be computed by looking only at homogeneous ideals, i.e., ideals of the form $I=\oplus_{k\geq 0}I_{k}T^{k}$ where $I_{k}\subseteq R$ and $I_{k}=R$ for $k\gg 0$.

Proof.

By Proposition 2.1.6, we have

[TABLE]

where the last equality follows from Proposition 2.1.3(2) since $R[T]_{{\mathfrak{m}}+(T)}$ and $R[[T]]$ agree up to completion. The conclusion follows from the definition of $\operatorname{c^{gr}_{LM}}$. ∎

Lemma 2.1.8**.**

Let $(R,\mathfrak{m})$ be a local ring and $x$ be a parameter element. Then we have $\operatorname{c^{gr}_{LM}}(\operatorname{gr}_{(x)}(R))\leq\operatorname{c^{gr}_{LM}}((R/xR)[T])$.

Proof.

We have an isomorphism $(x^{n})/(x^{n+1})\cong R/((x)+(0:x^{n}))$. Thus we have a natural surjective graded homomorphism $(R/xR)[T]\to\operatorname{gr}_{(x)}(R)$. Moreover, it is clear that $\dim((R/xR)[T])=\dim(\operatorname{gr}_{(x)}(R))=\dim(R)$. Thus $\operatorname{c^{gr}_{LM}}(\operatorname{gr}_{(x)}(R))\leq\operatorname{c^{gr}_{LM}}((R/xR)[T])$ by the graded version of Proposition 2.1.3(5). ∎

We next generalize [Mum77, Proposition 3.5] and show that Lech–Mumford constant does not decrease under specialization.

Proposition 2.1.9**.**

Let $(R,\mathfrak{m})$ be a local ring and $x$ be a parameter element. Then we have $\operatorname{c_{LM}}(R)\leq\operatorname{c_{LM}}(R/xR)$. Moreover, if the equality holds and $\dim(R)>1$, then the supremum in the definition of $\operatorname{c_{LM}}(R)$ is not attained.

Proof.

By Proposition 2.1.5 and Lemma 2.1.8, we have that

[TABLE]

Thus it suffices to show that $\operatorname{c^{gr}_{LM}}(S[T])\leq\operatorname{c_{LM}}(S)$ for a local ring $(S,\mathfrak{m})$. This was proved by Mumford in [Mum77, Proposition 3.5] and we include a proof here for completeness.

Fix a homogeneous ideal $I=\oplus_{k\geq 0}I_{k}T^{k}$ in $S[T]$ of finite colength. We want to bound $\operatorname{e}(I)$. Suppose that $I_{N+1}=S$ and note that $I^{n}$ contains the ideal

[TABLE]

Thus using that $I_{k}\subseteq I_{k+1}$ we obtain that

[TABLE]

Let $d=\dim(S)$. The inclusion of ideals above implies that

[TABLE]

It follows that

[TABLE]

Therefore $\operatorname{c^{gr}_{LM}}(S[T])\leq\operatorname{c_{LM}}(S)$. Finally, for the last claim, we simply note that the middle inequality in the above formula is strict unless $d=0$. This completes the proof. ∎

Corollary 2.1.10**.**

Let $(R,\mathfrak{m})$ be a local ring of dimension $d$. Then we have

[TABLE]

Proof.

The first inequality follows from Proposition 2.1.9. For the second inequality, let $I$ be an $\mathfrak{m}$-primary ideal in $R$ and consider the ideal $J=IR[[T]]+(T)$. It is easy to see that $\ell(R[[T]]/J)=\ell(R/I)$ and $\operatorname{e}(J)=\operatorname{e}(I)$. Thus

[TABLE]

This completes the proof. ∎

Remark 2.1.11*.*

It is not true that $\operatorname{c_{LM}}(R/xR)\leq d\operatorname{c_{LM}}(R)$ for a linear form $x\in{\mathfrak{m}}-{\mathfrak{m}}^{2}$, where $d=\dim(R)$. We will see in Theorem 3.1.5 in Section 3 that the $E_{8}$ singularity $R=\mathbb{C}[[x,y,z]]/(x^{2}+y^{3}+z^{5})$ satisfies $\operatorname{c_{LM}}(R)=1$, but it follows from Proposition 2.1.3(4) that $\operatorname{c_{LM}}(R/xR)=\operatorname{e}(R/xR)=3$. On the other hand, we do not know whether $\operatorname{c_{LM}}(R/xR)\leq d\operatorname{c_{LM}}(R)$ for a general linear form $x\in{\mathfrak{m}}-{\mathfrak{m}}^{2}$. This holds when $d=2$, because in this case

[TABLE]

where the first equality follows from Proposition 2.1.3(4) and the second equality follows from the fact that $x$ is a general linear form.

We conclude this subsection by recalling another result of Mumford, which extends the main step in Proposition 2.1.9 by bounding the multiplicity of an ideal in the power series using mixed multiplicities of its components. This result will be used several times in Section 6 and Section 7.

Proposition 2.1.12** ([Mum77, Proposition 4.3]).**

Let $(R,{\mathfrak{m}})$ be a local ring and

[TABLE]

be an $({\mathfrak{m}},T)$-primary ideal of $R[[T]]$. Set $I_{N}=R$. Then for all sequences of integers $0=r_{0}<r_{1}<\cdots<r_{l}=N$ we have

[TABLE]

Proof.

One can verify directly that

[TABLE]

It follows that

[TABLE]

By the definition of mixed multiplicities, we have

[TABLE]

Since $\operatorname{e}(I)=\lim_{m\to\infty}\frac{(d+1)!}{m^{d+1}}\ell(R[[T]]/I^{m})$, it suffices to show that

[TABLE]

This formula follows by setting $N=m-d,a=j,b=d-j$ and expanding the binomial coefficients in the identity

[TABLE]

obtained by comparing the coefficients at $x^{N}$ of the formal power series identity

[TABLE]

2.2. Stability of local rings

As an immediate consequence of Proposition 2.1.3 and Proposition 2.1.9, for any local ring $(R,{\mathfrak{m}})$, we have the following inequalities

[TABLE]

We thus define

[TABLE]

Definition 2.2.1**.**

We say a local ring $(R,{\mathfrak{m}})$ is

• •

Lech-stable if $\operatorname{c_{LM}}(R)=1$,

• •

semistable if $\operatorname{c_{LM}}(R[[T]])=1$,

• •

lim-stable if $\operatorname{c^{\infty}_{LM}}(R)=1$,

• •

stable if $R$ is semistable and the supremum in the definition of $\operatorname{c_{LM}}(R[[T]])$ is not attained.

Remark 2.2.2*.*

We have the following implications of these stability conditions, which are straightforward from Proposition 2.1.9 (see ($\dagger$ ‣ 2.2)):

[TABLE]

We will see in Proposition 2.2.5, Proposition 2.2.6 and Theorem 4.2.1 a complete characterization of these stability notions for local rings of dimension [math] and $1$. We also have a good understanding of them in dimension $2$, see Theorem 3.1.5, Theorem 5.3.6 and Theorem 5.5.3; and for many class of rings of combinatorial nature, see Theorem 6.2.1, Theorem 6.2.5, and Example 7.2.4. From these results one easily obtains examples of semistable singularities that are not stable, and examples of stable singularities that are not Lech-stable. On the other hand, quite embarrassingly, we do not know/have a single example of a lim-stable local ring that is not semistable: in fact, even in some simple examples, computing the entire chain ($\dagger$ ‣ 2.2) is challenging (for us), see Example 7.4.3 for a partial attempt.

Remark 2.2.3*.*

Let $(R,{\mathfrak{m}})$ be a local ring of dimension $d$. By considering ${\mathfrak{m}}$ and ${\mathfrak{m}}^{2}$ in the definition of $\operatorname{c_{LM}}(R)$, we see that if $R$ is Lech-stable, then

[TABLE]

Similarly, if $R$ is semistable, then

[TABLE]

Proposition 2.2.4**.**

Let $(R,{\mathfrak{m}})$ be a local ring that is lim-stable. Then ${\widehat{R}}^{\textup{un}}$ is reduced. In particular, Cohen–Macaulay semistable singularities are reduced.

Proof.

By Proposition 2.1.3 (2) and (1), we may replace $R$ by ${\widehat{R}}^{\textup{un}}$ to assume that $R$ is complete and unmixed. Since $R$ is lim-stable, unwinding the definition we have that for any $\epsilon>0$, there exists $n$ such that

[TABLE]

where the second inequality follows from Proposition 2.1.3(3). Letting $\epsilon\to 0$ we obtain that $\operatorname{e}(R)=\operatorname{e}({R}_{\textup{red}})$. Since $R$ is unmixed, if $R$ is not reduced then the nilradical $\sqrt{0}\neq 0$ have full dimension and thus $\operatorname{e}(R)=\operatorname{e}({R}_{\textup{red}})+\operatorname{e}({\mathfrak{m}},\sqrt{0})>\operatorname{e}(R)$ which is a contradiction. Thus $R$ is reduced as wanted. The last conclusion follows from the fact that Cohen–Macaulay rings are unmixed and semistability implies lim-stability. ∎

We now give characterizations of various notions of stability of local rings in small dimensions. We start with the case that $\dim(R)=0$.

Proposition 2.2.5**.**

Let $(R,\mathfrak{m})$ be an Artinian local ring. The following are equivalent:

1. (1)

$R$* is regular (i.e., $R$ is a field),* 2. (2)

$R$* is Lech-stable,* 3. (3)

$R$* is semistable,* 4. (4)

$R$* is lim-stable.*

Furthermore, $R$ is never stable.

Proof.

Obviously we have $(1)\Rightarrow(2)\Rightarrow(3)\Rightarrow(4)$ (and these hold in any dimension). To see $(4)\Rightarrow(1)$, note that since $R$ is Artinian, Proposition 2.2.4 shows that if $R$ is lim-stable then $R$ is reduced and thus regular. For the last part, note that since $\dim(R[[T]])=1$, we have $\operatorname{c_{LM}}(R[[T]])=\operatorname{e}(R[[T]])$ (see Proposition 2.1.3) and it is easy to see that the supremum is achieved at the maximal ideal of $R[[T]]$. Thus $R$ is not stable. ∎

We end this subsection by providing a characterization of Lech-stable and stable singularities of dimension one. The semistable and lim-stable singularities in dimension one will be treated later, after computing the necessary surface examples.

Proposition 2.2.6**.**

Let $(R,\mathfrak{m})$ be a local ring of dimension one. Then the following conditions are equivalent:

1. (1)

$R$* is Lech-stable,* 2. (2)

$R$* is stable,* 3. (3)

${\widehat{R}}^{\textup{un}}$* is a regular local ring.*

Proof.

We first note that by Proposition 2.1.3 (2) and (2), we have $\operatorname{c_{LM}}(R)=\operatorname{c_{LM}}({\widehat{R}}^{\textup{un}})$ and $\operatorname{c_{LM}}(R[[T_{1},\dots,T_{r}]])=\operatorname{c_{LM}}({\widehat{R}}^{\textup{un}}[[T_{1},\dots,T_{r}]])$ for all $r>0$. Hence to prove the proposition, we may replace $R$ by ${\widehat{R}}^{\textup{un}}$ to assume that $R$ is a Cohen–Macaulay complete local ring of dimension one.

We already know that $(1)\Rightarrow(2)$, and clearly if $R$ is regular, then it is Lech-stable and thus $(3)\Rightarrow(1)$. It remains to prove $(2)\Rightarrow(3)$. Suppose $R$ is stable, then $R$ is semistable and thus by Remark 2.2.3, we know that $\operatorname{e}(R)\leq 2$. Since $R$ is a one-dimensional Cohen–Macaulay local ring, $\operatorname{edim}(R)\leq\operatorname{e}(R)+1-1\leq 2$ by Abhyankar’s inequality. If $\operatorname{edim}(R)=2$, then we have equalities throughout. In particular, $\operatorname{e}(R)=2$ and it is easy to see that $\operatorname{c_{LM}}(R[[T]])=1$ is attained at the maximal ideal of $R[[T]]$, in contradiction with $R$ being stable. Therefore we must have $\operatorname{edim}(R)=1$, i.e., $R$ is regular. ∎

2.3. Localization and faithfully flat extension

In this subsection, we study the behavior of the Lech–Mumford constant under certain ring homomorphisms. We start with a key lemma.

Lemma 2.3.1**.**

Let $(R,{\mathfrak{m}})$ be a local ring and $\mathfrak{p}\in\operatorname{Spec}(R)$ such that $\operatorname{ht}\mathfrak{p}+\dim(R/\mathfrak{p})=\dim(R)$. Let $x_{1},\dots,x_{r}$ be part of a system of parameters of $R$ whose images in $R/\mathfrak{p}$ form a system of parameters of $R/\mathfrak{p}$. Then we have

[TABLE]

Proof.

We use induction on $r$ and we first assume that $r=1$. Let $d\coloneqq\dim(R)$ and thus $\dim(R_{\mathfrak{p}})=d-1$. Let $J$ be a $\mathfrak{p}R_{\mathfrak{p}}$-primary ideal of $R_{\mathfrak{p}}$ and let $I$ be the contraction of $J$ to $R$. Note that $I$ is a $\mathfrak{p}$-primary ideal by construction and thus $x\coloneqq x_{1}$ is a nonzerodivisor on $R/I$. It follows that

[TABLE]

where the second equality follows from the associativity formula for multiplicity. Now let $I_{n}$ be the contraction of $J^{n}$ to $R$, we clearly have $I^{n}\subseteq I_{n}$, thus we have

[TABLE]

where the first equality follows from the fact that $I_{n}$ is $\mathfrak{p}$-primary (and thus $x$ is a nonzerodivisor on $R/I_{n}$) and the second equality follows again from the associativity formula for multiplicity. Letting $n\to\infty$ and taking limit, we obtain that $\operatorname{e}(I,R/(x))\geq\operatorname{e}(x,R/\mathfrak{p})\operatorname{e}(J,R_{\mathfrak{p}}).$ Therefore we have

[TABLE]

It follows that $\operatorname{c_{LM}}(R_{\mathfrak{p}})\leq\operatorname{c_{LM}}(R/(x))$. This completes the proof when $r=1$.

Now suppose $r\geq 2$. We can choose a minimal prime $\mathfrak{q}$ of $\mathfrak{p}+(x_{1})$ such that $\operatorname{ht}\mathfrak{q}+\dim(R/\mathfrak{q})=\dim(R)$. It follows from the choice of $\mathfrak{q}$ that the images of $x_{2},\dots,x_{r}$ form a system of parameters of $R/\mathfrak{q}$, and that $\operatorname{ht}(\mathfrak{p}R_{\mathfrak{q}})+1=\dim(R_{\mathfrak{q}})$ and $\operatorname{ht}(\mathfrak{q}/(x_{1}))+\dim(R/\mathfrak{q})=\dim(R/x_{1}R)$. Therefore we have

[TABLE]

where the first inequality follows from the $r=1$ case applied to $R_{\mathfrak{q}}$, and the second inequality follows by the inductive hypothesis applied to $R/x_{1}R$. This completes the proof. ∎

We now prove the effect of localization on Lech–Mumford constants.

Theorem 2.3.2**.**

Let $(R,\mathfrak{m})$ be a local ring and $\mathfrak{p}\in\operatorname{Spec}(R)$ such that $\operatorname{ht}\mathfrak{p}+\dim(R/\mathfrak{p})=\dim(R)$. Then we have

1. (1)

$\operatorname{c_{LM}}(R)\geq\operatorname{c_{LM}}(R_{\mathfrak{p}}[[T_{1},\ldots,T_{r}]])$* where $r=\dim(R/\mathfrak{p})$;* 2. (2)

$\operatorname{c^{\infty}_{LM}}(R)\geq\operatorname{c^{\infty}_{LM}}(R_{\mathfrak{p}})$.

In particular, if $R$ is equidimensional, catenary, and lim-stable, then so is $R_{\mathfrak{p}}$ for all $\mathfrak{p}\in\operatorname{Spec}(R)$.

Proof.

Let $S\coloneqq R[T_{1},\dots,T_{r}]_{({\mathfrak{m}}+(T_{1},\dots,T_{r}))}$ and let $P\coloneqq\mathfrak{p}+(T_{1},\dots,T_{r})$ be a prime ideal of $S$. We can choose $x_{1},\dots,x_{r}$ part of a system of parameters of $R$ such that their images in $R/\mathfrak{p}$ form a system of parameters of $R/\mathfrak{p}$. It follows that that $\operatorname{ht}(P)+\dim(S/P)=\dim(S)$ and that $x_{1}-T_{1},\dots,x_{r}-T_{r}$ is part of a system of parameters of $S$ such that their images in $S/P$ form a system of parameters of $S/P$. Therefore we have

[TABLE]

where the first equality follows from Proposition 2.1.3(2) as $S_{P}$ and $R_{\mathfrak{p}}[[T_{1},\dots,T_{r}]]$ agree upon completion at their maximal ideals, and the inequality above follows from Lemma 2.3.1. This completes the proof of the part $(1)$.

We next proceed with part $(2)$. Note that by the definition of $\operatorname{c^{\infty}_{LM}}(R)$, for any $\epsilon>0$, there exists $n$ such that $\operatorname{c^{\infty}_{LM}}(R)\geq\operatorname{c_{LM}}(R[[T_{1},\dots,T_{n}]])-\epsilon$. It follows from part $(1)$ that

[TABLE]

for any $\epsilon>0$. Thus, we conclude that $\operatorname{c^{\infty}_{LM}}(R)\geq\operatorname{c^{\infty}_{LM}}(R_{\mathfrak{p}})$, as required. ∎

Remark 2.3.3*.*

One cannot expect that $\operatorname{c_{LM}}(R)\geq\operatorname{c_{LM}}(R_{\mathfrak{p}})$ in general: for example let $R=k[[x,y,z,t]]/(xyz)$ and $\mathfrak{p}=(x,y,z)$, then we will see from Theorem 6.2.1 and Proposition 6.2.3 from Section 6 that $\operatorname{c_{LM}}(R)=1$ while $\operatorname{c_{LM}}(R_{\mathfrak{p}})=3/2$.

Remark 2.3.4*.*

We expect the following generalization of Theorem 2.3.2. Suppose $(R,{\mathfrak{m}})$ is a local ring such that $\widehat{R}$ is equidimensional (more generally, one can work on an excellent and biequidimensional scheme). Then we suspect that the function $\Phi\colon\operatorname{Spec}(R)\to\mathbb{R}$ defined by $\Phi(\mathfrak{p})=\operatorname{c_{LM}}(R_{\mathfrak{p}}[[T_{1},\dots,T_{\dim(R/\mathfrak{p})}]])$, and the function $\Phi^{\prime}\colon\operatorname{Spec}(R)\to\mathbb{R}$ defined by $\Phi^{\prime}(\mathfrak{p})=\operatorname{c^{\infty}_{LM}}(R_{\mathfrak{p}})$ are upper semicontinuous.

Corollary 2.3.5**.**

Let $(R,\mathfrak{m})$ be a local ring and let $\mathfrak{p}\in\operatorname{Spec}(R)$ be such that $\operatorname{ht}\mathfrak{p}+\dim(R/\mathfrak{p})=\dim(R)$. Then $\dim(R)!\operatorname{c_{LM}}(R)\geq(\operatorname{ht}\mathfrak{p})!\operatorname{c_{LM}}(R_{\mathfrak{p}})$.

Proof.

Let $d=\dim(R)$, $r=\dim(R/\mathfrak{p})$ and $h=\operatorname{ht}\mathfrak{p}$. By Theorem 2.3.2 we have

[TABLE]

where the last line above is due to Corollary 2.1.10. ∎

We next study the behavior of Lech–Mumford constant under faithfully flat extension.

Proposition 2.3.6**.**

Let $(R,\mathfrak{m})\to(S,\mathfrak{n})$ be a flat local extension of local rings. Then we have $\operatorname{c_{LM}}(S)\geq\operatorname{c_{LM}}(R[[T_{1},\ldots,T_{r}]])$ where $r=\dim(S)-\dim(R)$. Furthermore, we have $\operatorname{c^{\infty}_{LM}}(S)\geq\operatorname{c^{\infty}_{LM}}(R)$.

Proof.

Since $S$ is faithfully flat over $R$, $\dim(S)=\dim(R)+\dim(S/\mathfrak{m}S)$. Let $\mathfrak{p}$ be a minimal prime of $\mathfrak{m}S$ such that $\dim(S/\mathfrak{p})=\dim(S/\mathfrak{m}S)=r$. By Theorem 2.3.2, we have $\operatorname{c_{LM}}(S)\geq\operatorname{c_{LM}}(S_{\mathfrak{p}}[[T_{1},\dots,T_{r}]])$. By replacing $R\to S$ by $R[[T_{1},\dots,T_{r}]]\to S_{\mathfrak{p}}[[T_{1},\dots,T_{r}]]$, we may assume that $\dim(R)=\dim(S)$. In this case, all we need to show is that $\operatorname{c_{LM}}(R)\leq\operatorname{c_{LM}}(S)$. But for any ${\mathfrak{m}}$-primary ideal $I$ of $R$, we have

[TABLE]

It follows that

[TABLE]

The proof of $\operatorname{c^{\infty}_{LM}}(S)\geq\operatorname{c^{\infty}_{LM}}(R)$ is entirely similar and we omit the details. ∎

Putting Theorem 2.3.2 and Proposition 2.3.6 together allows us to prove a graded version of the localization inequality.

Corollary 2.3.7**.**

Let $R$ be a $\mathbb{N}$-graded ring generated in degree one over a local ring $(R_{0},{\mathfrak{m}}_{0})$ and let ${\mathfrak{m}}={\mathfrak{m}}_{0}+R_{>0}$. Let $\mathfrak{p}$ be a homogeneous prime ideal in $R$ such that $\operatorname{ht}\mathfrak{p}+\dim(R/\mathfrak{p})=\dim(R)$. Then we have $\operatorname{c^{gr}_{LM}}(R)=\operatorname{c_{LM}}(R_{{\mathfrak{m}}})\geq\operatorname{c_{LM}}(R_{(\mathfrak{p})}[[T_{1},\ldots,T_{r}]])$ where $r=\dim(R/\mathfrak{p})$ and $R_{(\mathfrak{p})}$ is the homogeneous localization.

Proof.

By a similar reduction as in the proof of Theorem 2.3.2, we may assume that $r=\dim(R/\mathfrak{p})=1$. Recall that $R_{(\mathfrak{p})}$ consists of equivalence classes of elements of the form $a/b$, where $a,s$ are homogeneous, $\deg a=\deg b$, and $b\notin\mathfrak{p}$. Thus, if $z\notin\mathfrak{p}$ with $\deg z=1$, then $R_{(\mathfrak{p})}$ is the localization of $A\coloneqq R[z^{-1}]_{0}$ at $\mathfrak{q}=A\cap\mathfrak{p}$. Note that $R_{z}\cong A[z,z^{-1}]$ and $\mathfrak{q}R_{z}=\mathfrak{p}R_{z}$. By Theorem 2.3.2, $\operatorname{c_{LM}}(R_{{\mathfrak{m}}})\geq\operatorname{c_{LM}}(R_{\mathfrak{p}}[[T]])$. Finally, since $R_{\mathfrak{p}}\cong A[z,z^{-1}]_{\mathfrak{q}[z,z^{-1}]}\cong A_{\mathfrak{q}}(z)$, by Proposition 2.3.6,

[TABLE]

This completes the proof. ∎

We also have the following corollary as a consequence of the results discussed so far. This corollary will allow us to reduce to the case of infinite residue field when studying the Lech–Mumford constant.

Corollary 2.3.8**.**

Let $(R,{\mathfrak{m}})$ be a local ring and let $R(t)\coloneqq R[t]_{{\mathfrak{m}}R[t]}$. Then $\operatorname{c_{LM}}(R)=\operatorname{c_{LM}}(R(t))$.

Proof.

Let $S\coloneqq R[t]_{{\mathfrak{m}}+(t)}$ and let $P\coloneqq{\mathfrak{m}}S$. Then clearly $0pt(P)+\dim(S/P)=\dim(S)$ and $t$ is a parameter in $S$ whose image is a system of parameters on $S/P$. Thus by Lemma 2.3.1, $\operatorname{c_{LM}}(R(t))=\operatorname{c_{LM}}(S_{P})\leq\operatorname{c_{LM}}(S/(t))=\operatorname{c_{LM}}(R)$. On the other hand, $\operatorname{c_{LM}}(R)\leq\operatorname{c_{LM}}(R(t))$ by Proposition 2.3.6. ∎

Given Corollary 2.3.8, one might wonder whether the Lech–Mumford constant is preserved under arbitrary field extensions. Here we record an example showing that this is not always the case.

Example 2.3.9**.**

Let $R=\overline{\mathbb{F}}_{2}[x,y,t]_{\mathfrak{m}}/(x^{2}+ty^{2})$ where ${\mathfrak{m}}=(x,y,z)$. By Proposition 3.1.8 in Section 3, we have $\operatorname{c_{LM}}(R)=\operatorname{c_{LM}}(\widehat{R})=1$. It follows from Theorem 2.3.2 that

[TABLE]

However, $\operatorname{c_{LM}}(\overline{\mathbb{F}_{2}(t)}[[x,y,z]]/(x^{2}+ty^{2}))=2$ by Proposition 2.1.3(3).

On the other hand, we do not know whether the Lech–Mumford constant is preserved under separable field extensions.

Question 2.3.10*.*

Let $(R,{\mathfrak{m}},k)\to(R^{\prime},{\mathfrak{m}}^{\prime},k^{\prime})$ be a flat local extension of complete local rings such that ${\mathfrak{m}}^{\prime}={\mathfrak{m}}R^{\prime}$ and $k^{\prime}$ is separable over $k$. Then is it true that $\operatorname{c_{LM}}(R)=\operatorname{c_{LM}}(R^{\prime})$?

We want to point out that if we restrict to ideals of fixed colength, i.e., to the invariants

[TABLE]

then it is not true that $L_{N}(R)$ is invariant under separable field extensions (i.e., the analogous question has a negative answer).

Proposition 2.3.11**.**

Consider a local ring of the form $R=\mathbb{F}_{5}[[x,y,z]]/(x^{4}+y^{4}+z^{4}+g(x,y,z))$ or $\mathbb{Q}[[x,y,z]]/(3x^{3}+4x^{3}+5x^{3}+g(x,y,z))$, where $g$ consists of higher order terms. Then

[TABLE]

where $k$ is the corresponding coefficient field.

Proof.

We observe that if $\ell(R/I)=2$ then $\operatorname{e}(I)>\operatorname{e}(R)$ if and only if $I$ is integrally closed. Therefore, $L_{2}(R)=\operatorname{e}(R)/4$ if and only if there is no integrally closed ideal $I$ of colength $2$.

By the proof of [MQS23, Proposition 4] (or taking $J={\mathfrak{m}}^{n}$ for $n\gg 0$ in the statement), integrally closed ideals of colength $2$ in a local ring $(S,{\mathfrak{n}},\ell)$ are parametrized by $\ell$-rational points in $\operatorname{Proj}\operatorname{gr}_{\mathfrak{n}}(S)$. In our cases, $\operatorname{Proj}\operatorname{gr}_{\mathfrak{m}}(R)$ does not have any $k$-rational point. For the curve $\operatorname{Proj}(\mathbb{F}_{5}[x,y,z]/(x^{4}+y^{4}+z^{4}))$ this follows from the fact that $\{0,1\}$ are the only forth powers in $\mathbb{F}_{5}$ and for Selmer’s curve $\operatorname{Proj}({\mathbb{Q}}[x,y,z]/(3x^{3}+4y^{3}+5z^{3}))$, an elementary exposition can be found in [Conb].

On the other hand, $\operatorname{Proj}(\operatorname{gr}_{\mathfrak{m}}(R\mathbin{\widehat{\otimes_{k}}}\overline{k}))$ always has $\overline{k}$-rational points as $\overline{k}$ is algebraically closed. Thus

[TABLE]

We next record the following observation regarding the behavior of the Lech–Mumford constant under blowing up.

Proposition 2.3.12**.**

Let $(R,\mathfrak{m})$ be a local domain and let $\pi\colon X\to\operatorname{Spec}(R)$ be the blowup of an ideal $I$. For any closed point $x\in X$ we have $\operatorname{c_{LM}}(\mathcal{O}_{X,x})\leq\operatorname{c^{gr}_{LM}}(\operatorname{gr}_{I}(R))$.

Proof.

Since $R$ is local, every closed point $x\in X$ lies in the exceptional fiber $E\coloneqq\pi^{-1}(V(I))=\operatorname{Proj}(\operatorname{gr}_{I}(R))$. Let $\pi$ be a local generator of $I\mathcal{O}_{X,x}$. Since $\pi$ is a nonzerodivisor on $\mathcal{O}_{X,x}$, we have $\operatorname{gr}_{(\pi)}(\mathcal{O}_{X,x})\cong(\mathcal{O}_{X,x}/\pi\mathcal{O}_{X,x})[T]=\mathcal{O}_{E,x}[T]$. By Proposition 2.1.5, we have that

[TABLE]

Since $\mathcal{O}_{E,x}=(\operatorname{gr}_{I}(R))_{(P)}$ for a homogeneous prime $P$ such that $\dim(\text{gr}_{I}(R)/P)=1$, by Corollary 2.3.7, we have

[TABLE]

Remark 2.3.13*.*

Tracing the proof of Proposition 2.3.12 and Corollary 2.3.7 more carefully, it applies to any numerical measure of singularities $f$ such that:

1. (1)

$f(R_{\mathfrak{p}})\leq f(R_{\mathfrak{q}})$ whenever $\mathfrak{p}\subseteq\mathfrak{q}$ in $\operatorname{Spec}(R)$, 2. (2)

$f(R)\leq f(\operatorname{gr}_{(\pi}(R))$ for a regular element $\pi\in R$, 3. (3)

$f(R)\leq f(R(t))$ where $R(t)=R[t]_{{\mathfrak{m}}R[t]}$.

In particular, it applies to the Hilbert–Samuel and Hilbert–Kunz multiplicities. In the case of Hilbert–Samuel multiplicity, our proof resembles the argument of Orbanz in [Orb82] who generalized a classical result of Dade [Dad60]: the multiplicity does not increase when blowing up a regular equimultiple prime ideal (i.e., $R/\mathfrak{p}$ is regular and $\operatorname{e}(R_{\mathfrak{p}})=\operatorname{e}(R_{\mathfrak{q}})$ for any $\mathfrak{p}\subseteq\mathfrak{q}$), since these assumptions imply that $\operatorname{e}(\operatorname{gr}_{\mathfrak{p}}(R))\leq\operatorname{e}(R_{\mathfrak{p}})$.

We also point out that Proposition 2.3.12 might fail without passing to the associated graded ring.

Example 2.3.14**.**

Let $R=\mathbb{C}[x,y,z]/(x^{2}+y^{3}+z^{6})$. By Theorem 3.2.8 and Example 3.2.2 in Section 3, $\operatorname{c_{LM}}(R)=8/7$. After blowing up the closed point, the chart corresponding to $z\neq 0$ is defined by the equation $a^{2}+b^{3}z+z^{4}$. At the origin it has a simple elliptic singularity of degree $2$ (see Example 3.2.2), so the Lech–Mumford constant of the origin is $4/3$ by Theorem 3.2.3 in Section 3.

Finally, we study the behavior of the Lech–Mumford constant under finite extensions.

Proposition 2.3.15**.**

Let $(R,\mathfrak{m},k)$ be a local domain and let $(S,\mathfrak{n},\ell)$ be a finite extension of $R$. Then $\operatorname{rank}_{R}(S)\operatorname{c_{LM}}(R)\geq\operatorname{c_{LM}}(S)$. In particular, if $(S,\mathfrak{n},\ell)$ is a finite birational extension of $R$, then $\operatorname{c_{LM}}(R)\geq\operatorname{c_{LM}}(S)$.333This appeared without proof in [Mum77, Proposition 3.10], which asserts that $\operatorname{c_{LM}}(R)\geq\operatorname{c_{LM}}(S)$ when $S$ is birational and integral over $R$, we have not been able to verify this fact without mild assumptions.

Proof.

Let $J$ be an $\mathfrak{n}$-primary ideal in $S$ and denote $I\coloneqq J\cap R$. Then

[TABLE]

On the other hand, since $R$ is a domain, we have

[TABLE]

By combining these inequalities we obtain that

[TABLE]

Therefore, $\operatorname{c_{LM}}(S)\leq\operatorname{rank}_{R}(S)\operatorname{c_{LM}}(R)$ and the proposition follows. ∎

2.4. Weak semicontinuity

We now turn our attention to the behavior of Lech–Mumford constants in the fibers of a finite type morphism. We will work in the following setting and notation.

Setting 2.4.1*.*

Let $T\to R$ be a finite type flat homomorphism of rings and suppose there is an ideal $I\subseteq R$ such that the composition $T\to R\to R/I$ is an isomorphism (geometrically, $\operatorname{Spec}(R)\to\operatorname{Spec}(T)$ is a flat family together with a section $\sigma\colon\operatorname{Spec}(T)\to\operatorname{Spec}(R)$).

For each ${\mathfrak{p}}\in\operatorname{Spec}(T)$ we will denote $R({\mathfrak{p}})\coloneqq R\otimes_{T}T_{\mathfrak{p}}/{\mathfrak{p}}T_{\mathfrak{p}}$. Note that $R({\mathfrak{p}})$ is a finitely generated algebra over the field $k({\mathfrak{p}})\coloneqq T_{\mathfrak{p}}/{\mathfrak{p}}R_{\mathfrak{p}}$ and $IR({\mathfrak{p}})$ is a maximal ideal of $R({\mathfrak{p}})$. We denote by $\overline{R(\mathfrak{p})}$ the local ring obtained by the localization of $R(\mathfrak{p})\otimes_{k(\mathfrak{p})}\overline{k(\mathfrak{p})}$ at $IR(\mathfrak{p})$.

We will study the function ${\mathfrak{p}}\mapsto\operatorname{c_{LM}}(\overline{R(\mathfrak{p})})$ culminating in a weak semicontinuity result of Proposition 2.4.10. First we record the following lemma on equidimensionality of fibers, which should be well-known.

Lemma 2.4.2**.**

In Setting 2.4.1 suppose that $T$ is a domain and $R(0)$ is equidimensional444Here $R(0)$ is a finite type $K$-algebra where $K$ is the fraction field of $T$, thus equidimensional means that each irreducible component of $R(0)$, as an affine domain over $K$, has the same dimension. (e.g., $R$ is a domain). Then the function $\operatorname{Spec}(T)\ni{\mathfrak{p}}\mapsto 0ptIR({\mathfrak{p}})$ is constant.

Proof.

The proof follows the outline of [Gro66, Corollaire 14.2.2] (see also [Cona]), which shows that if $X\to Y$ is an open dominant locally finite type morphism of irreducible Noetherian schemes, then all fibers $X_{y}$ are of pure dimension and $\dim X_{y}$ is constant for all $y\in Y$.

We will show that for each $0\neq{\mathfrak{p}}\in\operatorname{Spec}(T)$, $\operatorname{ht}(IR({\mathfrak{p}}))=\operatorname{ht}(IR(0))$. We first reduce to the case where $T=(V,\pi,\kappa)$ is a DVR and ${\mathfrak{p}}=(\pi)$. By the Krull–Akizuki theorem, there exists a DVR $(V,\pi,\kappa)$ over $T$ in the fraction field of $T$ whose center on $T$ is ${\mathfrak{p}}$. The generic fiber of $V\to R_{V}\coloneqq R\otimes_{T}V$ is the same as $R(0)$ while the special fiber is $R_{V}/\pi R_{V}\cong R({\mathfrak{p}})\otimes_{k({\mathfrak{p}})}\kappa$. Since $k({\mathfrak{p}})\to\kappa$ is a field extension, it is clear that $0ptIR({\mathfrak{p}})=0ptI(R_{V}/\pi R_{V})$. Thus we may replace $T$ by $V$ and $R$ by $R_{V}$ to assume that $T=V$.

Now we assume that $(V,\pi,\kappa)\to R$ is a finite type flat map. Observe that $\pi\notin P$ for any minimal prime $P$ of $R$ since $\pi$ is a regular element of $V$ and the map is flat. Since the generic fiber is obtained by inverting $\pi$, we have a one-to-one correspondence between minimal primes of $R$ and $R(0)$. Moreover, we have that $R/P$ is torsion-free and thus flat over $V$. Hence, by [Gro66, Corollaire 14.2.2], for every minimal prime $P$ of $R$, $R/(P,\pi)$ is equidimensional (as a finite type $\kappa$-algebra) and

[TABLE]

where the second equality is by equidimensionality of $R(0)$ and the third equality follows as $IR(0)$ is a maximal ideal of $R(0)$. Since $I(R/\pi R)$ is a maximal ideal of $R/\pi R$, ${\mathfrak{n}}\coloneqq(I,\pi)$ is a maximal ideal of $R$ and we have

[TABLE]

for some minimal prime $P$ of $R$, where the last equality above follows from the equidimensionality of $R/(P,\pi)$. It follows that $\operatorname{ht}(I(R/\pi R))=\operatorname{ht}(IR(0))$ as wanted. ∎

Remark 2.4.3*.*

Consider the map $f\colon T\coloneqq k[[t]]\to R\coloneqq k[[x,y,z]]/(xy,xz)$ that sends $t\mapsto x+y$. Since $x+y$ is a regular element of $R$, $f$ is flat. Setting $I=(y,z)$, we have $R/I\cong T$ which produce a section of $f$. On the other hand, $0pt(I(R/tR))=1>0=0pt(I(R[1/t]))$. This shows that the equidimensional assumption on $R(0)$ is necessary in Lemma 2.4.2.

We next record some consequences of Lemma 2.3.1,

Corollary 2.4.4**.**

Let $(D,\pi)$ be a DVR and $D\to R$ be a finite type flat homomorphism of rings with a section given by $P\subseteq R$. Suppose that $\dim(R_{P})=0ptP(R/\pi R)$ (for example, this holds if $R[1/\pi]$ is equidimensional by Lemma 2.4.2). Then $\operatorname{c_{LM}}(R_{P})\leq\operatorname{c_{LM}}((R/\pi R)_{(P,\pi)})$.

Proof.

Note that $P$ is a prime ideal of $R$ and $(P,\pi)$ is a maximal ideal of $R$. The assumption implies that

[TABLE]

Now the conclusion follows immediately from Lemma 2.3.1. ∎

Corollary 2.4.5**.**

Let $R=S/I$ where $S=k[x_{1},\dots,x_{n}]$ is a polynomial ring over a field $k$ and let ${\mathfrak{m}}=(x_{1},\dots,x_{n})$. Let $J\subseteq{\mathfrak{m}}S[t]$ be an ideal such that

1. (1)

$S[t,t^{-1}]/JS[t,t^{-1}]\cong R[t,t^{-1}]$, and 2. (2)

$t$* is a nonzerodivisor on $S[t]/J$.*

Then $\operatorname{c_{LM}}(R_{\mathfrak{m}})\leq\operatorname{c_{LM}}((S[t]/(t,J))_{\mathfrak{m}}).$

Proof.

First, we want to show that

[TABLE]

Note that the second equality holds since $t$ is a nonzerodivisor on $S[t]/J$, and the last equality holds by the following claim.

Claim 2.4.6**.**

Let $(T,{\mathfrak{n}})$ be a catenary local ring and $x\in{\mathfrak{n}}$ be a parameter. Then $\dim(T_{x})=\dim(T)-1$.

Proof.

Clearly, $\dim(T_{x})\leq\dim(T)-1$, so it suffices to show the opposite inequality. Let ${\mathfrak{p}}$ be a minimal prime of $T$ such that $\dim(T/{\mathfrak{p}})=\dim(T)$. Note that $x\notin{\mathfrak{p}}$ because it is a parameter, so it suffices to prove the claim in the case when $T$ is a domain.

We may now extend $x$ to a system of parameters $x_{1},\ldots,x_{d-1},x_{d}=x$ and let ${\mathfrak{q}}$ be a minimal prime of $(x_{1},\ldots,x_{d-1})$. Since $T$ is a catenary local domain, we have $\dim(T_{\mathfrak{q}})=\dim(T)-1$. Since $x\notin{\mathfrak{q}}$ by the construction, the claim follows. ∎

Therefore, to prove (2.4.5.1) it remains to show that $\dim(R_{\mathfrak{m}})=\dim\left((S[t]/J)_{({\mathfrak{m}},t)}[t^{-1}]\right)$. We may present $(S[t]/J)_{({\mathfrak{m}},t)}[1/t]$ as a localization of $R_{\mathfrak{m}}[t,t^{-1}]$ at a multiplicative closed subset $W$ that avoids ${\mathfrak{m}}$. Thus it suffices to show that $W$ intersects all maximal ideals of $R_{\mathfrak{m}}[t,t^{-1}]$. But those ideals can be lifted to ideals of the form $({\mathfrak{m}},f(t))$ for some polynomial $f(t)\in S[t]/J$ and it is clear that they are trivialized in $(S[t]/J)_{({\mathfrak{m}},t)}[1/t]$.

We now apply Corollary 2.4.4 to the map $k[t]_{(t)}\to(S[t]/J)\otimes_{k[t]}k[t]_{(t)}$ (with $P={\mathfrak{m}}$) and obtain by Corollary 2.3.8 that

[TABLE]

We include two major applications of Corollary 2.4.5. We will use these results to compute examples in later sections.

Corollary 2.4.7**.**

Let $R=S/I$ where $S=k[x_{1},\dots,x_{n}]$ is a polynomial ring over a field $k$ and let ${\mathfrak{m}}=(x_{1},\dots,x_{n})$. Suppose we give weights to the variables such that $\deg(x_{i})=d_{i}\in\mathbb{N}$ and let $\operatorname{in}(I)$ be the initial ideal of $I$ with respect to the weights. Then we have

[TABLE]

Proof.

Consider the $\mathbb{Z}$-graded ring $S[t]$ by assigning $\deg(t)=-1$. Let $J\coloneqq\hom(I)\subseteq S[t]$ be the homogenization of $I$ in $S[t]$. The conclusion follows from Corollary 2.4.5 by noting that $S[t,t^{-1}]/JS[t,t^{-1}]\cong R[t,t^{-1}]$ and $(S[t]/(t,J))_{\mathfrak{m}}=(S/\operatorname{in}(I))_{\mathfrak{m}}$. ∎

Corollary 2.4.8**.**

Let $R=S/I$ where $S=k[x_{1},\dots,x_{n}]$ is a standard graded polynomial ring over a field $k$ with ${\mathfrak{m}}=(x_{1},\dots,x_{n})$, and $I\subseteq R$ is a homogeneous ideal. Fix a monomial order $\prec$ on $S$ and let $\operatorname{in}_{\prec}(I)$ be the initial ideal with respect to $\prec$. Then we have $\operatorname{c_{LM}}(R_{\mathfrak{m}})\leq\operatorname{c_{LM}}((S/\operatorname{in}_{\prec}(I))_{\mathfrak{m}})$.

Proof.

We follow the standard construction as in [Eis95, 15.16 and 15.17]. We choose an appropriate weight function $w$ so that $\operatorname{in}_{w}(I)=\operatorname{in}_{\prec}(I)$. Let $J\coloneqq\hom(I)\subseteq S[t]$ be the $w$-homogenization of $I$ in $S[t]$. The desired inequality follows from Corollary 2.4.5 by noting that $S[t,t^{-1}]/JS[t,t^{-1}]\cong R[t,t^{-1}]$ and $(S[t]/(t,J))_{\mathfrak{m}}=(S/\operatorname{in}_{\prec}(I))_{\mathfrak{m}}$. ∎

To prove the weak semicontinuity result on Lech–Mumford constant in a flat family, we need a lemma that reduces a general field extension to an algebraic field extension.

Lemma 2.4.9**.**

Let $k$ be a field and $R=k[x_{1},\ldots,x_{m}]/I$ be a finitely generated $k$-algebra with ${\mathfrak{m}}=(x_{1},\ldots,x_{m})$. Then for any field extension $k\subseteq\ell$ we have

[TABLE]

Proof.

First note that any ${\mathfrak{m}}$-primary ideal $J$ of $\ell[x_{1},\ldots,x_{m}]/I$ has finitely many generators, which are polynomials in $x_{1},\dots,x_{m}$ with finitely many coefficients in $\ell$. Hence, we can find a finitely generated field extension $k\subset\ell^{\prime}$ inside $\ell$ and an ${\mathfrak{m}}$-primary ideal $J^{\prime}\subseteq R^{\prime}\coloneqq R\otimes_{k}\ell^{\prime}$ such that $J=J^{\prime}(R\otimes_{k}\ell)$. Since the multiplicities and the colengths of $J$ and $J^{\prime}$ are equal, we may assume that $\ell$ is a finitely generated field extension of $k$. We may further assume that $\ell$ is separably generated over $k$: by [Sta25, Tag 04KM], there exists a commutative diagram:

[TABLE]

where $k\subseteq k^{\prime}$ and $\ell\subseteq\ell^{\prime}$ are finite purely inseparable and $k^{\prime}\to\ell^{\prime}$ is separably generated. We may then replace $k$ by $k^{\prime}$, $R$ by $R\otimes_{k}k^{\prime}$ and $\ell$ by $\ell^{\prime}$: this will not change the right hand side and will only possibly increase the left hand side of the desired inequality.

We now write $\ell=k(t_{1},\ldots,t_{n})[x]/(f(x))$ where $t_{1},\ldots,t_{n}$ are a transcendental basis of $\ell$ over $k$ and $f(x)$ is a monic separable polynomial. By collecting denominators of $f(x)$, we may assume that $f(x)\in k[t_{1},\ldots,t_{n},g^{-1}][x]$ for some $g\in k[t_{1},\dots,t_{n}]$. Let $A\coloneqq k[t_{1},\ldots,t_{n}]$. By the assumption, $k[t_{1},\ldots,t_{n},g^{-1}][x]/(f(x))$ is generically étale over $A$, so we may find $0\neq h\in A$ such that $A_{h}[x]/(f(x))$ is étale over $A_{h}$ (we may assume that $g$ divides $h$ so that $f(x)$ is well-defined in $A_{h}[x]$). Thus for any maximal ideal $h\notin{\mathfrak{n}}\subseteq A$ the quotient ring $(A_{h}/{\mathfrak{n}})[x]/(f(x))\cong(A/{\mathfrak{n}})[x]/(f(x))$ is a finite étale extension of $A/{\mathfrak{n}}$, so it is a product of finite separable field extensions of $A/{\mathfrak{n}}$.

By lifting a maximal ideal corresponding to one component, say $L$, of $(A_{h}/{\mathfrak{n}})[x]/(f(x))$, we find a maximal ideal $\mathfrak{N}$ of $R\otimes_{k}A_{h}[x]/(f(x))$ such that the extension of $({\mathfrak{m}},{\mathfrak{n}})$ to $S\coloneqq(R\otimes_{k}A_{h}[x]/({f}(x)))_{\mathfrak{N}}$ is the maximal ideal $\mathfrak{N}$, ${\mathfrak{m}}S$ is a prime ideal ($S/{\mathfrak{m}}S$ is the localization of $A_{h}[x]/(f(x))$ at one of its maximal ideal, so it is a regular local ring), and $S/{\mathfrak{N}}=L$ is an algebraic field extension of $k$.

Let $g_{1},\ldots,g_{n}\in A$ be elements whose images are minimal generators of ${\mathfrak{n}}A_{\mathfrak{n}}$. By Lemma 2.3.1 applied to $S$ we have that $\operatorname{c_{LM}}(S_{{\mathfrak{m}}S})\leq\operatorname{c_{LM}}(S/(g_{1},\ldots,g_{n})).$ Now, observe that

[TABLE]

while

[TABLE]

Thus we have $\operatorname{c_{LM}}((R\otimes_{k}\ell)_{\mathfrak{m}})\leq\operatorname{c_{LM}}((R\otimes_{k}L)_{\mathfrak{m}})\leq\operatorname{c_{LM}}((R\otimes_{k}\overline{k})_{\mathfrak{m}})$ where the last inequality follows from the fact that $L$ is algebraic over $k$. ∎

We can now prove the main result of this subsection.

Proposition 2.4.10**.**

Let $T\to R$ be a finite type flat map of rings so that $T$ is a domain and the generic fiber $R(0)$ is equidimensional. Let $I\subseteq R$ be an ideal such that the composition $T\to R\to R/I$ is an isomorphism. Let $\mathfrak{p}\subseteq\mathfrak{q}$ be two prime ideals of $T$. Then we have

[TABLE]

Proof.

By base change to $(T/{\mathfrak{p}})_{\mathfrak{q}}$ and using induction on its dimension (via Lemma 2.4.2), we may assume that $T$ is a one-dimensional local domain with maximal ideal ${\mathfrak{q}}$ and fraction field $F$. Since every $I\overline{R(0)}$-primary ideal of $\overline{R(0)}$ is extended from $(R(0)\otimes_{F}L)_{I}$ for some finite field extension $L$ of $F$, it suffices to show that for any finite field extension $F\subseteq L$ we have $\operatorname{c_{LM}}((R(0)\otimes_{F}L)_{I})\leq\operatorname{c_{LM}}(\overline{R(\mathfrak{q})})$. By localizing the integral closure of $T$ in $L$ at a maximal ideal, we may find a DVR $(V,\pi)$ such that the finite type flat map $V\to R^{\prime}\cong R\otimes_{T}V$ with a section $IR^{\prime}$ has generic fiber $R(0)\otimes_{F}L$. Therefore, by Corollary 2.4.4

[TABLE]

Since $R^{\prime}/\pi R^{\prime}\cong R({\mathfrak{q}})\otimes_{k({\mathfrak{q}})}V/\pi V$, it follows from Lemma 2.4.9 that

[TABLE]

Remark 2.4.11*.*

We expect that the Lech–Mumford constant function in a flat family should be upper semicontinuous. More precisely, we suspect that, with notations as in Proposition 2.4.10, fix $\mathfrak{q}\in\operatorname{Spec}(T)$, then for any $\epsilon>0$, there exists $f\notin\mathfrak{q}$ so that

[TABLE]

for all $\mathfrak{p}\in\operatorname{Spec}(T)$ that do not contain $f$.

3. Examples: surface singularities

In this section, we study Lech-Mumford constant of certain two-dimensional singularities.

3.1. Lech-stable singularities

We aim to determine which surface singularities are Lech-stable. We begin with a key lemma.

Lemma 3.1.1**.**

Let $(R,\mathfrak{m})$ be a Cohen–Macaulay local ring of dimension two with an infinite residue field. Suppose $I\subseteq R$ is an ${\mathfrak{m}}$-primary ideal such that $I^{2}\subseteq J$ for some minimal reduction $J$ of $I$ (e.g., $I$ is stable). Then

[TABLE]

Furthermore, if every $\mathfrak{m}$-primary integrally closed ideal is stable, then either $R$ is regular or $\operatorname{c_{LM}}(R)=\operatorname{e}(R)/2$.

Proof.

All our assumptions are preserved when passing from $R$ to $\widehat{R}$, so we may replace $R$ by $\widehat{R}$ to assume that $R$ has a canonical module $\omega_{R}$. Let $J$ be a minimal reduction of $I$ such that $I^{2}\subseteq J$. Now in $S\coloneqq R/J$ we have

[TABLE]

Since $S$ is Artinian, $\ell(\omega_{S})=\ell(S)=\ell(R/J)=\operatorname{e}(J)=\operatorname{e}(I)$ and putting these together yields that

[TABLE]

We now prove the second part. The proof of the first part and Proposition 2.1.3(1) show that $\operatorname{c_{LM}}(R)\leq(\operatorname{type}(R)+1)/2$. Let $\mathfrak{q}$ be a minimal reduction of ${\mathfrak{m}}$ such that ${\mathfrak{m}}^{2}\subseteq\mathfrak{q}$. Since we may assume that $R$ is not regular, ${\mathfrak{m}}\neq\mathfrak{q}$ and thus we have ${\mathfrak{m}}/\mathfrak{q}=(\mathfrak{q}:{\mathfrak{m}})/\mathfrak{q}$. It follows that

[TABLE]

This shows that $\operatorname{c_{LM}}(R)=(\operatorname{type}(R)+1)/2=\operatorname{e}(R)/2$. ∎

We can now compute the Lech–Mumford constant for two-dimensional pseudo-rational singularities.

Corollary 3.1.2**.**

Let $(R,\mathfrak{m})$ be a pseudo-rational local ring of dimension two. Then either $R$ is regular or $\operatorname{c_{LM}}(R)=\operatorname{e}(R)/2$. In particular, if $R$ is Gorenstein and pseudo-rational of dimension two (i.e., $R$ is a rational double point), then $R$ is Lech-stable.

Proof.

Suppose $R$ is pseudo-rational, then so is $R(t)=R[t]_{{\mathfrak{m}}R[t]}$. Thus by Corollary 2.3.8 we may assume that $R$ has an infinite residue field. Now the first statement follows from Theorem 1.3.5 and Lemma 3.1.1. The second statement follows from the first statement and the fact that two-dimensional Gorenstein pseudo-rational singularities of are hypersurfaces of multiplicity two, see [Rei97, Theorem 4.17] or [CRMP*+*21, Lemma 4.1]. ∎

Our goal in this subsection is to classify Lech-stable Cohen–Macaulay singularities in dimension two. We start with the normal case. In this case, the converse to Corollary 3.1.2 was originally obtained by Goto–Iai–Watanabe [GIW01, Proposition 7.5]. We include a short proof here for completeness (and to emphasize that the result holds in arbitrary characteristic). We will use Kato’s Riemann–Roch formula which connects colength and multiplicity using intersection theory. We provide a proof since we cannot find a good reference in the generality stated below.

Theorem 3.1.3** ([Kat76] or [OWY16, Theorem 2.8]).**

Let $(R,\mathfrak{m})$ be a two-dimensional excellent normal local ring admitting a dualizing complex. Let $f\colon Y\to\operatorname{Spec}(R)$ be a resolution of singularities. Suppose $E$ is an exceptional effective divisor on $Y$ with $I\coloneqq\Gamma(Y,\mathcal{O}_{Y}(-E))\subseteq R$. Then

[TABLE]

where $p_{g}(R)\coloneqq h^{1}(Y,\mathcal{O}_{Y})$.

Proof.

We follow the approach of [Lip69, Lemma 23.1]. By taking the cohomology exact sequence associated with the short exact sequence

[TABLE]

and noting that $\Gamma(Y,\mathcal{O}_{Y})=R$, we obtain that

[TABLE]

By adjunction and the Riemann–Roch for curves we obtain that

[TABLE]

The conclusion follows by combining the two equations above. ∎

Theorem 3.1.4** (Goto–Iai–Watanabe).**

Let $(R,{\mathfrak{m}})$ be a two-dimensional complete normal local ring of dimension two. Then the following are equivalent:

1. (1)

$R$* is Lech-stable.* 2. (2)

$R$* is Gorenstein and pseudo-rational.*

Proof.

First note that $(2)\Rightarrow(1)$ is contained in Corollary 3.1.2 and we only need to prove $(1)\Rightarrow(2)$.

So we assume that $R$ is Lech-stable, we know that $\operatorname{e}(R)\leq 2$ by Remark 2.2.3. It follows from Abhyankar’s inequality that $\operatorname{edim}(R)\leq 3$, and thus $R$ is either regular or a hypersurface. In particular, $R$ is Gorenstein. Let $\pi\colon Y\to X\coloneqq\operatorname{Spec}(R)$ be the minimal resolution, we know that $K_{Y/X}\coloneqq K_{Y}-\pi^{*}K_{X}\leq 0$. If $K_{Y/X}=0$, then $R^{1}\pi_{*}\mathcal{O}_{Y}\cong R^{1}\pi_{*}\omega_{Y}=0$ and thus $R$ is pseudo-rational. Thus we may assume that $K_{Y/X}<0$. Let $I\subseteq R$ be an ${\mathfrak{m}}$-primary ideal such that $I\mathcal{O}_{Y}=\mathcal{O}_{Y}(-E)$ is $\pi$-ample (for example, any $I$ such that $Y\cong\operatorname{Bl}_{I}(R)$). By Theorem 3.1.3 and noting that $h^{1}(Y,\mathcal{O}_{Y}(-nE))=0$ for $n\gg 0$ as $-E$ is $\pi$-ample, we have

[TABLE]

for all $n\gg 0$. Since $K_{Y/X}<0$ and $-E$ is $\pi$-ample, we know that $K_{Y}\cdot(-E)=K_{Y/X}\cdot(-E)<0$ and thus for $n\gg 0$, we have

[TABLE]

In particular, this shows that $\operatorname{c_{LM}}(R)>1$ contradicting that $R$ is Lech-stable. ∎

We next state our classification of Lech-stable Cohen–Macaulay surface singularities.

Theorem 3.1.5**.**

Let $(R,\mathfrak{m},k)$ be a Lech-stable complete Cohen–Macaulay local ring of dimension two such that $k$ is separably closed. Suppose that $\operatorname{char}(k)=0$ or $\operatorname{char}(k)>5$. Then $R$ is Lech-stable if and only if $R$ is either regular or a possibly non-normal ADE singularity in the following sense: there exists a complete regular local ring $S$ of dimension three with the maximal ideal $\mathfrak{n}=(x,y,z)$ such that $\widehat{R}\cong S/fS$ with $f$ one of the following:

[TABLE]

Moreover, these equations define Lech-stable singularities (even if $k$ is not separably closed).

Remark 3.1.6*.*

1. (1)

We expect that even when the residue characteristic is $2,3,5$, Lech-stable Cohen–Macaulay singularities are still the rational double points and their non-normal limits as classified by Artin [Art77]. 2. (2)

We also expect that, without assuming $k$ is separably closed in Theorem 3.1.5, the completion of the strict Henselization $\widehat{R^{\operatorname{sh}}}$ should be isomorphic to $S/fS$ as in the theorem. But our theorem does not imply this because it is not clear to us that $\widehat{R^{\operatorname{sh}}}$ is Lech-stable, see Question 2.3.10. 3. (3)

The Cohen–Macaulay assumption in Theorem 3.1.5 is necessary for having a viable classification since, by Proposition 2.3.15, any finite birational extension of a Lech-stable singularity is still Lech-stable.

To prove Theorem 3.1.5, we will first extend Corollary 3.1.2 from rational double points to their non-normal limits $A_{\infty}$ and $D_{\infty}$. We shall do this by showing that a version of the Briançon–Skoda theorem holds in these rings. The first ingredient is the characterization of stable ideals in Theorem 1.3.3 and the second is the $\mathfrak{m}$-adic stability of the associated graded rings due to Srinivas–Trivedi (see [ST96], and [MQS20b] for more general results).

Theorem 3.1.7** ([ST96, Theorem 3] or [MQS20b, Main Theorem]).**

Let $(R,\mathfrak{m})$ be a local ring, $I=(f_{1},\ldots,f_{k})$ be an ideal generated by a regular sequence, and $J$ be an ideal such that $I+J$ is $\mathfrak{m}$-primary. Then there exists $N$ such that for any ideal $I^{\prime}=(f_{1}+g_{1},\ldots,f_{k}+g_{k})$ with $g_{i}\in\mathfrak{m}^{N}$ we have

[TABLE]

In particular, the Hilbert–Samuel functions of $J(R/I)$ and $J(R/I^{\prime})$ are the same.

We can now prove the promised Briançon–Skoda theorem for $A_{\infty}$ and $D_{\infty}$ singularities.

Proposition 3.1.8**.**

Let $(S,{\mathfrak{m}})$ be a complete regular local ring of dimension three with an infinite residue field and let $x,y,z$ be a regular system of parameters of $S$ (i.e., ${\mathfrak{m}}=(x,y,z)$). Let $R=S/(f)$ where $f=xy$ or $f=x^{2}+y^{2}z$. Then for any integrally closed $\mathfrak{m}$-primary ideal $I$ of $R$, we have $I^{2}=JI$ for some minimal reduction $J$ of $I$. As a consequence, $R$ is Lech-stable.

Proof.

By Theorem 3.1.7, we know that there exists $N\gg 0$ such that

[TABLE]

[TABLE]

Since $S/(xy+z^{N})$ and $S/(x^{2}+y^{2}z+yz^{N})$ are rational double points in all characteristics,555These are $A_{N}$ and $D^{0}_{2N}$ in Artin’s list [Art77], and they are rational singularities in all characteristics. we know that $\operatorname{e}_{2}(IS)=0$ by Theorem 1.3.5 and Theorem 1.3.3. It follows that $\operatorname{e}_{2}(I)=0$ and thus $I$ is stable by Theorem 1.3.3 again. The last assertion then follows from Lemma 3.1.1. ∎

Remark 3.1.9*.*

In the notation of Proposition 3.1.8, set $R^{\prime}=S/(xy+z^{N})$ when $f=xy$ and $R^{\prime}\coloneqq S/(x^{2}+y^{2}z+yz^{N})$ when $f=x^{2}+y^{2}z$. Another way to see that $R$ is Lech-stable is to observe that

[TABLE]

where the first equality follows from Theorem 3.1.7 and the last equality follows from Theorem 1.3.5 as $R^{\prime}$ has rational double point as in the proof of Proposition 3.1.8. Since this is true for all ${\mathfrak{m}}$-primary ideals $I\subseteq R$, we have $\operatorname{c_{LM}}(R)=1$ and thus $R$ is Lech-stable. This avoids the use of Theorem 1.3.3, but does not shed light on the Briançon–Skoda property of $A_{\infty}$ and $D_{\infty}$.

Based on Theorem 3.1.7 and Proposition 3.1.8, it seems natural to ask whether the Lech–Mumford constant is stable under small perturbations. We do not know this even for hypersurfaces so we record it here as a question.

Question 3.1.10*.*

Let $(S,{\mathfrak{m}})$ be a complete regular local ring and $0\neq f\in S$. Then does there exist $N$ such that for all $g\in{\mathfrak{m}}^{N}$, we have $\operatorname{c_{LM}}(S/f)=\operatorname{c_{LM}}(S/(f+g))$?

We next record a useful lemma, essentially due to Mumford, which will be needed in the proof of Theorem 3.1.5.

Lemma 3.1.11**.**

Let $(S,\mathfrak{m})$ be a three-dimensional regular local ring and let $x,y,z$ be a regular system of parameters. Suppose $R=S/(x^{2}+g)$ such that $g\in(y,z^{2})^{3}$. Then the ideal $I=(y,z^{2})^{2}+x\mathfrak{m}$ is integral over $(y^{2},z^{4})$.

Proof.

It is enough to check that $I$ is integral over $(y,z^{2})^{2}$. To see this, simply note that

[TABLE]

Now we are ready to prove Theorem 3.1.5.

Proof of Theorem 3.1.5.

We begin by proving the last claim: the singularities in the list are Lech-stable. It is well-known that the equations defining $A_{n},D_{n},E_{6},E_{7},E_{8}$ are rational double points when the residue characteristic is zero or larger than five (see [CRMP*+*21]). Thus by Corollary 3.1.2 they are Lech-stable. We have also shown in Proposition 3.1.8 that $A_{\infty}$ and $D_{\infty}$ are Lech-stable.

It remains to show that if $R$ is Lech-stable, then $\widehat{R}$ is isomorphic to one of the singularities in the list. We may assume that $R$ is complete. Since $R$ is Lech-stable, we know that $\operatorname{e}(R)\leq 2$ by Remark 2.2.3. It follows from Abhyankar’s inequality that $\operatorname{edim}(R)\leq 3$, and thus $R$ is either regular or a hypersurface of multiplicity two, i.e., $R=S/fS$ where $S$ is a complete regular local ring of dimension three. We will analyze the possible shape of $f$, following the strategy in [CRMP*+*21, Section 3].

First of all, by [CRMP*+*21, Lemma 3.1] we may write $f=x^{2}+g$, $f=x^{2}+y^{2}+g$, or $f=x^{2}+y^{2}+z^{2}+g$ with $g\in\mathfrak{n}^{3}$. Next, following [CRMP*+*21, Proposition 3.2], the case that $f=x^{2}+y^{2}+z^{2}+g$ can be transformed to $f=x^{2}+y^{2}+z^{2}$ after a change of generators. Similarly, following [CRMP*+*21, Proposition 3.3], the case $f=x^{2}+y^{2}+g$ can be transformed to $f=x^{2}+y^{2}+z^{n}$ for $n\geq 3$ or $f=x^{2}+y^{2}$.

Now in the remaining case that $f=x^{2}+g$ where $g\in{\mathfrak{n}}^{3}$, the argument in [CRMP*+*21, Lemma 3.5] shows that after a change of generators we may assume that $g\in(y,z)^{3}$. Furthermore, we must have $\operatorname{ord}_{{\mathfrak{m}}}(g)\leq 3$, since otherwise $x$ is integral over $\mathfrak{m}^{2}$, so $\ell(R/\overline{{\mathfrak{m}}^{2}})\leq 3$, and thus

[TABLE]

contradicting that $R$ is Lech-stable. Therefore we may assume that

[TABLE]

where $a,b,c,d$ are either units or zero (but not all of them are zero) and $h\in(y,z)^{4}$. Then by the same argument as in [CRMP*+*21, Lemma 3.5], we may reduce to the case that $f$ is one of the following three forms: $f=x^{2}+y^{2}z+z^{3}+h$, $f=x^{2}+yz^{2}+h$, or $f=x^{2}+z^{3}+h$ with $h\in(y,z)^{4}$. In the first case, [CRMP*+*21, Proposition 3.6] brings us to a $D_{4}$ singularity. In the second case, the proof of [CRMP*+*21, Proposition 3.7] shows that $f$ is either a $D_{n}$ singularity for some $n\geq 5$ or a $D_{\infty}$ singularity.

Finally, we arrive at $f=x^{2}+y^{3}+h$ with $h\in(y,z)^{4}$. In this case, following the argument of [CRMP*+*21, Proposition 3.9], after a change of generators we may write $f=x^{2}+y^{3}+z^{4}$, $f=x^{2}+y^{3}+yz^{3}$, or $f=x^{2}+y^{3}+z^{5}$ unless there exist $\eta_{1}$ and $\eta_{2}$ such that

[TABLE]

see the last paragraph in the proof of [CRMP*+*21, Proposition 3.9]. However, this last case cannot happen since in this case we have $f-x^{2}\in(y,z^{2})^{3}$ and hence by Lemma 3.1.11, $\ell(R/\overline{(y^{2},z^{4})})\leq\ell(R/((y,z^{2})^{2}+x{\mathfrak{m}}))\leq 7$ and thus

[TABLE]

contradicting that $R$ is Lech-stable. This completes the proof. ∎

Remark 3.1.12*.*

Via Theorem 3.1.5 and Proposition 2.1.9, we obtain examples of Lech-stable singularities in higher dimensions that are not regular. For example, the ADE singularities are Lech-stable in all dimensions. One particular family of such examples are compound Du Val (cDV) singularities

[TABLE]

where $f$ is one of the ADE equations in Theorem 3.1.5 and $g$ is arbitrary. It was shown by Reid in [Rei83, 1.1] that isolated cDV singularities are precisely Gorenstein terminal singularities in dimension three.

3.2. Minimally elliptic singularities

Using the rich theory of surface singularities, we are able to compute the Lech–Mumford constant of singularities closest to rational singularities. We refer the reader to [Art66, Lau77, Rei97, Ish14, OWY16] for additional background information.

Definition 3.2.1**.**

Let $(R,\mathfrak{m})$ be a two-dimensional excellent normal local ring containing an algebraically closed field $k\cong R/{\mathfrak{m}}$. Let $X\to\operatorname{Spec}(R)$ be the minimal resolution with reduced exceptional divisor $E\coloneqq E_{0}+\cdots+E_{r}$ where each $E_{i}$ is irreducible. The fundamental cycle $Z_{f}$ is the unique nonzero minimal effective divisor on $X$ supported in $E$ such that $Z_{f}E_{i}\leq 0$ for all $i$ (the existence of $Z_{f}$ follows from the Hodge index theorem).

We say that $R$ has a minimally elliptic singularity if $R$ is Gorenstein with geometric genus $p_{g}(R)\coloneqq\ell(H^{1}(X,\mathcal{O}_{X}))=1$. The number $e\coloneqq-Z_{f}^{2}$ is called the degree of $R$.

If, in addition, the reduced exceptional divisor $E=E_{0}$ is a non-singular elliptic curve, then we say that $R$ is a simple elliptic singularity. In this case $Z_{f}=E_{0}$ and thus the degree of $R$ is $-E_{0}^{2}$.

Laufer ([Lau77, Theorem 3.13]) showed that minimally elliptic singularities of degree at most three are hypersurfaces and listed the dual graphs of their minimal resolutions. We list some examples from [Lau77, Section 5, Tables 1-3].

Example 3.2.2**.**

In $\mathbb{C}[[x,y,z]]$, we have:

1. (1)

$z^{2}+y^{3}+x^{6}$ is a simple elliptic singularity of degree $1$; 2. (2)

$z^{2}+xy^{3}+x^{4}$ is a simple elliptic singularity of degree $2$; 3. (3)

$z^{3}+y^{3}+x^{3}$ is a simple elliptic singularity of degree $3$; 4. (4)

$z^{2}+y^{3}+x^{7}$ is a minimally elliptic singularity of degree $1$, its exceptional set is a rational curve with a cusp singularity; 5. (5)

$z^{2}+y^{3}+x^{8}$ is a minimally elliptic singularity of degree $1$, its exceptional set consists of three non-singular rational curves meeting transversely at the same point; 6. (6)

$z^{2}+x(y+x)(y^{2}+x^{3})$ is a minimally elliptic singularity of degree $2$, its exceptional set is a rational curve with a node singularity; 7. (7)

$xz^{2}+y^{3}+x^{4}$ is a minimally elliptic singularity of degree $3$, its exceptional set is a rational curve with a cusp singularity.

Our techniques allow us to compute the Lech–Mumford constant for minimally elliptic singularities whose exceptional set is irreducible or whose degree is $1$. In particular, the list in Example 3.2.2 is covered by our theorems.

Theorem 3.2.3**.**

Let $(R,\mathfrak{m})$ be a minimally elliptic singularity of degree $e$. Let $X\to\operatorname{Spec}(R)$ be the minimal resolution, and suppose that $E_{0}$ is the only irreducible exceptional divisor (e.g., $R$ is simple elliptic). Then, the following hold:

[TABLE]

Proof.

Let $I\subseteq R$ be an integrally closed $\mathfrak{m}$-primary ideal. By choosing a resolution $Y\to\operatorname{Spec}(R)$ that dominates the blowup of $I$, we have that $I\mathcal{O}_{Y}=\mathcal{O}_{Y}(-Z)$ where $Z$ is an anti-nef effective exceptional divisor on $Y$ and $\Gamma(Y,\mathcal{O}_{Y}(-Z))=I$ (as $I$ is integrally closed).

By [OWY16, Theorem 3.1], we know that $h^{1}(Y,\mathcal{O}_{Y}(-Z))\leq p_{g}(R)=1$. If $h^{1}(Y,\mathcal{O}_{Y}(-Z))=1$, then $I$ is a $p_{g}$-ideal in the sense of [OWY16, Definition 3.2] and thus by [OWY16, Corollary 3.6], $I$ is stable. Therefore by Lemma 3.1.1, $\operatorname{e}(I)\leq(\operatorname{type}(R)+1)\ell(R/I)=2\ell(R/I)$. In other words, in this case we have

[TABLE]

Now we suppose $h^{1}(Y,\mathcal{O}_{Y}(-Z))=0$. By Theorem 3.1.3, we have

[TABLE]

Since $X$ is the minimal resolution, we have a map $f\colon Y\to X$. Let $W\coloneqq f_{*}Z$. Since $E_{0}$ is the only irreducible exceptional divisor on $X$, we have that $W=nE_{0}$ for some $n>0$. By [Lau77, Theorem 3.4] or [Rei97, Theorem 4.21], $-E_{0}=-Z_{f}$ is numerically equivalent to the canonical divisor $K_{X}$.

We first consider the case $e\geq 2$. By [Lau77, Theorem 3.13] or [Rei97, Theorem 4.23] we know that ${\mathfrak{m}}\mathcal{O}_{X}=\mathcal{O}_{X}(-E_{0})$. Thus by Theorem 3.1.3, we have

[TABLE]

Therefore $h^{1}(X,\mathcal{O}_{X}(-E_{0}))=0$, and thus by [OWY16, Proposition 2.5], $h^{1}(X,\mathcal{O}_{X}(-W))=h^{1}(X,\mathcal{O}_{X}(-nE_{0}))=0$. Let $I_{W}\coloneqq\Gamma(X,\mathcal{O}_{X}(-W))=\Gamma(X,\mathcal{O}_{X}(-nE_{0}))=\overline{{\mathfrak{m}}^{n}}$. By Theorem 3.1.3, we have

[TABLE]

We next claim that

[TABLE]

Note that since $W=f_{*}Z$, we have $I\mathcal{O}_{X}=\mathfrak{I}\cdot\mathcal{O}_{X}(-W)$ for some ideal sheaf $\mathfrak{I}\subseteq\mathcal{O}_{X}$ and

[TABLE]

It follows that $Z=f^{*}W+V$ where $V\geq 0$ is $f$-anti-nef (since $\mathcal{O}_{Y}(-V)=\mathfrak{I}\mathcal{O}_{Y}$ is $f$-relatively globally generated). Write $K_{Y/X}=K_{Y}-f^{*}K_{X}$, then by Theorem 3.1.3

[TABLE]

Hence, to prove (3.2.3.1), it is enough to show that

[TABLE]

This amounts to proving that

[TABLE]

To see this, note that $K_{Y/X}$ is $f$-exceptional and effective since $X$, $Y$ are non-singular. Since $V$ is $f$-anti-nef, it follows that $K_{Y/X}V\leq 0$ and thus the right-hand side is non-negative. On the other hand, we have $K_{X}W=n(-E_{0}^{2})=ne\geq 2$ and thus the left-hand side is non-positive. This completes the proof of (3.2.3.1). It remains to optimize

[TABLE]

for various $n>0$. Equivalently, we need to minimize $2/(n^{2}e)-1/n$ for $n\geq 1$. By analyzing the derivative, we see that the function on positive real numbers $x\mapsto 2/(x^{2}e)-1/x$ has its only local minimum at $x=4/e$. It follows easily that we have

[TABLE]

This completes the proof when $e\geq 2$.

Now we suppose $e=1$. By [Rei97, Lemma 4.23], the line bundle $\mathcal{L}=\mathcal{O}_{X}(-nE_{0})|_{E_{0}}$ is base point free when $n\geq 2$ and thus $\mathcal{O}_{X}(-nE_{0})$ is globally generated on $X$. This implies that $I_{nE_{0}}\coloneqq\Gamma(X,\mathcal{O}_{X}(-nE_{0}))$ is represented on $X$ when $n\geq 2$ (see [OWY16, 2.3, 2.6]). Furthermore, by [Rei97, 4.25, Corollary and Remark], we know $R$ is a hypersurface and we can choose a generating set $x,y,z$ of ${\mathfrak{m}}$ such that $I_{2E_{0}}=(x^{2},y,z)$ and $I_{3E_{0}}=(x^{3},xy,y^{2},z)$. In particular, we have $\ell(R/I_{2E_{0}})=2$ and $\ell(R/I_{3E_{0}})=4$. By Theorem 3.1.3, we have

[TABLE]

[TABLE]

Hence $h^{1}(X,\mathcal{O}_{X}(-2E_{0}))=h^{1}(X,\mathcal{O}_{X}(-3E_{0}))=0$ and thus by [OWY16, Proposition 2.5] (applied to the globally generated line bundle $\mathcal{O}_{X}(-2E_{0})$), we have $h^{1}(X,\mathcal{O}_{X}(-nE_{0}))=0$ for all $n\geq 2$.

Now if $W=f_{*}Z=nE_{0}$ for some $n\geq 2$. Then we claim that

[TABLE]

To see this, note that by the same notation as in the $e\geq 2$ case, it reduces to showing that $V^{2}(K_{X}W-2)\leq(W^{2})(K_{Y/X}V)$. But as $n\geq 2$, $K_{X}W\geq-2E_{0}^{2}=2$, the left-hand side above is non-positive while the right-hand side is non-negative (since $V$ is $f$-anti-nef and $K_{Y/X}$ is $f$-exceptional and effective). Thus we have

[TABLE]

and it is easy to check that the maximum is achieved when $n=4$.

Finally, if $W=f_{*}Z=E_{0}$, then there exists $g\in I\setminus I_{2E_{0}}$. By Lech’s inequality (Theorem I),

[TABLE]

By [Rei97, 4.25, Theorem, Corollary and Remark], we can choose a generating set $x,y,z$ of ${\mathfrak{m}}$ such that if we assign weights $1,2,3$ to $x,y,z$ respectively, the associated graded ring $G$ of the Noetherian graded family $\{I_{nE_{0}}\}_{n}$ coincides with the associated graded ring of this weight filtration. Moreover, $G\cong k[x,y,z]/(f)$ where $f=z^{2}+y^{3}+h$ and $h$ is homogeneous of degree $6$ whose terms involve $x$. Since $I_{2E_{0}}=(x^{2},y,z)$ and $g\notin I_{2E_{0}}$, the image of $g$ in $G$ agrees with $x$ up to units. It follows that $\operatorname{e}(R/(g))=\operatorname{e}(G)=2$ and thus $\operatorname{e}(I)/2\ell(R/I)\leq 1$ in this case. This completes the proof when $e=1$. ∎

It is natural to ask whether the Lech–Mumford constant of a normal surface singularity $R$ can always be computed from the minimal resolution. If this is the case, then the computation of $\operatorname{c_{LM}}(R)$ becomes a finite-dimensional optimization problem on the lattice of cycles on the minimal resolution. Kei-ichi Watanabe pointed to us the following lemma that gives a partial affirmative answer to this question.

Lemma 3.2.4** (Watanabe).**

Let $(R,\mathfrak{m})$ be a two-dimensional excellent normal local ring containing an algebraically closed field $k\cong R/{\mathfrak{m}}$ and let $X\to\operatorname{Spec}(R)$ be the minimal resolution. Suppose $I\subseteq R$ is an integrally closed ${\mathfrak{m}}$-primary ideal and $Y\to\operatorname{Spec}(R)$ is a resolution such that $I\mathcal{O}_{Y}=\mathcal{O}_{Y}(-Z)$ where $Z$ is an anti-nef effective divisor on $Y$. Let $f\colon Y\to X$ be the canonical map and let $W\coloneqq f_{*}Z$. Suppose $\mathcal{O}_{X}(-W)$ is globally generated. Then we have

[TABLE]

where $I_{W}\coloneqq\Gamma(X,\mathcal{O}_{X}(-W))$.

In particular, if every anti-nef cycle on $X$ is globally generated, then $\operatorname{c_{LM}}(R)$ can be computed from the minimal resolution.

Proof.

First of all, since $W=f_{*}Z$, we have $I\mathcal{O}_{X}=\mathfrak{I}\cdot\mathcal{O}_{X}(-W)$ for some ideal sheaf $\mathfrak{I}\subseteq\mathcal{O}_{X}$ so that $\mathcal{O}_{X}/\mathfrak{I}$ is supported at points, and we have

[TABLE]

Thus $Z=f^{*}W+V$ where $V\geq 0$ is $f$-exceptional and $\mathcal{O}_{Y}(-V)=\mathfrak{I}\mathcal{O}_{Y}$ is $f$-relatively globally generated. It follows that

[TABLE]

where the inequality follows from Lech’s inequality (Theorem I) in the regular case.666Here the notation $\operatorname{e}(\mathfrak{J},\mathcal{O}_{X})$ should be interpreted as $\sum_{x}\operatorname{e}(\mathfrak{J}_{x},\mathcal{O}_{X,x})$ where $x$ runs over all (finitely many) points on $X$ that are the support of $f_{*}\mathcal{O}_{V}$, and we are using Lech’s inequality for each $\operatorname{e}(\mathfrak{J}_{x},\mathcal{O}_{X,x})$ and noting that each $\mathcal{O}_{X,x}$ is a two-dimensional regular local ring.

Since $f_{*}\mathcal{O}_{V}$ is supported at points, from the short exact sequence

[TABLE]

we obtain that $R^{1}f_{*}\mathcal{O}_{Y}(-f^{*}W)\twoheadrightarrow R^{1}f_{*}\mathcal{O}_{V}\cong H^{1}(V,\mathcal{O}_{V})$. But we have

[TABLE]

since $X$, $Y$ are regular. Thus $H^{1}(V,\mathcal{O}_{V})=0$. Now after taking global sections, the short exact sequence above induces an exact sequence

[TABLE]

In particular, we have that $h^{1}(Y,\mathcal{O}_{Y}(-Z))\geq h^{1}(Y,\mathcal{O}_{Y}(-f^{*}W))$. On the other hand, since $\mathcal{O}_{Y}(-V)$ is $f$-relatively globally generated, each $f$-exceptional prime divisor is not in the base locus of $\mathcal{O}_{Y}(-V)$, and since $V$ is $f$-exceptional, (the strict transform of) each exceptional prime divisor on $X$ also cannot be in the base locus of $\mathcal{O}_{Y}(-V)$. Thus $\mathcal{O}_{Y}(-V)$ has no fixed component and thus $h^{1}(Y,\mathcal{O}_{Y}(-Z))\leq h^{1}(Y,\mathcal{O}_{Y}(-f^{*}W))$ by [OWY16, Proposition 2.5]. It follows that the surjection $\alpha$ must be an isomorphism and we have

[TABLE]

Thus we have

[TABLE]

Since $\operatorname{e}(I)=-W^{2}-V^{2}$ and $\operatorname{e}(I_{W})=-W^{2}$, we have

[TABLE]

where the first inequality follows from (3.2.4.1). ∎

One can use Lemma 3.2.4 to give an alternative proof of Theorem 3.2.3 when $e\geq 2$. The point is that, for minimally elliptic singularity of degree $e\geq 2$ whose exceptional divisor $E_{0}$ on the minimal resolution is irreducible, we know that $\mathcal{O}_{X}(-nE_{0})$ is globally generated for all $n\geq 0$ and thus Lemma 3.2.4 tells us immediately that $\operatorname{c_{LM}}(R)$ can be computed from $\operatorname{e}(I_{W})/2\ell(R/I_{W})$ where $W=nE_{0}$ for some $n$, and we come down to optimize $n^{2}e/(n^{2}e-ne+2)$ when $n$ varies, as in the proof of Theorem 3.2.3. In fact, we can use Lemma 3.2.4 to compute $\operatorname{c_{LM}}(R)$ for all two-dimensional standard graded normal domains.

Corollary 3.2.5**.**

Let $(R,{\mathfrak{m}},k)$ be (the localization of) a two-dimensional standard graded normal domain over an algebraically closed field $k$. If $R$ is not regular, then $\operatorname{c_{LM}}(R)=\operatorname{e}(R)/2$ and $\operatorname{c_{LM}}(R)$ is computed by ${\mathfrak{m}}$.

Proof.

The hypothesis implies that $X\coloneqq\operatorname{Bl}_{\mathfrak{m}}(R)\to\operatorname{Spec}(R)$ is the minimal resolution, and moreover, the exceptional divisor $E$ on $X$ is irreducible (we have $E\cong\operatorname{Proj}(R)$). It is also easy to see that $\mathcal{O}_{X}(-nE)={\mathfrak{m}}^{n}\mathcal{O}_{X}$ is globally generated and that ${\mathfrak{m}}^{n}=\overline{{\mathfrak{m}}^{n}}$ for all $n\geq 0$. By Lemma 3.2.4, $\operatorname{c_{LM}}(R)$ is computed from cycles on $X$, i.e., we only need to find the supremum of $\operatorname{e}({\mathfrak{m}}^{n})/2\ell(R/{\mathfrak{m}}^{n})$ when $n$ varies.

By taking ${\mathfrak{m}}$, we see that $\operatorname{c_{LM}}(R)\geq\operatorname{e}(R)/2$. Thus it suffices to show that $\ell({\mathfrak{m}}^{n}/{\mathfrak{m}}^{n+1})\geq 2n+1$ for all $n\geq 0$, since this would imply that for all $n\geq 1$

[TABLE]

and thus $\operatorname{e}({\mathfrak{m}}^{n})/2\ell(R/{\mathfrak{m}}^{n})\leq\operatorname{e}(R)/2$ as wanted. By contradiction, suppose $\ell({\mathfrak{m}}^{n_{0}}/{\mathfrak{m}}^{n_{0}+1})\leq 2n_{0}$ for some $n_{0}$. Note that $\ell({\mathfrak{m}}/{\mathfrak{m}}^{2})\geq 3$ because $R$ is singular, so $n_{0}\geq 2$. Now by Macaulay’s theorem on Hilbert function (see [BH93, Theorem 4.2.10], which originates from [Mac27]), for any $d\in\mathbb{N}$, if we write

[TABLE]

where $d\geq j\geq 1$ and $k_{d}>k_{d-1}>\cdots>k_{j}\geq j$, then

[TABLE]

Now if $n_{0}=2$, then we can write

[TABLE]

and it follows by induction that

[TABLE]

and thus $\operatorname{e}(R)=1$ contradicting that $R$ is singular. If $n_{0}\geq 3$, then we have

[TABLE]

and again it follows by induction that

[TABLE]

Therefore, $\operatorname{e}(R)=2$ and thus $R$ is a hypersurface (of degree $2$) by Abhyankar’s inequality. But then a direct computation shows that the Hilbert function of $R$ must be $\ell({\mathfrak{m}}^{n}/{\mathfrak{m}}^{n+1})=2n+1$, a contradiction. ∎

Corollary 3.2.6**.**

Let $(R,{\mathfrak{m}},k)$ be a two-dimensional local ring so that $\operatorname{gr}_{\mathfrak{m}}(R)$ is geometrically normal. Then $\operatorname{c_{LM}}(R)=\operatorname{e}(R)/2$ and $\operatorname{c_{LM}}(R)$ is computed by ${\mathfrak{m}}$.

Proof.

First of all, we know $\operatorname{c_{LM}}(R)\geq\operatorname{e}(R)/2$ by considering the maximal ideal. By Proposition 2.1.5 and Proposition 2.1.6, we know that $\operatorname{c_{LM}}(R)\leq\operatorname{c_{LM}}(\operatorname{gr}_{\mathfrak{m}}(R))\leq\operatorname{c_{LM}}(\operatorname{gr}_{\mathfrak{m}}(R)\otimes_{k}\overline{k})$, the latter is a two-dimensional standard graded normal domain by hypothesis and thus $\operatorname{c_{LM}}(\operatorname{gr}_{\mathfrak{m}}(R)\otimes_{k}\overline{k})=\operatorname{e}(R)/2$ by Corollary 3.2.5 (note that multiplicity does not change upon passing to the associated graded ring and base change the field). ∎

We next use a different argument to handle all minimally elliptic singularities of degree one: the key point is that we can compute the Lech–Mumford constant for the “non-normal limit” for this family. We treat this case algebraically.

Lemma 3.2.7**.**

Let $(R,{\mathfrak{m}})=K[[x,y,z]]/(z^{2}+y^{3})$ where $K$ is a field. Then $\operatorname{c_{LM}}(R)=8/7$, and $8/7$ is achieved only by ideals $\overline{(y^{2},x^{4n})}$ where $n\geq 1$ is an integer.

Proof.

First of all, it is straightforward to see that $\operatorname{e}(\overline{(y^{2},x^{4n})})=16n$ and $\ell(R/\overline{(y^{2},x^{4n})})=7n$ (for example, using integral dependence as in Lemma 3.1.11) and thus we achieved $8/7$. Next we note that $R\cong K[[x,t^{2},t^{3}]]$. By Proposition 2.1.6 $\operatorname{c_{LM}}(R)$ can be computed from monomial ideals $I\subset R$, thus it suffices to show that

[TABLE]

for all integrally closed ${\mathfrak{m}}$-primary monomial ideals $I\subset R$.

Let $S\coloneqq K[[x,t]]$ be the integral closure of $R$. We will transfer the computation to $S$. Set $J\coloneqq\overline{IS}$, then $J$ is an integrally closed ${\mathfrak{m}}$-primary monomial ideal of $S$ and $I=J\cap R$ by [HS06, Proposition 1.6.1]. Since the difference between $R$ and $S$ is the vector space spanned by monomials $\{tx^{n}\}_{n\geq 0}$, we have $\operatorname{e}(I)=\operatorname{e}(J)$ and

[TABLE]

Our goal is to show that

[TABLE]

At this point, we analyze both sides of (3.2.7.1) using data of the Newton polytope associated to $J$. More specifically, recall that $\operatorname{e}(J)$ is twice the volume of the complement of the Newton polytope, $\ell(S/J)$ is the number of integer points in the complement of the Newton polytope, and $\ell(S/(J:t)+(t))$ is the number of integer points in the complement of the Newton polytope whose $t$-coordinate is $1$.

Suppose $\{(a_{i},b_{i})\}_{i=0}^{i=k}$ are all integer points on the boundary of the Newton polytope associated to $J$ so that $a_{0}=0$, $b_{0}=\ell(S/J+(t))$ and $a_{k}=\ell(S/J+(x))$, $b_{k}=0$. By dividing the complement of the Newton polytope by the lines $t=a_{i}$ we may write it as a union of $k$ trapezoids $T_{i}$, $0\leq i\leq k-1$, where $T_{i}$ is defined by the end points $(a_{i},0),(a_{i},b_{i}),(a_{i+1},b_{i+1}),(a_{i+1},0)$.

a_{1}$$a_{2}$$a_{3}$$t$$b_{3}$$b_{2}$$b_{1}$$b_{0}$$T_{0}$$T_{1}$$T_{2}$$T_{3}x

Since $\operatorname{e}(J)$ is twice the volume of the complement of the Newton polytope, we have

[TABLE]

Since $\ell(S/J)$ is the total number of integer points inside the union of all the $T_{i}$’s, and by our assumption, $k$ is the number of integer points on the boundary of the Newton polytope, by Pick’s theorem we have

[TABLE]

It follows that

[TABLE]

Moreover, it is easy to see that the number of integer points on the intersection of the line $t=1$ and the complement of the Newton polytope is

[TABLE]

Thus we may rewrite (3.2.7.1) as

[TABLE]

and since $a_{k}\geq k$, it is enough to show that

[TABLE]

Next we note that, since $J=\overline{IS}$ with $I\subseteq R$, none of the vertices defining the Newton polytope of $J$ has the $t$-coordinate $1$. Thus if $a_{1}=1$, then $(a_{2},b_{2})$ must be on the same edge, so that

[TABLE]

Therefore, we can omit the point $(a_{1},b_{1})$ when $a_{1}=1$777We caution the readers that, omitting the point $(a_{1},b_{1})$ will decrease $k$, however, this will not affect (3.2.7.3) by the fact that $(b_{0}-b_{1})(a_{2}-a_{0})=(b_{0}-b_{2})(a_{1}-a_{0})$. to assume that $a_{1}\geq 2$. Now we set $x_{i}=a_{i+1}-a_{i}$, note that each $x_{i}$ is a positive integer and $x_{0}\geq 2$ (as $a_{1}\geq 2$). Since the Newton polytope is convex, for all $i$ we have

[TABLE]

Thus our goal becomes proving

[TABLE]

under the condition (3.2.7.4) and that $x_{0}\geq 2$. We rewrite the above as

[TABLE]

Using the quadratic formula, we further rewrite the above as

[TABLE]

Now if $x_{0}\geq 4$, then the coefficient $x_{1}+x_{0}-16/x_{0}$ at $b_{1}$ is non-negative and thus (3.2.7.5) follows. Observe that none of these ideals will attain $8/7$. Namely, for the equality to hold in (3.2.7.5) we must take $x_{0}=4$ and $b_{1}=0$, otherwise $x_{1}>0$. But then $\operatorname{e}(J)=4b_{0}$ and (3.2.7.3) has strict inequality.

We still have two cases remaining: $x_{0}=2$ or $x_{0}=3$. Note that by (3.2.7.4),

[TABLE]

By plugging this into (3.2.7.5) and rearranging the terms, it is enough to show that

[TABLE]

Suppose that $x_{0}=3$. Then the coefficient at $b_{1}$ is $x_{1}+1/x_{1}-2\geq 0$ and at $b_{2}$ is $x_{1}+x_{2}-1/x_{1}\geq 0$. Hence (3.2.7.6) holds and becomes equality if and only if $x_{1}=1,x_{2}=0$. In this case, (3.2.7.4) shows that $b_{0}>4b_{1}$. Note that the equality is not possible, otherwise the integer point $(2b_{1},2)$ is on the line through $(b_{0},0)$ and $(b_{1},3)$ contradicting the definition of $b_{1}$. Thus (3.2.7.3) is a strict inequality:

[TABLE]

Last assume that $x_{0}=2$. The first possibility for the equality to hold in (3.2.7.6) is to have $b_{1}=0$. In this case, $\operatorname{e}(J)=2b_{0}$ and (3.2.7.3) is always strict. Now, $b_{1}>0$ and the coefficient of $b_{1}$ in (3.2.7.6) is $x_{1}+4/x_{1}-4\geq 0$. If $b_{2}=0$, then the equality in (3.2.7.6) may only happen when $x_{1}=2$. We have $\operatorname{e}(J)=2b_{0}+4b_{1}$ and $b_{0}\geq 2b_{1}$ by (3.2.7.4). First, suppose that $b_{0}-b_{1}$ is odd. Then

[TABLE]

and we have a strict inequality in (3.2.7.2). Second, if $b_{0}-b_{1}$ is even, then, in fact, $b_{0}>2b_{1}$, otherwise $(b_{0},0),(b_{1},2),(0,4)$ are on the same line and we would have an uncounted integer point $(b_{1}/2,3)$. Hence, in this case (3.2.7.3) is a strict inequality.

Thus, we arrived to the case where $x_{0}=2$ and $b_{2}>0$. Now, the coefficient of $b_{2}$ in (3.2.7.6) is $x_{1}+x_{2}-4/x_{1}$. Since $x_{1}$ is a positive integer, this coefficient is always non-negative unless $x_{0}=2$ and $x_{1}=1$. If the coefficients are non-negative, we must make them [math], so $x_{1}=2$ and $x_{2}=0$, but this would contradict $b_{2}>0$.

Hence, we now proceed to analyze the last case where $x_{0}=2$ and $x_{1}=1$. Summing up, the coefficient at $b_{1}$ in (3.2.7.6) is $1$ and the coefficient of $b_{2}$ is $x_{2}-3$. By (3.2.7.4) again, we have

[TABLE]

Plugging these into (3.2.7.6), it is enough to show that

[TABLE]

is nonnegative. This is always true since the coefficients of all $b_{i}$’s above are non-negative. It remains to note that the equality in (3.2.7.6) now forces that $x_{2}=1$ and $b_{3}=0$. We compute $\operatorname{e}(J)=2b_{0}+3b_{1}+2b_{2}$ and observe that if any of the convexity inequalities (3.2.7.4) is strict, then

[TABLE]

showing that (3.2.7.3) is a strict inequality. Hence, we may assume that all those points lay on the same line, giving that $b_{0}=4b_{2}$ and $b_{1}=2b_{2}$. ∎

Theorem 3.2.8**.**

Let $(R,\mathfrak{m})$ be a minimally elliptic singularity of degree $1$. Then we have $\operatorname{c_{LM}}(R)=8/7$.

Proof.

Let $X\to\operatorname{Spec}(R)$ be the minimal resolution with the fundamental cycle $Z_{f}$, and let $J_{n}\coloneqq\Gamma(X,\mathcal{O}_{X}(-nZ_{f}))$. By [Rei97, 4.25, Theorem, Corollary, and Remark], $\{J_{n}\}_{n}$ is a graded family and we have $\operatorname{gr}_{J_{\bullet}}(R)\cong k[x,y,z]/(z^{2}+y^{3}+xf(x,y,z))$ where $z^{2}+y^{3}+xf(x,y,z)$ is homogeneous if we assign $x$, $y$, $z$ weights $1$, $2$, $3$ respectively. It follows that $\operatorname{gr}_{(x)}\operatorname{gr}_{J_{\bullet}}(R)\cong k[x,y,z]/(z^{2}+y^{3})$. Thus applying Proposition 2.1.5 twice (and using Proposition 2.1.6), we have

[TABLE]

where the last equality follows from Lemma 3.2.7.

It remains to find an ${\mathfrak{m}}$-primary ideal of $R$ such that $8/7$ is achieved. Note that by [Rei97, Lemma 4.23], we know that $\mathcal{O}_{X}(-nZ_{f})$ is globally generated for all $n\geq 2$ and that by [Rei97, Corollary 4.25], $J_{2}=\Gamma(X,\mathcal{O}_{X}(-2Z_{f}))$ satisfies $\ell(R/J_{2})=2$. By Theorem 3.1.3, we have

[TABLE]

where we used that $K_{X}$ is numerically equivalent to $-Z_{f}$ and $-Z_{f}^{2}=1$. It follows that $h^{1}(X,\mathcal{O}_{X}(-2Z_{f}))=0$ and hence $h^{1}(X,\mathcal{O}_{X}(-4Z_{f}))=0$ by [OWY16, Proposition 2.5]. Finally, we apply Theorem 3.1.3 to $J_{4}=\Gamma(X,\mathcal{O}_{X}(-4Z_{f}))$, we obtain that

[TABLE]

Since $\operatorname{e}(J_{4})=-(4Z_{f})^{2}=16$, we have

[TABLE]

This completes the proof of the theorem. ∎

3.3. Asymptotic Lech’s inequality

For most surface examples studied in the previous two subsections, the Lech–Mumford constant is attained (i.e., the supremum in the definition is actually a maximum). We expect the following more general conjecture in this direction.

Conjecture 3.3.1**.**

Let $(R,{\mathfrak{m}})$ be a local ring. Suppose $\widehat{R}$ has an isolated singularity, i.e., $\widehat{R}_{P}$ is regular for all $P\in\operatorname{Spec}(\widehat{R})-\{{\mathfrak{m}}\}$. If $\operatorname{c_{LM}}(R)>1$, then the supremum in the definition of $\operatorname{c_{LM}}(R)$ is attained. In particular, $\operatorname{c_{LM}}(R)\in{\mathbb{Q}}$.

Remark 3.3.2*.*

1. (1)

Without the assumption that $\operatorname{c_{LM}}(R)>1$, the first conclusion of Conjecture 3.3.1 is false: consider $R=\mathbb{C}[[x,y,z,w]]/(x^{2}+y^{2}+z^{2}+w^{2})$, then $R$ has an isolated singularity, but as $\operatorname{c_{LM}}(R/w)=1$ by Theorem 3.1.5, Proposition 2.1.9 implies that $\operatorname{c_{LM}}(R)=1$ and the supremum in the definition of $\operatorname{c_{LM}}(R)$ is not attained. 2. (2)

Without the isolated singularity assumption, the first conclusion of Conjecture 3.3.1 is also false. Let $A$ be any Artinian local ring that is not a field and let $R\coloneqq A[[T_{1},\dots,T_{d}]]$. By Theorem 2.1.4 or Proposition 2.1.3(3), $\operatorname{c_{LM}}(R)=\operatorname{e}(R)>1$ for every $d$. But the supremum in the definition of $\operatorname{c_{LM}}(R)$ is not attained by Proposition 2.1.9 for $d\geq 2$.

On the other hand, we do not know any example that $\operatorname{c_{LM}}(R)>1$ and $\widehat{R}$ is reduced, but the supremum in the definition of $\operatorname{c_{LM}}(R)$ is not attained. Nor do we have an example where $\operatorname{c_{LM}}(R)$ is irrational. However, beyond the isolated singularity case, we have limited evidence. In fact, if we examine the descending chain ($\dagger$ ‣ 2.2):

[TABLE]

we observe that as long as this chain is not strictly decreasing, by Proposition 2.1.9 there must exist some $n$ such that the supremum in the definition of $\operatorname{c_{LM}}(R[[T_{1},\dots,T_{n}]])$ is not attained. Thus, one way to produce or to seek examples where $\operatorname{c_{LM}}(R[[T_{1},\dots,T_{n}]])>1$ and is not attained is to start with a non-lim-stable singularity $R$ and compute the chain ($\dagger$ ‣ 2.2) above. However, while there are plenty of non-lim-stable singularities (for example, see Theorem 4.2.1 in Section 4 or Theorem 5.3.6 in Section 5), computing these higher Lech–Mumford constants seems extremely difficult. We will see in Example 7.4.3 that even producing an example $(R,{\mathfrak{m}})$ with $\operatorname{c_{LM}}(R)>\operatorname{c_{LM}}(R[[T_{1}]])>\operatorname{c_{LM}}(R[[T_{1},T_{2}]])$ requires nontrivial effort.

We also do not have an example of a local ring $(R,{\mathfrak{m}})$ with $\operatorname{c_{LM}}(R[[T_{1},\ldots,T_{n}]])=1$ for some $n\geq 2$ but $R$ is not semistable. In fact, as we mentioned earlier, we do not even know an example of a lim-stable but not semistable singularity.

Despite the difficulty discussed above, we point out the following result providing strong evidence towards Conjecture 3.3.1.

Proposition 3.3.3**.**

Conjecture 3.3.1 holds if $R$ contains a field of characteristic $p>0$ and $R/{\mathfrak{m}}$ is perfect.

Proof.

Since $\operatorname{c_{LM}}(R)>1$, we can fix an $\epsilon>0$ such that $\operatorname{c_{LM}}(R)-\epsilon>1$. By [HMQS20, Corollary 4.4], we know that there exists $N>0$ such that

[TABLE]

for all ${\mathfrak{m}}$-primary ideals $I\subseteq R$ such that $\ell(R/I)>N$. On the other hand, if $\ell(R/I)\leq N$, then by Lech’s inequality (Theorem I), $\operatorname{e}(I)\leq d!\operatorname{e}(R)\ell(R/I)$. It follows that $\operatorname{e}(I)/(d!\ell(R/I))$ can only take values in finitely many rational numbers when $\ell(R/I)\leq N$ and thus $\operatorname{c_{LM}}(R)$ is attained. ∎

Proposition 3.3.3 is essentially a consequence of [HMQS20, Corollary 4.4]. For non-isolated singularities, one cannot expect the conclusion of [HMQS20, Corollary 4.4] (or the statement of [HMQS20, Conjecture 1.2 (a)]) to hold, as already observed in [HMQS20, Example 4.8]. In fact, Lemma 3.2.7 also exhibits an example that the Lech–Mumford constant is achieved by sufficiently deep ideals.

Note that, when $R$ is singular, $R[[T]]$ never has an isolated singularity. In this case we do not know whether $\operatorname{c_{LM}}(R[[T]])$ can always be approximated (or even achieved) by sufficiently deep ideals.

Question 3.3.4*.*

Let $(R,{\mathfrak{m}})$ be a local ring of dimension $d$. Is it always true that

[TABLE]

Remark 3.3.5*.*

1. (1)

Question 3.3.4 has an affirmative answer whenever $\operatorname{c_{LM}}(R[[T]])$ is computed by $(J,T)$ for some ${\mathfrak{m}}$-primary ideal $J\subseteq R$. By [HMQS20, Lemma 5.10], it is enough to show that

[TABLE]

But this is straightforward by considering $I=(J,T^{N+1})$ for each $N$. 2. (2)

On the other hand, it rarely happens that $\operatorname{c_{LM}}(R[[T]])$ can be computed by $(J,T)$ (this fails in simple examples, see Lemma 3.2.7). In fact, if $\operatorname{c_{LM}}(R[[T]])$ is computed by $(J,T)$, then set $\rho$ to be the Rees period of $J$ and consider the ideal $\overline{(J,T^{\rho})}$. By Corollary 1.2.8, we find that

[TABLE]

It follows that $\sum_{j=1}^{\rho}\ell(R/J^{\frac{j}{\rho}})\geq\rho\cdot\ell(R/J)$ and thus we must have $J=J^{\frac{j}{\rho}}$ for each $1\leq j\leq\rho$. But since $J^{\frac{1}{\rho}}={\mathfrak{m}}$, we see that $J={\mathfrak{m}}$. Thus $\operatorname{c_{LM}}(R[[T]])$ is computed by the maximal ideal $({\mathfrak{m}},T)$ and $\operatorname{c_{LM}}(R[[T]])=\operatorname{e}(R)/(d+1)!$.

We end this subsection by proving that Conjecture 3.3.1 holds for Gorenstein normal rings of dimension two, in arbitrary characteristic. This includes most of the examples studied in the previous two subsections.

We will need the following lemma, which is well-known to experts (see [MST*+*22, Corollary 4.2] and [EIM16, Corollary 3.16] for more general results in all dimensions). We include a short and characteristic-free proof for completeness.

Lemma 3.3.6**.**

Let $(R,{\mathfrak{m}})$ be a two-dimensional complete normal local domain. Then there exists $N>0$ such that for any ${\mathfrak{m}}$-primary ideal $I\subseteq R$, we have ${\mathfrak{m}}^{N}I^{2}\subseteq J$ for any reduction $J$ of $I$.

Proof.

We may replace $R$ by the completion of $R(t)$ to assume $R$ has an infinite residue field. It is enough to prove the result when $J=(x,y)$ is a minimal reduction of $I$. Now let $\pi\colon Y\to X\coloneqq\operatorname{Spec}(R)$ be a resolution of singularities such that $I\mathcal{O}_{Y}=(x,y)\mathcal{O}_{Y}=\mathcal{O}_{Y}(-E)$. We have an exact Koszul complex $0\to\mathcal{O}_{Y}\to\mathcal{O}_{Y}(-E)^{\oplus 2}\xrightarrow{\cdot(x,y)}\mathcal{O}_{Y}(-2E)\to 0.$ Pushforward to $X$ and note that $\pi_{*}\mathcal{O}_{Y}(-nE)=\overline{I^{n}}$, we obtain that $\overline{I}^{\oplus 2}\xrightarrow{\cdot(x,y)}\overline{I^{2}}\to R^{1}\pi_{*}\mathcal{O}_{Y}$. Since $R$ is normal and $\dim(R)=2$, $R^{1}\pi_{*}\mathcal{O}_{Y}$ is independent of the resolution and is annihilated by ${\mathfrak{m}}^{N}$ for some fixed $N>0$. It follows that ${\mathfrak{m}}^{N}I^{2}\subseteq{\mathfrak{m}}^{N}\overline{I^{2}}\subseteq(x,y)=J$ as wanted. ∎

Proposition 3.3.7**.**

Let $(R,\mathfrak{m})$ be a two-dimensional complete Gorenstein normal domain. Then we have

[TABLE]

As a consequence, if $\operatorname{c_{LM}}(R)>1$, then the supremum in the definition of $\operatorname{c_{LM}}(R)$ is attained. In particular, $\operatorname{c_{LM}}(R)\in{\mathbb{Q}}$.

Proof.

We replace $R$ by the completion of $R(t)$ to assume that $R$ has an infinite residue field. We fix $K>0$ given by Lemma 3.3.6. Now let $I$ be an ${\mathfrak{m}}$-primary ideal and let $J=(x,y)$ be a minimal reduction of $I$. In $S\coloneq R/J$ we have by Lemma 3.3.6 that

[TABLE]

Since $S$ if Gorenstein, we get a further bound

[TABLE]

Filtering now gives us the inequality

[TABLE]

By writing $R$ as a quotient of a complete regular local ring $A$ one obtains from [HMQS20, Corollary 3.5] that $\mu(I)/\ell(R/I)$ tends to [math] when $\ell(R/I)\to\infty$. It follows that

[TABLE]

Therefore we have

[TABLE]

The rest of the assertions follow from the same argument as in Proposition 3.3.3. ∎

4. Estimates on higher Lech–Mumford constants

4.1. Lower bound on $\operatorname{c^{\infty}_{LM}}(R)$ via graded families of ideals

To obtain estimates for $\operatorname{c^{\infty}_{LM}}(R)$, we use the following elementary construction, which transforms a graded family of ideals of $R$ into a controllable graded family of ideals in the ring of power series. This construction is applicable in arbitrary dimensions, and the results presented in this subsection will be also utilized in Section 5.

Lemma 4.1.1**.**

Let $(R,\mathfrak{m})$ be a local ring and $\{I_{n}\}_{n}$ be a graded family of ${\mathfrak{m}}$-primary ideals where we set $I_{0}=R$. Fix non-negative integers $k,N$ and consider the following ${\mathfrak{m}}+(t_{1},\dots,t_{r})$- primary ideal in $R[[t_{1},\ldots,t_{r}]]$

[TABLE]

Then we have

[TABLE]

Proof.

The second formula easily follows after noting that the colength of a graded ideal (with the monomial grading on $t_{i}$) can be computed in each component, and that the number of monomials in $r$ variables of degree at most $d$ is $\binom{d+r}{r}$. The first formula follows by reordering the terms in the following way

[TABLE]

∎

We next observe that the multiplicity of the ideal $J_{N,k}$ constructed in Lemma 4.1.1 can be controlled under rather mild assumptions on the graded family $\{I_{n}\}_{n}$.

Lemma 4.1.2**.**

Let $(R,\mathfrak{m})$ be a local ring of dimension $d\geq 1$ and let $\{I_{n}\}_{n}$ be a graded family of ${\mathfrak{m}}$-primary ideals. If there exists $A\in\mathbb{R}$ such that $\ell(R/I_{n})=\frac{A}{d!}n^{d}+O(n^{d-1})$, then for the ideal $J_{N,k}\subseteq R[[t_{1},\ldots,t_{r}]]$ constructed in Lemma 4.1.1, we have

[TABLE]

Proof.

Since $\{I_{n}\}_{n}$ is a graded family of ideals, it is easy to check that for each fixed $k$, $\{J_{N,k}\}_{N}$ form a graded family of ${\mathfrak{m}}+(t_{1},\dots,t_{r})$-primary ideals in $R[[t_{1},\dots,t_{r}]]$. In particular, we have $J_{N,k}^{m}\subseteq J_{Nm,k}$ for every $m$. It follows from Lemma 4.1.1 that

[TABLE]

Now we estimate that

[TABLE]

Putting these together, we obtain that

[TABLE]

where the last equality follows, for example, by [Mum77, Lemma 4.5] (or see the combinatorial identity at the end of the proof of Proposition 2.1.12). It is now clear that we have $\operatorname{e}(J_{N,k})\geq Ak^{r}N^{r+d}$. ∎

The following is our key proposition, providing an estimate on $\operatorname{c^{\infty}_{LM}}(R)$.

Proposition 4.1.3**.**

Let $(R,\mathfrak{m})$ be a local ring of dimension $d\geq 1$. Suppose that $\{I_{n}\}_{n}$ is a graded family of ${\mathfrak{m}}$-primary ideals and we set $I_{0}=R$. Suppose there exists $A\in\mathbb{R}$ such that

[TABLE]

Then for every $x\in\mathbb{R}_{>0}$, we have

[TABLE]

Proof.

Let $S\coloneqq R[[t_{1},\ldots,t_{r}]]$ and consider the ${\mathfrak{m}}+(t_{1},\dots,t_{r})$-primary ideal

[TABLE]

constructed in Lemma 4.1.1. By Lemma 4.1.1 and Lemma 4.1.2, we may bound

[TABLE]

for any $N$. Setting $N=r/x$ yields

[TABLE]

At this point, we note that $\left(1-\frac{y}{r}\right)^{r}\leq e^{y}$ for all $0\leq y\leq r$. Thus we have

[TABLE]

and the conclusion follows by letting $r\to\infty$. ∎

4.2. Semistability in dimension one

Now we prove our main theorem in dimension one. Our result is a refinement and generalization of Mumford’s characterization of (semi)stability in dimension one (see [Mum77, Proposition 3.14]).

Following [Kol13, 1.41], we say that a local ring $R$ of dimension one is a node if $R\cong S/(f)$ where $(S,{\mathfrak{m}})$ is a regular local ring of dimension two, $f\in{\mathfrak{m}}^{2}$, and the image of $f$ in ${\mathfrak{m}}^{2}/{\mathfrak{m}}^{3}$ is not a square.

Theorem 4.2.1**.**

Let $(R,\mathfrak{m},k)$ be a local ring of dimension one. Then the following conditions are equivalent:

1. (1)

$R$* is semistable,* 2. (2)

$R$* is lim-stable,* 3. (3)

${\widehat{R}}^{\textup{un}}$* is either a regular local ring or a node,* 4. (4)

${\widehat{R}}^{\textup{un}}$* is Gorenstein and seminormal.*

Proof.

We first note that by Proposition 2.1.3 (2) and (2), we know that $\operatorname{c_{LM}}(R)=\operatorname{c_{LM}}({\widehat{R}}^{\textup{un}})$ and $\operatorname{c_{LM}}(R[[T_{1},\dots,T_{r}]])=\operatorname{c_{LM}}({\widehat{R}}^{\textup{un}}[[T_{1},\dots,T_{r}]])$ for all $r>0$. Hence, to prove the theorem, we can replace $R$ by ${\widehat{R}}^{\textup{un}}$, thereby assuming that $R$ is a Cohen–Macaulay complete local ring of dimension one for the remainder of the argument.

We already know $(1)\Rightarrow(2)$ from ($\dagger$ ‣ 2.2). Moreover, $(3)\Leftrightarrow(4)$ follows from [GT80, Theorem 8.1 (ii)$\Leftrightarrow$(viii)].

We next show that $(3)\Rightarrow(1)$. Suppose $R$ is a node. By [GT80, Theorem 8.1 (x)], we know that (the localization of) ${\operatorname{gr}_{\mathfrak{m}}(R)}$ is still a node and it is enough to show that the latter is semistable by Proposition 2.1.5 and Proposition 2.1.6. Thus we may assume that $R$ contains a field. Now by [GT80, Theorem 8.1 (xii)] or [Kol13, 1.41], when $\operatorname{char}(k)\neq 2$, $R$ is an ordinary double point, i.e., $R$ is a simple normal crossing after an étale extension. In this case, the completion of the strict henselization of $R$ is isomorphic to $k^{\operatorname{sep}}[[x,y]]/(xy)$. By [Mum77, Proposition 3.14 (2)] (see also Proposition 3.1.8 for an alternative treatment), we have $\operatorname{c_{LM}}(k^{\operatorname{sep}}[[x,y,z]]/(xy))=1$. It follows that $R$ is semistable since $R\to k^{\operatorname{sep}}[[x,y]]/(xy)$ is faithfully flat (for example, see Proposition 2.3.6). Now if $\operatorname{char}(k)=2$, then by [GT80, Proposition 8.4] and the discussion above, we may assume that

[TABLE]

where $\lambda\in k$ does not have a square root in $k$. To show such $R$ is semistable, we may also assume $k$ is infinite by Corollary 2.3.8. We claim the following:

Claim 4.2.2**.**

With notations as above, for all integers $n\geq 1$ the ring

[TABLE]

is pseudo-rational.

Proof of Claim.

We can assign $x,y$ degree $n$ and $z$ degree $2$ to make $R_{n}$ an $\mathbb{N}$-graded ring over $k$. With this grading, $a(R_{n})=-2$. Thus to show $R_{n}$ is pseudo-rational, it is enough to show that $R_{n}$ is normal.888This is well-known in characteristic zero, for example see [KSSW09, Theorem 5.1] (which originates from [Wat83]). But the characteristic zero assumption in those references is only used to guarantee resolution of singularities. In our context, we can take $f\colon X\to\operatorname{Spec}(R_{n})$ where $X=\operatorname{Proj}(R\oplus R_{\geq 1}t\oplus R_{\geq 2}t^{2}\oplus\cdots)$. It is easy to check that, if $R_{n}$ is normal, then $X$ has pseudo-rational singularities (see [KSSW09, Proof of Lemma 5.4]). Thus to see $R_{n}$ is pseudo-rational, it is enough to observe that $R^{1}f_{*}\mathcal{O}_{X}\cong[H_{\mathfrak{m}}^{2}(R_{n})]_{\geq 0}=0$. Now clearly, $R$ satisfies $(\textnormal{S}_{2})$ as $R$ is a hypersurface. Since $R$ is $\mathbb{N}$-graded and $k$ is infinite, the defining ideal of the nonsingular locus of $R$ is homogeneous and thus to show $R_{n}$ is $(\textnormal{R}_{1})$, it is enough to show that $(R_{n})_{\mathfrak{m}}$ satisfies Serre’s condition $(\textnormal{R}_{1})$ where ${\mathfrak{m}}=(x,y,z)$.

We use induction on $n$. When $n=1$, $R_{1}\cong k[x,y]$ is regular and the conclusion is obvious. Consider the blowup $\operatorname{Bl}_{{\mathfrak{m}}}(R_{n})$. It has three affine charts isomorphic to

[TABLE]

We are aiming to show $(R_{n})_{\mathfrak{m}}$ is $(\textnormal{R}_{1})$. It suffices to prove that the three charts are normal at points that map to ${\mathfrak{m}}$. Now the third chart is normal by our inductive hypothesis (when $n=2$, it is easy to see that this chart is regular). In the first chart, ${\mathfrak{m}}=(x)$ and thus it is enough to observe that after modulo $(x)$, the ring becomes $k[y^{\prime},z^{\prime}]/(1+\lambda y^{\prime 2})$ which is regular as $\lambda$ does not have a square root in $k$. Similarly the second chart is regular after modulo ${\mathfrak{m}}=(y)$ and we are done. ∎

Now we complete the proof of $(3)\Rightarrow(1)$. For any $(x,y,t)$-primary ideal $I\subseteq R[[t]]$, we invoke Theorem 3.1.7 to see that $\operatorname{gr}_{I}(R[[t]])\cong\operatorname{gr}_{I}(R_{n})$ for some $n\gg 0$ (here we still use $I$ to denote the image of the ideal in $R_{n}$). In particular, $\ell(R[[t]]/I)=\ell(R_{n}/I)$ and $\operatorname{e}(I,R[[t]])=\operatorname{e}(I,R_{n})$. Since $R_{n}$ is Gorenstein and pseudo-rational by Claim 4.2.2, Corollary 3.1.2 implies that $\operatorname{c_{LM}}(R_{n})=1$. It follows that

[TABLE]

Thus $\operatorname{c_{LM}}(R[[t]])=1$ and $R$ is semistable.

It remains to prove $(2)\Rightarrow(3)$. Suppose $R$ is lim-stable, we first claim that $\operatorname{e}(R)\leq 2$. By Proposition 4.1.3 applied to the graded family $\{I_{n}={\mathfrak{m}}^{n}\}_{n}$, we find that

[TABLE]

Setting $x=1$ above and noting that $\ell({\mathfrak{m}}^{j}/{\mathfrak{m}}^{j+1})\leq\operatorname{e}(R)$ for all $j\geq 1$, since $R$ is Cohen–Macaulay, we have the bound

[TABLE]

If $\operatorname{e}(R)\geq 3$, then $1/(e-1)+1/\operatorname{e}(R)<$, contradicting that $R$ is lim-stable. Thus, $\operatorname{e}(R)\leq 2$. Since $R$ is a one-dimensional Cohen–Macaulay ring, by Abhyankar’s inequality we must have that $\operatorname{edim}(R)\leq\operatorname{e}(R)+1-1\leq 2$.

If $\operatorname{edim}(R)=1$, then $R$ is regular. So we suppose $\operatorname{edim}(R)=2$ and $\operatorname{e}(R)=2$. Then $R$ is a hypersurface, i.e., we have $R\cong S/(f)$ where $(S,\mathfrak{m})$ is a complete regular local ring of dimension two and $f\in{\mathfrak{m}}^{2}-{\mathfrak{m}}^{3}$. We need to show that the image of $f$ in ${\mathfrak{m}}^{2}/{\mathfrak{m}}^{3}$ is not a square. Suppose on the contrary that $f=z^{2}+g$ where $z$ is part of a regular system of parameters of $S$ and $g\in{\mathfrak{m}}^{3}$. Let $z,y$ be a full regular system of parameters of $S$. We want to construct a graded family of ideals and apply Proposition 4.1.3. There are two cases to consider:

1. (a)

$R$ is a domain. We let $V\coloneqq R^{\text{N}}$ be the normalization of $R$. Then $(V,\varpi,k^{\prime})$ is a DVR, and we set $I_{n}\coloneqq(\varpi^{n})\cap R$. 2. (b)

$R$ is not a domain. Then we can write $f=(z+g_{1})(z+g_{2})$ where $g_{1},g_{2}\in{\mathfrak{m}}^{2}$. We have $R^{\text{N}}\cong S/(z+g_{1})\times S/(z+g_{2})$ and we let $I_{n}\coloneqq(y^{n})R^{\text{N}}\cap R$.

In case (a), note that $2=\operatorname{e}(R)=\operatorname{e}_{R}({\mathfrak{m}},V)=[k^{\prime}:k]\operatorname{e}({\mathfrak{m}}V,V),$ thus we have two scenarios: either ${\mathfrak{m}}V=(\varpi)$ and $[k^{\prime}:k]=2$, or ${\mathfrak{m}}V=(\varpi^{2})$ and $V/(\varpi)\cong k$. In the first scenario, $z\in(\varpi^{2})$ and $yV=(\varpi)$. It follows that $\ell(R/I_{1})=\ell(I_{1}/I_{2})=1$ and that $\ell(I_{n}/I_{n+1})\leq 2$ for all $n$. By Proposition 4.1.3, we then have a bound

[TABLE]

contradicting that $R$ is lim-stable since

[TABLE]

In the second scenario, we have $I_{1}=I_{2}={\mathfrak{m}}$. It follows that we have

[TABLE]

By Proposition 4.1.3, we then have

[TABLE]

This bound contradicts again that $R$ is lim-stable since

[TABLE]

Finally, in case (b), note that in both $S/(z+g_{1})$ and $S/(z+g_{2})$, we have that $z\in(y^{2})$ since $g_{1},g_{2}\in{\mathfrak{m}}^{2}$. Thus $I_{1}={\mathfrak{m}}$ and $I_{2}=(y^{2},z)$. It follows that

[TABLE]

Thus by exactly the same argument as in the first scenario of case (a), we have $\operatorname{c^{\infty}_{LM}}(R)>1.1559$ contradicting that $R$ is lim-stable. This completes the proof of $(2)\Rightarrow(3)$. ∎

Corollary 4.2.3**.**

If $(R,{\mathfrak{m}})$ is equidimensional, $(\textnormal{S}_{2})$, and lim-stable, then $R$ is demi-normal.

Proof.

By definition ([Kol13, Definition 5.1]), demi-normality requires Serre’s condition $(\textnormal{S}_{2})$ and that codimension one points are either regular or nodes. These properties follow from Theorem 4.2.1 and Theorem 2.3.2 (note that equidimensional and $(\textnormal{S}_{2})$ implies catenary). ∎

4.3. Strict complete intersections and rings of large multiplicity

In this subsection, we collect some consequences of Proposition 4.1.3 in higher dimensions. First of all, we show that for strict complete intersections – i.e., local rings $(R,{\mathfrak{m}})$ such that $\operatorname{gr}_{\mathfrak{m}}(R)$ is a complete intersection (this includes all hypersurfaces and all standard graded complete intersections) – various stability conditions force the defining equations to have small orders. This result serves as one of our motivations for studying the relationship between stability of local rings and singularities in the Minimal Model Program (MMP), and will be significantly generalized in Section 5.

Corollary 4.3.1**.**

Let $(R,{\mathfrak{m}},k)$ be a strict complete intersection and write

[TABLE]

with $\deg f_{i}=D_{i}$. Then the following hold:

1. (1)

if $R$ is lim-stable, then $\sum_{i=1}^{c}D_{i}\leq n$; 2. (2)

if $R$ is Lech-stable, then $\sum_{i=1}^{c}D_{i}<n$.

Proof.

We first prove (1). We note that

[TABLE]

where the latter is the Hilbert series of the standard graded complete intersection $\operatorname{gr}_{\mathfrak{m}}(R)$. Thus by applying Proposition 4.1.3 to $\{{\mathfrak{m}}^{n}\}_{n}$, we obtain that

[TABLE]

for all $x\in\mathbb{R}_{>0}$. Using that

[TABLE]

we further estimate that

[TABLE]

Thus if $\sum_{i=1}^{c}D_{i}>n$, then $\operatorname{c^{\infty}_{LM}}(R)\geq L(x_{0})>1$ for any $x_{0}$ sufficiently small. It follows that if $R$ is lim-stable, then $\sum_{i=1}^{c}D_{i}\leq n$ as wanted.

We next prove (2). This is easier and we just need to rewrite $h_{\operatorname{gr}_{\mathfrak{m}}(R)}(t)$ via Laurent series in $(1-t)$:

[TABLE]

Note that this series is finite since $1$ is the only pole of $h_{\operatorname{gr}_{\mathfrak{m}}(R)}(t)$. From this we obtain that

[TABLE]

Since $\operatorname{e}({\mathfrak{m}}^{j})=(\prod_{i=1}^{c}D_{i})j^{n-c}$, our estimate for $\ell(R/{\mathfrak{m}}^{j})$ shows that whenever $\sum_{i=1}^{c}D_{i}\geq n$, we must have $\operatorname{e}({\mathfrak{m}}^{j})>(n-c)!\ell(R/{\mathfrak{m}}^{j})$ for all $j\gg 0$. Thus if $R$ is Lech-stable, then $\sum_{i=1}^{c}D_{i}<n$ as wanted. ∎

Remark 4.3.2*.*

Given an explicit strict complete intersection, we may get a lower bound for $\operatorname{c^{\infty}_{LM}}(R)$ by optimizing $L(s)$ in Corollary 4.3.1. For example, one easily obtains that $\operatorname{c^{\infty}_{LM}}(k[[x,y]]/(x^{3}+y^{3}))\geq 1.26$.

We next prove that Cohen–Macaulay local rings of sufficiently large multiplicity cannot be lim-stable.

Corollary 4.3.3**.**

For any positive integer $d$ there is a constant $C=C(d)$ with the following property: if $(R,{\mathfrak{m}})$ is a Cohen–Macaulay local ring of dimension $d$ such that $\operatorname{c_{LM}}(R)\geq C$ then $\operatorname{c^{\infty}_{LM}}(R)>1$. In particular, if $\operatorname{e}(R)\geq d!C$, then $\operatorname{c^{\infty}_{LM}}(R)>1$.

Proof.

By Theorem 4.2.1 and Theorem 2.1.4, we may take $C(1)=3$. Thus we may assume that $d\geq 2$. Since $R$ is Cohen–Macaulay, by [RVV01, Corollary 2.6] the Hilbert series of $\operatorname{gr}_{I}(R)$ is bounded above by $(\ell(R/I)+(\operatorname{e}(I)-\ell(R/I)t)(1-t)^{-d}$. By Proposition 4.1.3 applied to $\{I^{n}\}_{n}$ we thus estimate that

[TABLE]

It follows that

[TABLE]

Since $e^{x}(1-e^{-x})^{d}x^{-d}$ grows without a bound, it remains to fix any $x_{0}>0$ such that $e^{x_{0}}(1-e^{-x_{0}})^{d}x_{0}^{-d}>1$ and note that

[TABLE]

The last statement follows from the fact that if $\operatorname{e}(R)\geq d!C$, then $\operatorname{c_{LM}}(R)\geq C$. ∎

Remark 4.3.4*.*

While we know that $C(1)=3$ by Theorem 4.2.1, in higher dimensions we can only estimate that

[TABLE]

The values on the right-hand side grow quite fast, we have $C(2)\leq 8.375$, $C(3)\leq 77.664$, $C(4)\leq 754.48$, etc. As a consequence, we have that a degree $n$ Veronese subring of $k[x_{1},\ldots,x_{d}]$ is not lim-stable if $n\geq 17$ and $d=2$, $n\geq 22$ and $d=3$, and $n\geq 27$ and $d=4$, etc.

Corollary 4.3.5**.**

Let $(R,\mathfrak{m})$ be a two-dimensional Cohen–Macaulay local ring. If $R$ has minimal multiplicity and $\operatorname{e}(R)\geq 17$ or if $R$ does not have minimal multiplicity and $\operatorname{e}(R)\geq 13$, then $\operatorname{c^{\infty}_{LM}}(R)>1$.

Proof.

The first statement follows from Corollary 4.3.3 (see Remark 4.3.4). If $R$ does not have minimal multiplicity, then [RVV01, Theorem 3.3] implies that

[TABLE]

Thus we obtain from Proposition 4.1.3 that

[TABLE]

Setting $x=1$, it is easy to check that the right-hand side above is greater than one when $\operatorname{e}(R)\geq 13$. ∎

Finally, we prove the following result that extends [Sha81, Corollary 7.6 and Theorem 2] from semistable to lim-stable singularities. When $R$ is excellent and normal, this result will be revisited in Section 5.

Proposition 4.3.6**.**

Let $(R,{\mathfrak{m}})$ be a two-dimensional Cohen–Macaulay local ring. Suppose $R$ is lim-stable and $\operatorname{e}(R)=\operatorname{edim}(R)$. Then $\operatorname{gr}_{\mathfrak{m}}(R)$ is Cohen–Macaulay and reduced.

Proof.

If $\operatorname{gr}_{\mathfrak{m}}(R)$ is not Cohen–Macaulay, then by [Sal83, Theorem (i) and (ii)] (or [Sha81, Theorem 3]), the Hilbert polynomial of $\operatorname{gr}_{\mathfrak{m}}(R)$ is at most $n\operatorname{e}(R)-1$. Combining this with [RVV01, Theorem 3.3], we obtain that there exists $N\geq 0$ such that $\ell({\mathfrak{m}}^{n}/{\mathfrak{m}}^{n+1})\leq n\operatorname{e}(R)-1$ for all $n\geq N$ and $\ell({\mathfrak{m}}^{n}/{\mathfrak{m}}^{n+1})\leq n\operatorname{e}(R)$ for $1\leq n<N$. Therefore by Proposition 4.1.3 we have the bound

[TABLE]

Since $\lim_{x\to 0}f(x)=1$ and, by a straightforward computation, $\lim_{x\to 0}f^{\prime}(x)=\frac{1}{\operatorname{e}(R)}>0$, we obtain that $f(x)>1$ for all positive sufficiently small $x$. Therefore $\operatorname{c^{\infty}_{LM}}(R)>1$ which contradicts our assumption that $R$ is lim-stable. We have proved that $\text{gr}_{{\mathfrak{m}}}(R)$ is Cohen–Macaulay.

Next, by [Sha81, Theorem 3 (ii)] or by combining [Sal79b, Theorem 3.3, Lemma 3.4] with [Hun87, Theorem 2.11], the Hilbert function of $R$ is $\ell({\mathfrak{m}}^{n}/{\mathfrak{m}}^{n+1})=n\operatorname{e}(R)$ for all $n\geq 1$. Now by [PWY21, Corollary 3.6], ${\mathfrak{m}}^{n}$ has the strong Rees property for all $n$, i.e., $\ell({\mathfrak{m}}^{n}/{\mathfrak{m}}^{n+1})>\ell(I/{\mathfrak{m}}I)$ for any ideal ${\mathfrak{m}}^{n}\subsetneq I$. Thus for any $n$, we have that

[TABLE]

Without loss of generality, we may assume that $R$ is complete. By Proposition 2.2.4, $R$ is reduced, hence $\lim_{n\to\infty}\ell(R/\overline{{\mathfrak{m}}^{n}})/n^{2}=\operatorname{e}(R)/2$. Now if ${\mathfrak{m}}^{N}\neq\overline{{\mathfrak{m}}^{N}}$ for some $N$, then by applying Proposition 4.1.3 to the graded family $\{\overline{{\mathfrak{m}}^{n}}\}_{n}$ and using a similar argument as above, we obtain that

[TABLE]

In a neighborhood of [math] we may expand the numerator as $g_{1}(x)=\operatorname{e}(R)(1-x+7x^{2}/12)+o(x^{2})$ and the denominator as $g_{2}(x)=\operatorname{e}(R)(1-x+x^{2}/2)+o(x^{2})$. Thus we see that $\lim_{x\to 0}g(x)=1$ and $g^{\prime}(x)>0$ for all $x>0$ sufficiently small. Hence $\operatorname{c^{\infty}_{LM}}(R)>1$, which contradicts $R$ being lim-stable. Thus ${\mathfrak{m}}^{n}$ is integrally closed for all $n$, i.e., ${\mathfrak{m}}$ is normal.

Suppose the Rees period of ${\mathfrak{m}}$ is $\rho>1$. Fix any $0<c<\rho$ and let $I_{n}={\mathfrak{m}}^{\frac{n\rho-c}{\rho}}$. If $I_{n}\supsetneq{\mathfrak{m}}^{n}$, then we have

[TABLE]

by the strong Rees property of ${\mathfrak{m}}^{n}$. Thus, due to Proposition 1.2.4, we may apply Proposition 4.1.3 to the graded family $\{I_{n}\}_{n}$ and repeat the previous argument to arrive at a contradiction. Hence we have ${\mathfrak{m}}^{\frac{n\rho-c}{\rho}}={\mathfrak{m}}^{n}$ for all $0<c<\rho$ and all $n$. But this contradicts the definition of Rees period. Therefore we must have $\rho=1$ and thus ${\mathfrak{m}}^{\alpha}={\mathfrak{m}}^{\lfloor\alpha\rfloor}$ for all $\alpha\in{\mathbb{Q}}_{\geq 0}$. It follows that $\operatorname{gr}_{\mathfrak{m}}(R)$ is reduced by [HS06, Theorem 10.5.6]. ∎

5. Connections with singularities in MMP

In this section we will establish a connection between stability of local rings and singularities arising in the Minimal Model Program (MMP). Our results can be seen as a generalization of Mumford’s heuristic conjecture that semistable surface singularities are a restricted class of rational and strongly elliptic singularities and their non-normal limits (see [Mum77, Page 77]).

5.1. Singularities in the minimal model program

We start by recalling some basic concepts in birational geometry and MMP singularities, following [KM98] and [Kol13].

Let $(R,{\mathfrak{m}})$ be an excellent local ring admitting a dualizing complex. A pair $(X,\Delta)$ (over $\operatorname{Spec}(R)$) is a normal integral scheme $X$ of essentially of finite type over $R$ and an effective $\mathbb{Q}$-divisor $\Delta\geq 0$ on $X$. Note that $X$ admits a dualizing complex $\omega_{X}^{\bullet}=\pi^{!}\omega^{\bullet}_{R}$ where $\pi\colon X\to\operatorname{Spec}(R)$, in particular $X$ admits a canonical sheaf $\omega_{X}=H^{-\dim X}\omega_{X}^{\bullet}$. The canonical divisor $K_{X}$ is a divisor on $X$ so that $\mathcal{O}_{X}(K_{X})\cong\omega_{X}$, this is well-defined up to linear equivalence.

Let $(X,\Delta)$ be a pair. Let $E$ be a divisor over $X$, that is, there exists a projective birational morphism $f\colon Y\to X$ with $Y$ normal such that $E$ is a prime divisor on $Y$. The divisor $E$ is called exceptional over $X$ if the center of $E$ on $X$ has codimension at least two. Suppose $K_{X}+\Delta$ is ${\mathbb{Q}}$-Cartier. We define the discrepancy of $E$, $a(E;X,\Delta)$, as the coefficient of $E$ in $K_{Y}-f^{*}(K_{X}+\Delta)$, where we always assume that the canonical divisors $K_{X}$ and $K_{Y}$ are chosen so that $f_{*}K_{Y}=K_{X}$. When $X=\operatorname{Spec}(R)$, we write $a(E;R,\Delta)$ for $a(E;X,\Delta)$.

Definition 5.1.1**.**

Let $(X,\Delta)$ be a pair such that the coefficients of $\Delta$ are $\leq 1$ and $K_{X}+\Delta$ is ${\mathbb{Q}}$-Cartier. We say that $(X,\Delta)$ is

1. (1)

terminal if $a(E;X,\Delta)>0$ for all exceptional divisors $E$ over $X$; 2. (2)

canonical if $a(E;X,\Delta)\geq 0$ for all exceptional divisors $E$ over $X$; 3. (3)

log terminal if $a(E;X,\Delta)>-1$ for all divisors $E$ over $X$; 4. (4)

log canonical if $a(E;X,\Delta)\geq-1$ for all divisors $E$ over $X$.

When $X=\operatorname{Spec}(R)$, we will also say $(R,\Delta)$ is terminal (resp., canonical, log terminal, log canonical) if $(X,\Delta)$ is so. We will suppress $\Delta$ from the notation if $\Delta=0$.

Remark 5.1.2*.*

It is well-known that if $X$ is normal, ${\mathbb{Q}}$-Gorenstein and $X$ admits a log resolution $Y\to X$, then $X$ is terminal (resp., canonical) if and only if $a(E;X)>0$ (resp. $a(E;X)\geq 0$) for all exceptional prime divisors $E$ on $Y$. Similarly, if $(X,\Delta)$ is a pair such that $K_{X}+\Delta$ is ${\mathbb{Q}}$-Cartier and $(X,\Delta)$ admits a log resolution $Y\to(X,\Delta)$, then $(X,\Delta)$ is log terminal (resp., log canonical) if and only if $a(E;X,\Delta)>-1$ (resp., $a(E;X,\Delta)\geq-1$) for all prime divisors $E$ on $Y$.

Log canonical modifications will play a critical role in our proofs.

Definition 5.1.3**.**

Let $(X,\Delta)$ be a pair such that all coefficients of $\Delta$ are $\leq 1$. We say that a projective birational morphism $f\colon Y\to(X,\Delta)$ is a log canonical modification of $(X,\Delta)$ if $Y$ is normal and we have

• •

$(Y,f_{*}^{-1}\Delta+{E}_{\textup{red}})$ is log canonical,

• •

$K_{Y}+f_{*}^{-1}\Delta+{E}_{\textup{red}}$ is $f$-ample,

where ${E}_{\textup{red}}$ denotes the reduced exceptional divisor, i.e., the sum of all $f$-exceptional prime divisors with coefficient $1$.

It is well-known that the existence of log canonical modifications follows from the full log minimal model program (including the abundance conjecture), and when a log canonical modification of $(X,\Delta)$ exists, then it is unique, see [OX12, Kol13] for further discussions. We will use the following celebrated result of Odaka–Xu.

Theorem 5.1.4** ([OX12, Theorem 1.1] and [Kol13, Theorem 1.32]).**

Let $(X,\Delta)$ be a pair such that $X$ is essentially of finite type over a field of characteristic zero and all coefficients of $\Delta$ are $\leq 1$. If $K_{X}+\Delta$ is ${\mathbb{Q}}$-Cartier, then $(X,\Delta)$ admits a log canonical modification.

Our full results in characteristic zero require some generalizations of log canonical singularities beyond the ${\mathbb{Q}}$-Gorenstein setting. Though we will only require the numerically ${\mathbb{Q}}$-Gorenstein case (see below) in our applications, we discuss this theory in the more general setting of [dFH09], see also [BdFFU15, Has20]. Let $f\colon Y\to X$ be a projective birational morphism of normal integral schemes. The two types of $m$th limiting relative canonical divisors are defined as

[TABLE]

where $f^{\natural}(D)=\text{div}(\mathcal{O}_{X}(-D)\cdot\mathcal{O}_{Y})$ for a divisor $D$ on $X$. It is not difficult to see that

[TABLE]

for all $m,n$, with equalities throughout if $mK_{X}$ is Cartier. After taking limsup (resp., liminf) of $K^{-}_{m,Y/X}$ (resp., $K^{+}_{m,Y/X}$), we obtain $\mathbb{R}$-divisors $K^{-}_{Y/X}$ and $K^{+}_{Y/X}$.

For any prime divisor $E$ on $Y$, we write $a_{m}(E;X)$ for the coefficient of $E$ in $K^{-}_{m,Y/X}$ and $a(E;X)$ for the coefficient of $E$ in $K^{-}_{Y/X}$. Note that $a(E;X)=\limsup a_{m}(E;X)$ for every divisor $E$ over $X$.

Definition 5.1.5**.**

We say that

• •

$X$ is of log canonical type if there exists $m$ such that $a_{m}(E;X)\geq-1$ for all divisors $E$ over $X$, see [dFH09, Definition 7.1].

• •

$X$ is pseudo-log canonical if $a(E;X)\geq-1$ for all divisors $E$ over $X$, see [Has20, Definition 4.1, Proposition 4.7 and Theorem 4.8].

It is clear that log canonical type implies pseudo-log canonical and that they both agree with log canonical when $K_{X}$ is ${\mathbb{Q}}$-Cartier. On the other hand, there are examples of pseudo-log canonical singularities that are not of log canonical type [Has20, Example 4.11].

When $X$ is essentially of finite type over a field of characteristic zero, for every divisor $E$ over $X$ and every $m\geq 2$, there exists an effective ${\mathbb{Q}}$-divisor $\Delta\geq 0$ on $X$ such that $m\Delta$ is integral, $K_{X}+\Delta$ is ${\mathbb{Q}}$-Cartier, and $a_{m}(E;X)=a(E;X,\Delta)$, as shown in [dFH09, Theorem 5.4]. It follows that, in this case, we have

[TABLE]

Hence, $X$ is of log canonical type if and only if there exists a pair $(X,\Delta)$ such that $K_{X}+\Delta$ is ${\mathbb{Q}}$-Cartier and $(X,\Delta)$ is log canonical.

A ${\mathbb{Q}}$-divisor $D$ on $X$ is called numerically ${\mathbb{Q}}$-Cartier if, locally on $X$, there is a regular alteration $\pi\colon Z\to X$ and a $\pi$-numerically trivial ${\mathbb{Q}}$-divisor $G$ on $Z$ such that $\pi_{*}G=D$. We call $X$ numerically ${\mathbb{Q}}$-Gorenstein if $K_{X}$ is numerically ${\mathbb{Q}}$-Cartier. Note that, if $X$ is numerically ${\mathbb{Q}}$-Gorenstein and $f\colon Y\to X$ is any projective birational morphism with $Y$ normal, then we have a well-defined ${\mathbb{Q}}$-divisor $f_{\operatorname{num}}^{*}K_{X}$ such that $K_{Y}-f_{\operatorname{num}}^{*}K_{X}$ is $f$-exceptional and $(f_{\operatorname{num}}^{*}K_{X})\cdot F^{d-1}=0$ for all Cartier divisors $F$ on $Y$ that is $f$-exceptional. Explicitly, one may set $f_{\operatorname{num}}^{*}K_{X}\coloneqq g_{*}K$, where $g\colon Z\to Y$ is a sufficiently large regular alteration999Here we assume the existence of regular alterations, which holds when $X$ is essentially of finite type over a field or over a complete DVR, or when $\dim X\leq 3$. and $K$ is the numerically trivial ${\mathbb{Q}}$-divisor on $Z$ with $(f\circ g)_{*}K=K_{X}$. The property $(f_{\operatorname{num}}^{*}K_{X})\cdot F^{d-1}=0$ follows by the projection formula.

Since we can take $G=\pi^{*}D$, any ${\mathbb{Q}}$-Cartier divisor is numerically ${\mathbb{Q}}$-Cartier, in particular, ${\mathbb{Q}}$-Gorenstein singularities are numerically ${\mathbb{Q}}$-Gorenstein. More generally, we have

• •

If $\dim X=2$, then any ${\mathbb{Q}}$-divisor is numerically ${\mathbb{Q}}$-Cartier (in particular, $X$ is numerically ${\mathbb{Q}}$-Gorenstein), see [BdFFU15, Example 5.7] and [Tak21, Example 3.4].

• •

If $X$ is essentially of finite type over a field of characteristic zero, then $X$ is numerically ${\mathbb{Q}}$-Gorenstein if and only if $K^{-}_{Y/X}=K^{+}_{Y/X}$ for every resolution of singularities $Y\to X$, see [BdFFU15, Proposition 5.9]. In particular, when $X$ is numerically ${\mathbb{Q}}$-Gorenstein, $K^{-}_{Y/X}=K^{+}_{Y/X}$ is a ${\mathbb{Q}}$-divisor.

We record the following result on log canonicity for numerically ${\mathbb{Q}}$-Gorenstein varieties.

Theorem 5.1.6** ([Zha17, Proposition 3.5], see also [Has20, Corollary 5.2 and Lemma 5.4]).**

Let $X$ be a normal integral scheme essentially of finite type over a field of characteristic zero. Suppose $X$ is numerically ${\mathbb{Q}}$-Gorenstein and pseudo-log canonical. Then $X$ is ${\mathbb{Q}}$-Gorenstein and log canonical.

We end this subsection by proving the following slight generalization of [Has20, Theorem 1.1] for numerically ${\mathbb{Q}}$-Gorenstein rings that we will need later. We refer to [KM98], [BCHM10], and [Kol13] for basic language of MMP.

Theorem 5.1.7**.**

Let $(R,{\mathfrak{m}})$ be a numerically ${\mathbb{Q}}$-Gorenstein normal local domain essentially of finite type over a field of characteristic zero. Then there exists a projective birational morphism $h$: $W\to X\coloneqq\operatorname{Spec}(R)$ such that

1. (1)

$a(E_{i},X)<-1$* for any $h$-exceptional prime divisor $E_{i}$;* 2. (2)

each $h$-exceptional prime divisor $E_{i}$ on $W$ is ${\mathbb{Q}}$-Cartier; 3. (3)

$K_{W}+{E}_{\textup{red}}$* is ${\mathbb{Q}}$-Cartier and the pair $(W,{E}_{\textup{red}})$ is log canonical;* 4. (4)

there exists an $h$-ample exceptional Cartier divisor on $W$.

Proof.

We follow the approach from [Has20, Proof of Theorem 4.14]. Let $f\colon Y\to X$ be a log resolution. We divide the exceptional prime divisors based on their discrepancies with respect to $X$. Specifically, let:

1. (1)

$F_{j}$ be the exceptional divisors on $Y$ such that $a(F_{j};X)\geq-1$, 2. (2)

$G_{i}$ be the exceptional divisors on $Y$ such that $a(G_{i};X)<-1$.

Set $F=\sum F_{j}$ and consider the pair $(Y,F+\sum t_{i}G_{i})$ where each $t_{i}\in{\mathbb{Q}}$ and $0\leq t_{i}<1$.

We now run a $(K_{Y}+F+\sum t_{i}G_{i})$-MMP over $X$ with scaling of an ample divisor. After finitely many steps, we obtain a ${\mathbb{Q}}$-factorial dlt model $f^{\prime}\colon(Y^{\prime},F^{\prime}+\sum t_{i}G_{i}^{\prime})\to X$ where $K_{Y^{\prime}}+F^{\prime}+\sum t_{i}G_{i}^{\prime}$ is the limit of movable divisors over $X$. The divisors $F^{\prime},G_{i}^{\prime}$ are the birational transformations of $F$ and $G_{i}$ on $Y^{\prime}$. Moreover, $a(E^{\prime};X)\leq-1$ for any $f^{\prime}$-exceptional prime divisor $E^{\prime}$ on $Y^{\prime}$ (see [Has20, Step 2 in Proof of Theorem 4.14] or [Zha17, Theorem 3.1]).

We next claim that $(Y^{\prime},F^{\prime}+\sum t_{i}G_{i}^{\prime})$ admits a good minimal model over $X$. To establish this, it is enough to check that this pair satisfies the conditions of [Has20, Theorem 3.5]. Since $X$ is numerically ${\mathbb{Q}}$-Gorenstein, we know by construction that $K_{Y^{\prime}}-f^{\prime*}_{\operatorname{num}}K_{X}=-F^{\prime}-\sum a_{i}G_{i}^{\prime}$ where each $a_{i}>1$. Thus

[TABLE]

is $f^{\prime}$-numerically trivial. In particular, the divisor $-(K_{Y^{\prime}}+F^{\prime}+\sum t_{i}G_{i}^{\prime})$ is $f^{\prime}$-numerically equivalent to the effective divisor $\sum(a_{i}-t_{i})G_{i}^{\prime}$ since $a_{i}>1$ and $t_{i}<1$. Since no stratum of $\sum G_{i}^{\prime}$ could be an log canonical center of the pair $(Y^{\prime},F^{\prime}+\sum t_{i}G_{i}^{\prime})$, it follows that the pullback of $-(K_{Y^{\prime}}+F^{\prime}+\sum t_{i}G_{i}^{\prime})$ to the normalization of any log canonical center of $(Y^{\prime},F^{\prime}+\sum t_{i}G_{i}^{\prime})$ is pseudo-effective. Thus $(Y^{\prime},F^{\prime}+\sum t_{i}G_{i}^{\prime})$ admits a good minimal model $(Y^{\prime\prime},F^{\prime\prime}+\sum t_{i}G_{i}^{\prime\prime})$ over $X$ by [Has20, Theorem 3.5].

At this point, we let $Y^{\prime\prime}\to W_{\underline{t}}$ be the contraction over $X$ induced by the semi-ample divisor $K_{Y^{\prime\prime}}+F^{\prime\prime}+\sum t_{i}G_{i}^{\prime\prime}$. It is not hard to show that this contracts all exceptional prime divisors $E^{\prime\prime}$ with $a(E^{\prime\prime};X)=-1$ (see [Has20, Step 4 in Proof of Theorem 4.14]). Therefore, $(G_{i})_{W_{\underline{t}}}$, the birational transformation of $G_{i}^{\prime\prime}$, are all the exceptional prime divisors on $W_{\underline{t}}$. By [MZ23, Theorem 5.1], there are only finitely many ample models $W_{\underline{t}}$ as we vary $t_{i}$’s inside a rational polytope. Since each $K_{W_{\underline{t}}}+\sum t_{i}(G_{i})_{W_{\underline{t}}}$ is ${\mathbb{Q}}$-Cartier, when we vary the $t_{i}$’s in a fixed $P^{\circ}$ (the interior so that the ample model $W_{\underline{t}}$ is constant for all $\underline{t}\in P^{\circ}$), we see that each $(G_{i})_{W_{\underline{t}}}$ is ${\mathbb{Q}}$-Cartier. We also note that the pair $(W_{\underline{t}},\sum t_{i}(G_{i})_{W_{\underline{t}}})$ is log canonical. Thus, for $1-\epsilon<t_{i}$ for all $i$, the pair $(W_{\underline{t}},(1-\epsilon)\sum(G_{i})_{W_{\underline{t}}})$ is also log canonical, note that $K_{W_{\underline{t}}}+(1-\epsilon)\sum(G_{i})_{W_{\underline{t}}}$ is ${\mathbb{Q}}$-Cartier (because $K_{W_{\underline{t}}}+\sum t_{i}(G_{i})_{W_{\underline{t}}}$ and each $(G_{i})_{W_{\underline{t}}}$ is ${\mathbb{Q}}$-Cartier). Letting each $t_{i}\to 1$, we have that $\text{lct}(W_{\underline{t}},\sum(G_{i})_{W_{\underline{t}}})\geq 1-\epsilon$, and approaching to $1$ as $\min\{t_{i}\}\to 1$. By the ACC for log canonical thresholds [HMX14, Theorem 1.1], we can find a $\underline{t}$ so that $\text{lct}(W_{\underline{t}},\sum(G_{i})_{W_{\underline{t}}})=1$, i.e., the pair $(W_{\underline{t}},\sum(G_{i})_{W_{\underline{t}}})$ is log canonical.

Finally, set $W\coloneqq W_{\underline{t}}$ chosen as above with $h\colon W\to X$, and let $E_{i}\coloneqq(G_{i})_{W_{\underline{t}}}$ (these are all the $h$-exceptional prime divisors on $W$). We have already seen that $a(E_{i};X)<-1$, that each $E_{i}$ is ${\mathbb{Q}}$-Cartier, and that the pair $(W,{E}_{\textup{red}}=\sum E_{i})$ is log canonical. Moreover, we know that $K_{W}+\sum t_{i}E_{i}$ is $h$-ample. It follows that $K_{W}-h^{*}_{\operatorname{num}}K_{X}+\sum t_{i}E_{i}$ is $h$-exceptional, in particular ${\mathbb{Q}}$-Cartier and thus $h$-ample. Hence there exists an $h$-ample exceptional Cartier divisor on $W$. This completes the proof of the theorem. ∎

Remark 5.1.8*.*

In [Has20, Theorem 1.1], it was shown that for any pair $(X,\Delta)$, where $X$ is essentially of finite type over a field of characteristic zero and the coefficients of $\Delta$ are $\leq 1$, there exists a projective birational morphism $h\colon W\to X$ such that: $a(E_{i};X,\Delta)<-1$101010Here, a(E_{i};X,\Delta)\coloneqq\sup\{a(E_{i};X,\Delta+\Delta^{\prime})\mid\Delta^{\prime}\geq 0\text{ and }K_{X}+\Delta+\Delta^{\prime}\text{ is {\mathbb{Q}}-Cartier}\}. for any $h$-exceptional prime divisor $E_{i}$, ${E}_{\textup{red}}=\sum E_{i}$ is ${\mathbb{Q}}$-Cartier, and $(W,h_{*}^{-1}\Delta+{E}_{\textup{red}})$ is log canonical. The same result holds for $\mathbb{R}$-coefficients.

Theorem 5.1.7 slightly improves this result by ensuring that each $E_{i}$ is ${\mathbb{Q}}$-Cartier, assuming $X$ is numerically ${\mathbb{Q}}$-Gorenstein. In fact, the proof above can be generalized (adapting [Has20, Step 3 and Step 4 in Proof of Theorem 4.14]) to obtain a full generalization of [Has20, Theorem 1.1], also for $\mathbb{R}$-coefficients. Specifically, for any pair $(X,\Delta)$, where $X$ is essentially of finite type over a field of characteristic zero and the coefficients of $\Delta$ are $\leq 1$, there exists a projective birational morphism $h\colon W\to X$ such that: $a(E_{i};X,\Delta)<-1$ for any $h$-exceptional prime divisor $E_{i}$, each $E_{i}$ is ${\mathbb{Q}}$-Cartier, and $(W,h_{*}^{-1}\Delta+{E}_{\textup{red}})$ is log canonical. We leave the details to the interested reader.

Remark 5.1.9*.*

The morphism $W\to X$ constructed in Theorem 5.1.7 (and also in [Has20, Theorem 1.1]) is not necessarily a log canonical modification of $X$. In [Zha17, Theorem 3.6], the existence of log canonical modification $Y\to X$ was established under the assumption that $X$ is a normal integral numerically ${\mathbb{Q}}$-Gorenstein scheme essentially of finite type over a field of characteristic zero (this generalizes Theorem 5.1.4). However, it is unclear whether $Y$ admits an ample exceptional Cartier divisor in this setup.

5.2. Asymptotic Riemann-Roch formula and its consequence

In this subsection, we recall a well-known version of the asymptotic Riemann–Roch theorem. We provide a proof, as we have been unable to find a reference that covers the generality we require. We will follow notation of [Kol96, Chapter VI.2], and more generally [Ful98].

Theorem 5.2.1**.**

Let $X$ be a projective scheme over an Artinian local ring with $\dim X=d$ and let $L$ be a line bundle on $X$.

1. (1)

If $X$ is equidimensional and $(\textnormal{S}_{1})$, then we have

[TABLE]

where $[\omega_{X}]-[\mathcal{O}_{X}]$ is the class in $K_{d-1}(X)$ following **[Kol96, VI, Definition 2.1]**. 2. (2)

If $X$ is $(\textnormal{S}_{2})$ and $(\textnormal{G}_{1})$ and $D$ is a Weil divisor that is principal in codimension one, then we have

[TABLE]

Proof.

We will prove both (1) and (2) simultaneously. Let $D$ be a Weil divisor on $X$ that is principal in codimension one (in case (1), we simply take $D=0$). By [Kol96, VI, Theorem 2.13], for any coherent sheaf $G$ on $X$ (and thus for any $G\in D^{b}_{coh}(X)$), $\chi(X,G\otimes L^{\otimes n})$ is a polynomial in $n$ of degree at most the dimension of the support of $G$. Thus we can write

[TABLE]

It is well-known (see [Kol96, VI, Corollary 2.14]) that $A=L^{d}/d!$. We will compute $B$.

Let $\omega_{X}^{\bullet}$ be the normalized dualizing complex of $X$. Using Grothendieck duality, we have

[TABLE]

We now truncate $\omega_{X}^{\bullet}$. Consider now the exact triangle

[TABLE]

Since $X$ is $(\textnormal{S}_{1})$, we know that $\tau^{>-d}\omega_{X}^{\bullet}$ is supported in codimension at least two. As a result,

[TABLE]

Therefore, from (5.2.1.1) and the duality we compute that

[TABLE]

We now analyze the cases. In case (1), we have $\mathcal{O}_{X}(D)=\mathcal{O}_{X}$, and thus by (5.2.1.1) and (5.2.1.2) we compute that

[TABLE]

It thus follows from [Kol96, VI, Theorem 2.13] that

[TABLE]

Finally, in case (2), since $D$ is principal in codimension one, $R\underline{\operatorname{Hom}}(\mathcal{O}_{X}(D),\omega_{X}[d]\otimes L^{\otimes n})$ agrees with $\omega_{X}[d]\otimes\mathcal{O}_{X}(-D)\otimes L^{\otimes n}$ in codimension one. It follows from (5.2.1.2) that

[TABLE]

After combining this formuma with (5.2.1.1), we immediately express

[TABLE]

On the other hand, we also have the formula

[TABLE]

because for any effective divisor $E$ one has that

[TABLE]

By comparing the two computations of $\chi(X,\mathcal{O}_{X}(D)\otimes L^{\otimes n})-\chi(X,\mathcal{O}_{X}(K_{X}-D)\otimes L^{\otimes n})$ we immediately get the desired formula

[TABLE]

This completes the proof. ∎

We will need the following consequence of Theorem 5.2.1 on the asymptotic behavior of colengths of certain graded family of ideals. Note that we do not assume that $R$ is normal.

Proposition 5.2.2**.**

Let $(R,\mathfrak{m})$ be an excellent local domain of dimension $d\geq 2$ admitting a dualizing complex, and let $f\colon Y\to X\coloneqq\operatorname{Spec}(R)$ be a projective birational map such that $Y$ is $(\textnormal{S}_{2})$ and $(\textnormal{G}_{1})$, and $f_{*}\mathcal{O}_{Y}=\mathcal{O}_{X}$. Suppose $f$ is an isomorphism on the punctured spectrum, and there exists an effective Cartier divisor $E$ such that $-E$ is $f$-ample and $f$-exceptional. Let $F$ be an $f$-exceptional Weil divisor on $Y$ and set $I_{n}\coloneqq\Gamma(Y,\mathcal{O}_{Y}(-nE-F))\subseteq R$. Then we have

[TABLE]

Proof.

First note that, since $-E$ is $f$-ample and $f$-exceptional, $Y$ is the blowup of $X$ along some ${\mathfrak{m}}$-primary ideal $I$ (for example, one can take $I=\Gamma(Y,\mathcal{O}_{Y}(-mE))$ for $m$ sufficiently divisible). Taking cohomology of the short exact sequence

[TABLE]

we obtain that

[TABLE]

Since $-E$ is $f$-ample, we have $H^{i}(Y,\mathcal{O}_{Y}(-nE-F))\cong R^{i}f_{*}\mathcal{O}_{Y}(-nE-F)=0$ for all $i\geq 1$ and $n\gg 0$. Thus, we have asymptotic comparison $\ell(R/I_{n})=\chi(\mathcal{O}_{nE+F})+O(n^{d-2}).$

We now proceed to calculate $\chi(\mathcal{O}_{nE+F})$. From the short exact sequence

[TABLE]

we get that $\chi(\mathcal{O}_{nE+F})=\chi(\mathcal{O}_{nE})+\chi(F,\mathcal{O}_{F}(-nE)).$ By [Kol96, VI, Corollary 2.14] we compute $\chi(F,\mathcal{O}_{F}(-nE))=\frac{(-E|_{F})^{d-1}}{(d-1)!}n^{d-1}+O(n^{d-2})$. Therefore it suffices to show that

[TABLE]

Now note that for every $k\geq 1$ there is a short exact sequence

[TABLE]

Taking Euler characteristic gives $\chi(\mathcal{O}_{(k+1)E})=\chi(\mathcal{O}_{kE})+\chi(E,\mathcal{O}_{E}(-kE))$. By induction on $k$ formula (5.2.2.1) will follow once we prove the formula

[TABLE]

On the other hand, since $Y$ satisfies $(\textnormal{S}_{2})$, $E$ is equidimensional and $(\textnormal{S}_{1})$, allowing us to apply the asymptotic Riemann–Roch formula of Theorem 5.2.1. This yields

[TABLE]

We now analyze the term $([\omega_{E}]-[\mathcal{O}_{E}])\cdot(-E|_{E})^{d-2}$. To do so, consider the exact sequence

[TABLE]

Since $Y$ satisfies Serre’s $(\textnormal{S}_{2})$ condition, $h^{-(d-1)}(\omega_{Y}^{\bullet})$ has codimension at least three in $Y$. Therefore, its restriction to $E$ has codimension at least two in $E$. It follows that

[TABLE]

Therefore, essentially by the definition of intersection theory, we have

[TABLE]

Substituting back, we obtain (5.2.2.1) as desired. The proof is now complete. ∎

Remark 5.2.3*.*

Note that the assumptions of Proposition 5.2.2 are satisfied if $R$ is normal and $Y$ is the normalized blowup of $R$ along an ${\mathfrak{m}}$-primary ideal (this will be our main case of interest in the next subsection). We also note that, when $R$ is essentially of finite type over a field and $F$ is principal in codimension one, then one can also prove Proposition 5.2.2 by compactifying $X$ and applying Theorem 5.2.1 (2). However, we do not know how to use this approach in a more general situation (e.g., the mixed characteristic case).

The following immediate consequence of Proposition 5.2.2 relates some MMP singularities with Hilbert coefficients of ${\mathfrak{m}}$-primary ideals.

Corollary 5.2.4**.**

Let $(R,\mathfrak{m})$ be an excellent ${\mathbb{Q}}$-Gorenstein normal local domain of dimension $d\geq 2$ that admits a dualizing complex. Let $I$ be an arbitrary $\mathfrak{m}$-primary ideal and let $\overline{\operatorname{e}}_{1}(I)$ be the first normal Hilbert coefficient. The following hold:

1. (1)

If $R$ is canonical, then $(d-1)\operatorname{e}(I)\geq 2\overline{\operatorname{e}}_{1}(I),$ 2. (2)

if $R$ is terminal, then $(d-1)\operatorname{e}(I)>2\overline{\operatorname{e}}_{1}(I)$.

Proof.

Let $f\colon Y\to X\coloneqq\operatorname{Spec}(R)$ be the normalized blowup of $I$. Then $I\mathcal{O}_{Y}=\mathcal{O}_{Y}(-E)$ is an $f$-ample and $f$-exceptional Cartier divisor and we have $\Gamma(Y,\mathcal{O}_{Y}(-nE))=\overline{I^{n}}$. By the definition of normal Hilbert coefficient, we have

[TABLE]

Therefore, Proposition 5.2.2 provides the formula:

[TABLE]

Now when $R$ is canonical (resp., terminal), we have $a(F;X)\geq 0$ (resp., $a(F;X)>0$) for all exceptional prime divisors $F$ on $Y$, so $K_{Y}-f^{*}K_{X}\geq 0$ (resp., $K_{Y}-f^{*}K_{X}>0$). Since $-E$ is $f$-ample, it follows immediately that $(K_{Y}-f^{*}K_{X})\cdot(-E)^{d-1}\geq 0$ (resp., $>0$). ∎

The second application of Proposition 5.2.2 improves Proposition 1.2.4 by providing a formula for the second term of the Hilbert–Samuel quasi-polynomial for rational powers of ideals.

Corollary 5.2.5**.**

Let $(R,{\mathfrak{m}})$ be an excellent local domain of dimension $d\geq 2$ admitting a dualizing complex. Let $f$: $Y=\operatorname{Bl}_{I}(R)\to X\coloneqq\operatorname{Spec}(R)$ be the blowup of an ${\mathfrak{m}}$-primary ideal $I\subseteq R$. Suppose $X$ and $Y$ are $(\textnormal{S}_{2})$ and $(\textnormal{G}_{1})$, and that the generic point of each exceptional prime divisor on $Y$ is regular. Let $I\mathcal{O}_{Y}=\mathcal{O}_{Y}(-E)$ and let $\rho$ be the Rees period of $I$. Then we have

[TABLE]

Proof.

First of all, we have $0\to\mathcal{O}_{X}\to f_{*}\mathcal{O}_{Y}\to C\to 0$. Since $I$ is ${\mathfrak{m}}$-primary, $C$ has finite length. But as $R$ and $S$ are $(\textnormal{S}_{2})$, $C=0$ and thus $f_{*}\mathcal{O}_{Y}=\mathcal{O}_{X}$. Let $E_{1},\ldots,E_{n}$ be all the exceptional prime divisors of $f$ and write $E=\sum^{n}_{i=1}c_{i}E_{i}$. Note that $\rho=\operatorname{lcm}\{c_{1},\ldots,c_{n}\}$, and thus if we let $F=\lceil\frac{h}{\rho}E\rceil$, then by Proposition 1.2.5

[TABLE]

Thus we may apply Proposition 5.2.2 to obtain that, for all $k\gg 0$, we have

[TABLE]

which is easily seen to be equivalent to what we want to show. ∎

5.3. Lim-stable and log canonical singularities

We now prove our main result: lim-stable (and in particular, semistable) singularities are log canonical in the ${\mathbb{Q}}$-Gorenstein case, see Corollary 5.3.5. We also present generalizations beyond the ${\mathbb{Q}}$-Gorenstein setting, which are discussed in Theorem 5.3.6. We begin with the following lemma which is well-known to experts (for example, see [Oda13, Proof of Theorem 1.2, Step 2]).

Lemma 5.3.1**.**

Let $X$ be a numerically ${\mathbb{Q}}$-Gorenstein normal integral scheme. Let $\pi\colon Y=\operatorname{Bl}_{\mathcal{I}}X\to X$ be the blowup along an ideal sheaf $\mathcal{I}\subseteq\mathcal{O}_{X}$ locally of height at least two, and suppose $Y$ is normal with exceptional prime divisors $\{E_{i}\}_{i=1}^{n}$. Set $\mathcal{I}\mathcal{O}_{Y}=\mathcal{O}_{Y}(-\sum_{i=1}^{n}c_{i}E_{i})$. For any positive integer $m$ such that $b_{i}=m/c_{i}$ is an integer for all $i$, let $\pi^{\prime}\colon Y^{\prime}\to X^{\prime}\coloneqq X\times\mathbb{A}^{1}$ be the normalized blowup of $X^{\prime}$ along the ideal sheaf $\mathcal{I}+(t^{m})\subseteq\mathcal{O}_{X}[t]$. Then there is a bijection between exceptional prime divisors on $Y^{\prime}$ and on $Y$ such that for corresponding divisors $E^{\prime}_{i}$ on $Y^{\prime}$ and $E_{i}$ on $Y$ we have $a(E_{i}^{\prime};X^{\prime})=b_{i}(a(E_{i};X)+1)$.

In particular, if $a(E_{i};X)<-1$ for all $i$, then $a(E_{i}^{\prime};X^{\prime})<0$ for all $m$ sufficiently divisible.

Proof.

The computation of the discrepancy $a(E_{i};X)$ and $a(E_{i}^{\prime};X^{\prime})$ is local along each $E_{i}$. Thus after localizing at the generic point of $E_{i}$, we may assume that $Y=\operatorname{Spec}(V)$ is a DVR with uniformizer $s$, $E_{i}=\text{div}(s)$, and $\mathcal{I}\mathcal{O}_{Y}=(s^{c_{i}})$. The map $\pi^{\prime}$ then factors through $g\colon Y^{\prime}\to Y\times\mathbb{A}^{1}\cong\operatorname{Spec}(V[t])$, which is the normalized blowup along the ideal $(s^{c_{i}},t^{m})$. Since $b_{i}=m/c_{i}$ is an integer, it is easy to see that this normalized blowup is isomorphic to the blowup along the ideal $(s,t^{b_{i}})$. This blowup only creates one exceptional prime divisor (corresponding to $E_{i}^{\prime}$), and by a direct computation, we have $K_{Y^{\prime}/Y\times\mathbb{A}^{1}}=b_{i}E_{i}^{\prime}$ and $g^{*}E_{i}=b_{i}E_{i}^{\prime}+\widetilde{E}_{i}$ where $\widetilde{E}_{i}$ is the strict transform of $E_{i}$. It follows that the coefficient of $E_{i}^{\prime}$ in

[TABLE]

is equal to $b_{i}+b_{i}a(E_{i};X)$, that is, $a(E_{i}^{\prime};X^{\prime})=b_{i}(a(E_{i};X)+1)$ as claimed. ∎

We next record the following consequence of Proposition 4.1.3.

Theorem 5.3.2**.**

Let $(R,\mathfrak{m})$ be a local ring of dimension $d\geq 1$. Suppose $\{I_{n}\}_{n}$ is a Noetherian graded family of ${\mathfrak{m}}$-primary ideals such that $I_{1}={\mathfrak{m}}$. Set $I_{0}=R$. Then the Hilbert series of $\operatorname{gr}_{I_{\bullet}}(R)$ can be written as

[TABLE]

where $f(t)$ is a rational function with $f(1)\neq 0$ and all poles of $f(t)$ have order at most $d$. Furthermore, if $f^{\prime}(1)>\frac{d}{2}f(1)$, then we have $\operatorname{c^{\infty}_{LM}}(R)>1$.

Proof.

Our assumptions imply that $\operatorname{gr}_{I_{\bullet}}(R)$ is a finitely generated $\mathbb{N}$-graded ring over $k$. Hence there exist $z_{1},\dots,z_{d}$ with $\deg(z_{i})=a_{i}$ such that $\operatorname{gr}_{I_{\bullet}}(R)$ is finite over $k[z_{1},\dots,z_{d}]$. It follows that we can write

[TABLE]

where $p(t)\in\mathbb{Z}[t]$ and the first claim follows.

We now prove the second part. From the expression of $h(t)$, it is easy to check that

[TABLE]

Thus, applying Proposition 4.1.3 to $\{I_{n}\}_{n}$ yields, for any $x\in\mathbb{R}_{>0}$, the bound

[TABLE]

Now, if we let $x\to 0$ and use that $f(e^{-x})=f(1)-f^{\prime}(1)x+O(x^{2})$, then (5.3.2.1) becomes

[TABLE]

and the conclusion follows. ∎

As a corollary of Theorem 5.3.2, we generalize [Sha81, Theorem 3] and partially recover Proposition 4.3.6.

Corollary 5.3.3**.**

Let $(R,{\mathfrak{m}})$ be a Cohen–Macaulay ring of dimension two. If $R$ is lim-stable, then either $R$ has minimal multiplicity or $R$ has almost minimal multiplicity and $\operatorname{gr}_{\mathfrak{m}}(R)$ is Cohen-Macaulay.

Proof.

By Theorem 5.3.2 applied to the graded family $\{{\mathfrak{m}}^{n}\}_{n}$, $R$ is not lim-stable whenever $\operatorname{e}_{1}(R)=f^{\prime}(1)>f(1)=\operatorname{e}(R)$. By Northcott’s inequality, $\operatorname{e}_{1}(R)\geq\operatorname{e}(R)-1$ and it was shown by [Hun87, Theorem 2.1] that the equality holds if and only if $R$ has minimal multiplicity. Essentially by [Sal79b, Theorem 3.3] (the statement gives the Hilbert function, but the base case concerns only $\operatorname{e}_{1}$, see also [Ros11, Exercise 4.4]) $\operatorname{e}_{1}(R)=\operatorname{e}(R)$ if and only if $R$ has almost minimal multiplicity and the associated graded ring $\text{gr}_{\mathfrak{m}}(R)$ is Cohen-Macaulay. ∎

We are now ready to state and prove the main technical result.

Theorem 5.3.4**.**

Let $(R,{\mathfrak{m}})$ be an excellent numerically ${\mathbb{Q}}$-Gorenstein normal local domain admitting a dualizing complex. Suppose there exists a projective birational map $\pi$: $Y\to X\coloneqq\operatorname{Spec}(R)$ such that

• •

there exists a $\pi$-ample $\pi$-exceptional Cartier divisor, and

• •

$a(F;X)<-1$* for all exceptional prime divisors $F$ on $Y$.*

If $R$ is lim-stable, then $R$ is ${\mathbb{Q}}$-Gorenstein and log canonical.

Proof.

Let $-E$ be a $\pi$-ample and $\pi$-exceptional Cartier divisor on $Y$. By the negativity lemma [KM98, Lemma 3.39] we know that $E$ is effective. If a curve $C$ is contracted under $\pi$, then $C\cdot(-E)>0$. This implies that $C$ lies in the support of $E$, and consequently, the exceptional set is of pure codimension one. As a result $\pi$ is an isomorphism over the locus where $X$ is ${\mathbb{Q}}$-Gorenstein and log canonical. This is because none of the exceptional prime divisors can have centers on this locus, given the assumption on the discrepancy.

By Theorem 5.1.6, it suffices to show that $R$ is pseudo-log canonical. If this is not the case, we may choose a prime ideal $Q\in\operatorname{Spec}(R)$ of minimal height such that $R_{Q}$ is not pseudo-log canonical. By Theorem 2.3.2, we can replace $R$ by $R_{Q}$ and apply Theorem 5.1.6 again, which allows us to assume that $R$ is ${\mathbb{Q}}$-Gorenstein and log canonical on the punctured spectrum. After this replacement, both assumptions are preserved and $\pi$ is an isomorphism over the punctured spectrum of $R$. It follows that $\pi$ is the blowup of some ${\mathfrak{m}}$-primary ideal $I\subseteq R$.

We now consider the graded family of ideals $I_{n}\coloneqq\{I^{\frac{n}{\rho}}\}_{n}$, where $\rho$ is the Rees period of $I$. This is a Noetherian graded family of ${\mathfrak{m}}$-primary ideals with $I_{1}={\mathfrak{m}}$ by the definition of rational powers. By Theorem 5.3.2, we thus have

[TABLE]

where $f(t)$ is a rational function with $f(1)\neq 0$ and all poles of $f(t)$ have order at most $d\coloneqq\dim(R)$. Thus $\ell(R/I_{n})=({f(1)}/{d!})n^{d}+O(n^{d-1})$ and it follows from Proposition 1.2.4 that $f(1)=\operatorname{e}(I)/\rho^{d}$.

We next show that $f^{\prime}(1)>\frac{d}{2}f(1)$. We will deduce this by comparing the multiplicity and the colength of $R[[T]]$-ideals $\overline{(I+T^{m})^{n}}$, where $m$ is a sufficiently divisible and $n$ a sufficiently large positive integer. Note that by Corollary 1.2.8, we know that

[TABLE]

At this point, we observe that $\ell(R/I_{j})$ is the coefficient of $t^{j-1}$ in $h(t)/(1-t)$, and that $\sum_{j=1}^{n}\ell(R/I_{j})$ is the coefficient of $t^{n-1}$ in $h(t)/(1-t)^{2}=f(t)/(1-t)^{d+2}$. We can write

[TABLE]

where all poles of $g(t)$ have order at most $d$. We expand the first two terms above and obtain that the coefficient of $t^{n-1}$ in $f(t)/(1-t)^{d+2}$ is

[TABLE]

It follows that for $n\gg 0$,

[TABLE]

On the other hand, we know that $a(F;X)<-1$ for all exceptional prime divisors $F$ on $Y$ by assumption. Let $\pi^{\prime}\colon Y^{\prime}\to X^{\prime}\coloneqq\operatorname{Spec}(R[[T]])$ be the normalized blowup of $I+T^{m}$ (recall that $m$ is sufficiently divisible). By Lemma 5.3.1, $a(F^{\prime},X^{\prime})<0$ for all exceptional prime divisors $F^{\prime}$ on $Y^{\prime}$. Moreover, it is clear that $\pi^{\prime}$ is an isomorphism on the punctured spectrum of $R[[T]]$ and $(I+T^{m})\mathcal{O}_{Y^{\prime}}=\mathcal{O}_{Y^{\prime}}(-E^{\prime})$. In particular $-E^{\prime}$ is a $\pi^{\prime}$-ample $\pi^{\prime}$-exceptional Cartier divisor on $Y^{\prime}$. We now apply Proposition 5.2.2 to $Y^{\prime}\to X^{\prime}$ and note that $\Gamma(Y^{\prime},\mathcal{O}_{Y^{\prime}}(-nE^{\prime}))=\overline{(I+T^{m})^{n}}$. Thus we obtain that

[TABLE]

Since $\{\overline{(I+T^{m})^{n}}\}_{n}$ is a Noetherian graded family, it follows easily from the expression above that $\operatorname{e}(\overline{(I+T^{m})^{n}})=-(-E^{\prime})^{d+1}n^{d+1}$. Combining the two expressions yields the formula:

[TABLE]

By comparing the coefficients of $1/n$ in the denominators of (5.3) and (5.3), we see that

[TABLE]

Note that by definition,

[TABLE]

Since $-E^{\prime}|_{F^{\prime}}$ is ample on $F^{\prime}$ and $a(F^{\prime};X^{\prime})<0$, we have that $(K_{Y^{\prime}}-\pi^{\prime*}_{\operatorname{num}}K_{X^{\prime}})\cdot(-E^{\prime})^{d}<0.$ It follows that $f^{\prime}(1)>\frac{d}{2}f(1)$ and tbus $R$ is not lim-stable by Theorem 5.3.2, a contradiction. ∎

The following is an immediate consequence of Theorem 5.3.4 and the existence of log canonical modification in the ${\mathbb{Q}}$-Gorenstein setting (Theorem 5.1.4).

Corollary 5.3.5**.**

Let $(R,{\mathfrak{m}})$ be a ${\mathbb{Q}}$-Gorenstein normal local ring essentially of finite type over a field of characteristic zero. If $R$ is lim-stable (e.g., $R$ is semistable), then $R$ is log canonical.

Proof.

This follows from Theorem 5.3.4 and Theorem 5.1.4: if $f$: $Y\to X$ is the log canonical modification, then $K_{Y}+{E}_{\textup{red}}-f^{*}K_{X}$ is $f$-ample $f$-exceptional and $a(F;X)<-1$ for all exceptional prime divisors $F$ on $Y$ by the negativity lemma [KM98, Lemma 3.39] so that Theorem 5.3.4 can be applied. ∎

Beyond the ${\mathbb{Q}}$-Gorenstein setting, we have the following extension of Corollary 5.3.5. This is our main result on lim-stable singularities.

Theorem 5.3.6**.**

Let $(R,{\mathfrak{m}})$ be an excellent normal local domain admitting a dualizing complex. Suppose one of the following holds:

1. (1)

$\dim(R)\leq 2$, or 2. (2)

$R$* is essentially of finite type over a field of characteristic zero that is numerically ${\mathbb{Q}}$-Gorenstein.*

If $R$ is lim-stable, then $R$ is ${\mathbb{Q}}$-Gorenstein and log canonical.

Proof.

We just need to verify that the hypotheses of Theorem 5.3.4 are satisfied. In case $(2)$, this follows immediately from Theorem 5.1.7.

In case (1), we let $f\colon Y\to X\coloneqq\operatorname{Spec}(R)$ be the log canonical modification (which exists by the full log MMP in dimension two, see [Tan18]). Note that both $X$ and $Y$ are numerically ${\mathbb{Q}}$-Gorenstein by [Tak21, Example 3.4]. We need to show the existence of an ample exceptional Cartier divisor on $Y$. Let $g\colon Z\to Y$ be a log resolution that is an isomorphism over $Y\backslash{E}_{\textup{red}}$ (the latter is isomorphic to $X\backslash\{{\mathfrak{m}}\}$ which is regular). We will show that $Y$ is numerically log terminal in the sense that $a(F;Y)>-1$ for all $g$-exceptional prime divisors $F$ on $Z$.

To prove the claim, we start by writing

[TABLE]

where $a_{i}\geq-1$ for all $i$. This is possible since $(Y,{E}_{\textup{red}})$ is log canonical. We also write $K_{Z}-g_{\operatorname{num}}^{*}K_{Y}=\sum b_{i}F_{i}.$ It follows that $\sum(a_{i}-b_{i})F_{i}$ is $g$-numerically equivalent to $g_{*}^{-1}{E}_{\textup{red}}$. In particular, $\sum(a_{i}-b_{i})F_{i}$ is $g$-nef. By the negativity lemma [KM98, Lemma 3.39], we have $a_{i}-b_{i}\leq 0$ for all $i$. Now suppose that $a_{i}=b_{i}$ for some $i$ and let $y\coloneqq g(F_{i})$. Note that $y\in{E}_{\textup{red}}$ since $Z\to Y$ is an isomorphism over $Y\backslash{E}_{\textup{red}}$. We have

[TABLE]

so we must have equalities throughout. Hence $a_{j}=b_{j}$ for all $j$ such that $F_{i}\cap F_{j}\neq\emptyset$. Since the exceptional set $f^{-1}(y)$ is connected, repeating this procedure we obtain that $a_{i}=b_{i}$ for all $i$ such that $F_{i}\in f^{-1}(y)$. But then

[TABLE]

which is a contradiction to $y\in{E}_{\textup{red}}$. Therefore we must have $b_{i}>a_{i}\geq-1$ for all $i$, i.e., $a(F;Y)>-1$ for all $g$-exceptional prime divisors $F$. This implies $\lceil K_{Z}-g_{\operatorname{num}}^{*}K_{Y}\rceil$ is effective and thus $g_{*}\mathcal{O}_{Z}(\lceil K_{Z}-g_{\operatorname{num}}^{*}K_{Y}\rceil)=\mathcal{O}_{Y}$. Since $g_{\operatorname{num}}^{*}K_{Y}$ is $g$-numerically trivial by definition, by relative vanishing in dimension two (for example, see [Tan18, Theorem 3.4]), we have that $R^{1}g_{*}\mathcal{O}_{Z}(\lceil K_{Z}-g_{\operatorname{num}}^{*}K_{Y}\rceil)=0$. Thus the composition map

[TABLE]

is the identity. In particular, $\mathcal{O}_{Y}\to Rg_{*}\mathcal{O}_{Z}$ splits in $D(Y)$ and by duality $Y$ has pseudo-rational singularities. This in turn implies that $Y$ is ${\mathbb{Q}}$-factorial by [Lip69, Proposition 17.1]. It follows that $K_{Y}-f^{*}_{\operatorname{num}}K_{X}+{E}_{\textup{red}}$ is an $f$-ample $f$-exceptional ${\mathbb{Q}}$-Cartier divisor, and that $a(G;X)<-1$ for all exceptional prime divisors $G$ on $Y$ by the negativity lemma again. Thus all hypotheses of Theorem 5.3.4 are satisfied and the result follows. ∎

Remark 5.3.7*.*

There are some other situations in which Theorem 5.3.4 can be applied in low dimensions. Specifically, if $(R,{\mathfrak{m}})$ is essentially of finite type over a perfect field of characteristic $p>5$, and $R$ is normal and ${\mathbb{Q}}$-Gorenstein of dimension three, then we claim that log canonical modification of $X\coloneqq\operatorname{Spec}(R)$ exists. Hence Corollary 5.3.5 still holds.

To see the claim, note that by [BMP*+*23, Corollary 9.21], which builds on [HX15], [Bir16], we can find a $\mathbb{Q}$-factorial dlt modification $f\colon Z\to X$, i.e., $(Z,{E}_{\textup{red}})$ is dlt and $K_{Z}+{E}_{\textup{red}}$ is $f$-nef. By [HNT20, Theorem 1.4 (2)] (see also [Wal18, Theorem 1.1]), $K_{Z}+{E}_{\textup{red}}$ is $f$-semi-ample. Thus the contraction over $X$ induced by $K_{Z}+{E}_{\textup{red}}$ is the log canonical modification $Y$ of $X$. In fact, provided that the analog of [Wal18, Theorem 1.1] holds, this argument applies more generally to any three-dimensional excellent local ring $(R,{\mathfrak{m}})$ with $\operatorname{char}(R/{\mathfrak{m}})>5$ such that $R$ is normal, ${\mathbb{Q}}$-Gorenstein, and admits a dualizing complex. It is our understanding that the proof in [Wal18] goes through by replacing the use of positive characteristic MMP by [BMP*+*23] and using the gluing argument of [Pos23, Theorem 3.1], which works in mixed characteristic with minimal changes.

It is natural to expect that lim-stable singularities in characteristic zero are pseudo-log canonical (without assuming numerical ${\mathbb{Q}}$-Gorensteinness). In fact, we expect the following stronger statement.

Conjecture 5.3.8**.**

Let $(R,{\mathfrak{m}})$ be a normal local domain essentially of finite type over a field of characteristic zero. Suppose $R$ is lim-stable (e.g., $R$ is semistable), then $R$ is of log canonical type (in particular, $R$ is pseudo-log canonical).

We end this subsection by collecting some properties of lim-stable surface singularities, which strengthen some of the results obtained by Mumford and Shah, see [Mum77, Proposition 3.20] and [Sha81, Theorem 2]. The proof proceeds by combining our results in dimension two and the following consequence of the classification of log canonical surface singularities, see [Mat02, Remark 4-6-29] in characteristic zero and [Kol13, Example 3.27 and 3.39] in arbitrary characteristic.

Theorem 5.3.9**.**

Let $(R,{\mathfrak{m}})$ be a two-dimensional excellent normal local ring with a dualizing complex, with $R/{\mathfrak{m}}$ algebraically closed. If $R$ is log canonical and not a rational singularity, then $R$ is either a simple elliptic or a cusp singularity, in particular $R$ is Gorenstein.

Proposition 5.3.10**.**

Let $(R,\mathfrak{m})$ be a two-dimensional lim-stable Cohen–Macaulay local ring. Then $\operatorname{e}(R)\leq\operatorname{edim}(R)$ (i.e., either $R$ has minimal multiplicity or almost minimal multiplicity) and we have the following.

1. (1)

If $R$ has minimal multiplicity, then $\operatorname{gr}_{\mathfrak{m}}(R)$ is Cohen–Macaulay and its defining equations as a quotient of a polynomial ring, are quadratic. 2. (2)

If $\operatorname{e}(R)=\operatorname{edim}(R)$, then $\operatorname{gr}_{\mathfrak{m}}(R)$ is Cohen–Macaulay and reduced and its defining equations as a quotient of a polynomial ring are quadratic or cubic.

Furthermore, if $R$ is excellent normal with a dualizing complex, and $R/{\mathfrak{m}}$ is algebraically closed, then

1. (3)

$R$* is ${\mathbb{Q}}$-Gorenstein and log canonical;* 2. (4)

$R$* has minimal multiplicity if and only if $R$ has rational singularity;* 3. (5)

If $\operatorname{e}(R)=\operatorname{edim}(R)$, then $\operatorname{gr}_{\mathfrak{m}}(R)$ (and thus $R$) is Gorenstein and either $\operatorname{gr}_{\mathfrak{m}}(R)$ is a hypersurface of degree $3$, or its defining equations as a quotient of a polynomial ring are quadratic.

Proof.

The bound on the embedding dimension was obtained in Corollary 5.3.3. Note that $\operatorname{e}(R)\geq\operatorname{edim}(R)-1$ by Abhyankar’s inequality. If $\operatorname{e}(R)=\operatorname{edim}(R)-1$ then $R$ has minimal multiplicity and it is well-known that $\operatorname{gr}_{\mathfrak{m}}(R)$ is Cohen–Macaulay with quadratic defining equations, see [Sal79a, Theorem 1]. If $\operatorname{e}(R)=\operatorname{edim}(R)$, then the conclusion follows from Proposition 4.3.6 and [Sha81, Theorem 3].

Next, (3) follows from Theorem 5.3.6 and (4) follows from Theorem 5.3.9 (note that rational surface singularities have minimal multiplicity). For (5), note that if $R$ has almost minimal multiplicity, then it does not have rational singularity and thus by Theorem 5.3.9 $R$ is Gorenstein. Now we apply [Sal83, 3.1, 3.3] to obtain that, if $\operatorname{e}(R)\geq 4$, then $\operatorname{gr}_{\mathfrak{m}}(R)$ is Gorenstein with quadratic defining equations; otherwise $\operatorname{e}(R)=\operatorname{edim}(R)=3$ and thus $\operatorname{gr}_{\mathfrak{m}}(R)$ is a hypersurface of degree $3$. ∎

Remark 5.3.11*.*

We believe it should be possible to prove directly that if $(R,{\mathfrak{m}})$ is a semistable (or merely lim-stable) Cohen-Macaulay surface singularity such that $\operatorname{e}(R)=\operatorname{edim}(R)$, then $\text{gr}_{\mathfrak{m}}(R)$ is Gorenstein. Proposition 5.3.10 shows that this is true when $R$ is normal, but we do not see how to prove this without using the classification result (Theorem 5.3.9).

From [Sha81, Theorem 3 (ii)] we know that the Hilbert series of $\text{gr}_{\mathfrak{m}}(R)$ is given by

[TABLE]

Moreover, by [BH93, Corollary 4.4.6(a)] we know that $h_{\text{gr}_{\mathfrak{m}}(R)}(t)=h_{\omega_{\text{gr}_{\mathfrak{m}}(R)}}(t)$, meaning that the Hilbert series $\text{gr}_{\mathfrak{m}}(R)$ and the graded canonical module $\omega_{\text{gr}_{\mathfrak{m}}(R)}$ coincide. Thus it remains to show that there is a graded injection $\text{gr}_{\mathfrak{m}}(R)\to\omega_{\text{gr}_{\mathfrak{m}}(R)}$. In particular, it is immediate that $\text{gr}_{\mathfrak{m}}(R)$ is Gorenstein if it is a domain (see [BH93, Corollary 4.4.6 (c)]). However, in general, $\text{gr}_{\mathfrak{m}}(R)$ is only reduced, which is not sufficient to apply [BH93, Corollary 4.4.6 (c)].

5.4. Lech-stable and canonical singularities

We now present a connection between Lech-stable and canonical singularities.

Theorem 5.4.1**.**

Let $(R,{\mathfrak{m}})$ be an excellent normal local domain admitting a dualizing complex. Suppose one of the following holds:

1. (1)

$\dim(R)\leq 2$, or 2. (2)

$R$* is essentially of finite type over a field of characteristic zero, is numerically ${\mathbb{Q}}$-Gorenstein, and has canonical singularities on the punctured spectrum (e.g., $R$ has an isolated singularity).*

If $R$ is Lech-stable, then $R$ is ${\mathbb{Q}}$-Gorenstein and canonical.

Proof.

First of all, in both cases, we know that $R$ is lim-stable and thus $R$ is ${\mathbb{Q}}$-Gorenstein by Theorem 5.3.6. Let $f\colon Y\to X\coloneqq\operatorname{Spec}(R)$ be a canonical modification (see [Kol13, Theorem 1.31], which is based on [BCHM10]). This means $Y$ has canonical singularities and $K_{Y}$ is $f$-ample. It follows from ${\mathbb{Q}}$-Gorensteiness of $R$ that $K_{Y}-f^{*}K_{X}$ is an $f$-ample and $f$-exceptional ${\mathbb{Q}}$-Cartier divisor. Now since $f$ is an isomorphism on the punctured spectrum by the uniqueness of canonical modification, we know that $f$ is the blowup of some ${\mathfrak{m}}$-primary ideal $I\subseteq R$ and we set $I\mathcal{O}_{Y}=\mathcal{O}_{Y}(-E)$.

The rest of the argument follows the proof of Theorem 5.3.4. By Proposition 5.2.2 applied to $f$, and noting that $\Gamma(Y,\mathcal{O}_{Y}(-nE))=\overline{I^{n}}$, we may express that

[TABLE]

where $d\coloneqq\dim(R)$. Next, we write $K_{Y}-f^{*}K_{X}=\sum_{F}a(F;X)F$ where the sum runs over all exceptional prime divisors $F$ on $Y$. Since $K_{Y}-f^{*}K_{X}$ is $f$-amply and by the negativity lemma, we know that $a(F;X)<0$ for all such $F$. It follows that

[TABLE]

where the last inequality follows since $-E|_{F}$ is ample and $a(F;X)<0$. Therefore, for sufficiently large $n$, we have $d!\ell(R/\overline{I^{n}})<\operatorname{e}(\overline{I^{n}})$, which implies that $\operatorname{c_{LM}}(R)>1$. This contradicts our assumption that $R$ is Lech-stable. ∎

Remark 5.4.2*.*

Unlike Theorem 5.3.4 and Theorem 5.3.6, in Theorem 5.4.1 one cannot drop the assumption that $R$ has canonical singularities on the punctured spectrum, even when $R$ is Gorenstein. As an example, consider the flat family

[TABLE]

We know by Proposition 2.4.10 that

[TABLE]

where ${\mathfrak{m}}=(x,y,z,w)$ and the equality above follows from Theorem 6.2.1 in Section 6. It follows that $R\coloneqq\overline{\mathbb{C}(t)}[x,y,z,w]_{{\mathfrak{m}}}/(xyz+t(x^{3}+y^{3}+z^{3}))$ is a normal hypersurface that is Lech-stable. However, it is easy to see that $R$ is not canonical.

Remark 5.4.3*.*

The converse to Theorem 5.4.1 does not hold, even for Gorenstein rings. For $d\geq 3$, the degree $d$ Veronese subring $R$ of $\mathbb{C}[x_{1},\ldots,x_{d}]$ is an example of a Gorenstein canonical singularity such that $\operatorname{e}(R)=d^{d-1}>d!$, such $R$ cannot be Lech-stable by Remark 2.2.3. On the other hand, we do not know whether there are complete intersection canonical singularities that are not Lech-stable (conjecturally, such singularities have multiplicity at most $2^{d-1}$, see [HMTW08] and [Shi17] for related results).

5.5. Semi-log canonical singularities

In this subsection, we will partially generalize our results to the non-normal setting. We first recall that an $(\textnormal{S}_{2})$ and $(\textnormal{G}_{1})$ scheme $X$ is called semi-log canonical if $X$ is demi-normal, $K_{X}$ is ${\mathbb{Q}}$-Cartier and the pair $(X^{\prime},D^{\prime})$ is log canonical where $X^{\prime}$ is the normalization of $X$ and $D^{\prime}$ is the effective divisor defined by the conductor ideal, see [Kol13, Chapter 5.2] (note that $K_{X}$ is ${\mathbb{Q}}$-Cartier implies $K_{X^{\prime}}+D^{\prime}$ is ${\mathbb{Q}}$-Cartier).

The main result will use the following variant of Theorem 5.3.4 in the non-normal ${\mathbb{Q}}$-Gorenstein setting. Its proof relies on a careful analysis using rational powers of ideals and might also lead to an alternative and more algebraic approach of Theorem 5.3.4.

Theorem 5.5.1**.**

Let $\pi\colon Y\to X\coloneqq\operatorname{Spec}(R)$ be a projective birational map of $(\textnormal{S}_{2})$ and $(\textnormal{G}_{1})$ schemes, where $(R,\mathfrak{m})$ is an excellent local domain of dimension $d\geq 2$ admitting a dualizing complex. Suppose the following conditions hold:

1. (1)

$\pi$* is the blowup of some ${\mathfrak{m}}$-primary ideal $I\subseteq R$;* 2. (2)

$R$* is ${\mathbb{Q}}$-Gorenstein and $K_{Y}-\pi^{*}K_{X}=\sum_{i}a_{i}E_{i}$ with $a_{i}<-1$, where $E_{i}$ runs over all $\pi$-exceptional prime divisors on $Y$;* 3. (3)

the generic point of each $E_{i}$ is regular.

Then $\operatorname{c^{\infty}_{LM}}(R)>1$.

Proof.

The hypothesis implies that $\pi_{*}\mathcal{O}_{Y}=\mathcal{O}_{X}$ (see the proof of Corollary 5.2.5). Write $I\mathcal{O}_{Y}=\mathcal{O}_{Y}(-E)$. By Proposition 1.2.5, we know that $\Gamma(Y,\mathcal{O}_{Y}(-\lceil\frac{n}{\rho}E\rceil))=I^{\frac{n}{\rho}}$ where $\rho$ is the Rees period of $I$. Consider the Noetherian graded family $I_{n}\coloneqq\{I^{\frac{n}{\rho}}\}_{n}$. By Theorem 5.3.2,

[TABLE]

where $f(t)$ is a rational function with $f(1)\neq 0$ and all poles of $f(t)$ have order at most $d$. By Theorem 5.3.2, it is enough to show that $f^{\prime}(1)>\frac{d}{2}f(1)$.

We next express $f(t)$ in a different way. Observe that

[TABLE]

and that by Corollary 5.2.5, when $n=k\rho+h$, we have

[TABLE]

Note that this formula still holds for $h=\rho$. Thus the function $a(n)\coloneqq\ell(R/I_{n+1})$ satisfies the assumptions of Lemma 5.5.2 below: we may set $A=-(-E)^{d}$ and

[TABLE]

Now, by Lemma 5.5.2 the desired inequality $f’(1)>\frac{d}{2}f(1)$ is equivalent to showing that

[TABLE]

In order to prove this, it suffices to verify that the coefficient of each $E_{i}$ in

[TABLE]

is positive. We write $E=\sum c_{i}E_{i}$ and note that $c_{i}=\rho/d_{i}$ where $d_{i}\in\mathbb{N}$. Thus

[TABLE]

Thus the coefficient of $E_{i}$ in $E-K_{Y}+\pi^{*}K_{X}-\frac{2}{\rho}\sum_{h=1}^{\rho}\left\lceil\frac{hZ}{\rho}\right\rceil$ is

[TABLE]

The following lemma is used in the proof of Theorem 5.5.1 above.

Lemma 5.5.2**.**

Suppose that $a(n)\in\mathbb{C}$ is a quasi-polynomial sequence such that there is a constant $A$ and a periodic sequence $B(n)$ with period $\rho$ such that

[TABLE]

Then the generating function $\alpha(T)\coloneqq\sum_{n=0}^{\infty}a(n)T^{n}$ can be written as

[TABLE]

where $f(T)$ is a rational function such that $f(1)=A/\rho^{d}$, $f^{\prime}(1)\neq 0$, and

[TABLE]

Proof.

We may rewrite the generating function for the first term as

[TABLE]

We then treat the second term similarly, noting first that $B(k\rho+h)=B(h)$ since $B(n)$ is a periodic function with period $\rho$. Thus

[TABLE]

Since the $O(n^{d-2})$ term is given by a quasi-polynomial, by [Sta12, Proposition 4.4.1], the pole order of its generating function at $T=1$ is a most $d-1$. Combining the three parts, we now may express

[TABLE]

where $r(T)$ has no pole at $T=1$. Therefore

[TABLE]

It is now clear that $f(1)=A/\rho^{d}$ and we may further see that

[TABLE]

Thus it follows by a direct computation that

[TABLE]

We now prove our main result in this subsection, relating lim-stable and semi-log canonical singularities in the ${\mathbb{Q}}$-Gorenstein setting.

Theorem 5.5.3**.**

Let $(R,{\mathfrak{m}})$ be an excellent local domain that admits a dualizing complex and satisfies $(\textnormal{S}_{2})$ and $(\textnormal{G}_{1})$. Suppose that $R$ is ${\mathbb{Q}}$-Gorenstein and lim-stable. In addition, assume one of the following holds:

1. (1)

$\dim(R)=2$* and $\operatorname{char}(R/{\mathfrak{m}})\neq 2$,* 2. (2)

$R$* is essentially of finite type over a field of characteristic zero.*

Then $R$ is semi-log canonical.

Proof.

In both cases, $R$ is equidimensional, satisfies the condition $(\textnormal{S}_{2})$, and lim-stable. Therefore by Corollary 4.2.3, $R$ is demi-normal. We next claim that, in both cases, $R$ admits a semi-log canonical modification, i.e., a projective birational map $f\colon Y\to X\coloneqq\operatorname{Spec}(R)$ such that the pair $(Y,{E}_{\textup{red}})$ is semi-log canonical and $K_{Y}+{E}_{\textup{red}}$ is $f$-ample.

In case (2) this follows directly from [OX12, Corollary 1.2]. In case (1), essentially the same argument as in [OX12, Corollary 1.2] applies: the existence of log canonical modifications is guaranteed by the full log MMP for excellent surfaces [Tan18], and we use [Pos24, Theorem 4.1.1] for the gluing argument in dimension two. Here we note that [Pos24, Theorem 4.1.1] can be generalized to the relative setting as follows: for any excellent local ring $R$ admitting a dualizing complex with residue characteristic $\neq 2$, the normalization map provides a one-to-one correspondence between semi-log canonical surface pair $(S,\Delta)$ proper over $\operatorname{Spec}(R)$ and log canonical surface pairs $(\overline{S},\overline{D}+\overline{\Delta})$ proper over $\operatorname{Spec}(R)$, with an involution on $(\overline{D}^{n},\text{Diff}_{\overline{D}^{n}}\overline{\Delta})$ that is generically fixed point free on every component. Moreover, $K_{S}+\Delta$ is ample if and only if $K_{\overline{S}}+\overline{D}+\overline{\Delta}$ is ample. The proof is essentially the same, except that we replace [Pos24, Proposition 3.4.1] with [Wit22, Theorem 1.4] to argue that the geometric quotient exists (the characteristic zero fiber does not present any issues by [Kol13, Theorem 5.13]).

Now suppose $R$ is not semi-log canonical. We choose a prime ideal $Q\in\operatorname{Spec}(R)$ of minimal height such that $R_{Q}$ is not semi-log canonical. By Theorem 2.3.2, we may replace $R$ by $R_{Q}$ to assume $R$ is semi-log canonical on the punctured spectrum of $R$ (this step is only necessary in case (2)).

In both cases, let $\pi\colon Y\to X$ be the semi-log canonical modification. By the uniqueness of semi-log canonical modification $\pi$ is an isomorphism on the punctured spectrum. One easily checks that $Y\to X$ satisfies the assumptions of Theorem 5.5.1:

• •

$K_{Y}-\pi^{*}K_{X}+{E}_{\textup{red}}$ is $\pi$-ample $\pi$-exceptional ${\mathbb{Q}}$-Cartier divisor (so $Y$ is the blowup of some ${\mathfrak{m}}$-primary ideal of $R$);

• •

the coefficient of each exceptional prime divisor in $K_{Y}-\pi^{*}K_{X}$ is $<-1$ by the negativity lemma;

• •

the generic point of each exceptional prime divisor is regular, since this holds on the corresponding log canonical modification.

Thus, by Theorem 5.5.1 $R$ cannot be lim-stable, which is a contradiction. ∎

6. Examples: simple normal crossings and determinantal rings

In this section we present several refinements of Lech’s inequality and show how they can be used to compute the Lech–Mumford constant for special classes of rings.

6.1. Inclusion-exclusion formulas

As a technical tool we will need two weaker, diluted, versions of the inclusion-exclusion principle.

Proposition 6.1.1**.**

Let $S$ be a finite set and let $S_{1},\ldots,S_{d}$ be subsets of $S$ such that $S=\cup_{i=1}^{d}S_{i}$. Then we have

[TABLE]

Furthermore, we have inequalities

[TABLE]

Proof.

The first part is well-known: let $a_{k}$ be the number of elements in $S$ that are contained in exactly $k$ subsets $S_{i}$, then $|S|=\sum_{i=1}^{d}a_{i}$, $\sum_{i=1}^{d}|S_{i}|=\sum_{k=1}^{d}ka_{k}$, $\sum_{1\leq i<j\leq d}|S_{i}\cap S_{j}|=\sum_{k=2}^{d}\binom{k}{2}a_{k}$, etc., and the conclusion follows from the formula $\sum_{i=1}^{s}(-1)^{i+1}\binom{s}{i}=1$.

We now analyze the dilutions in a similar manner. First of all, the identity

[TABLE]

can be applied to obtain the first diluted formula:

[TABLE]

Secondly, we also have the identity

[TABLE]

It follows that

[TABLE]

Therefore

[TABLE]

Now clearly, we have $|S|=\sum_{i=1}^{d}a_{i}>A\geq B\geq\sum_{i=1}^{d}a_{i}/2=|S|/2$. ∎

Corollary 6.1.2**.**

Let $R=K[[x_{1},\ldots,x_{n}]]$, $I$ be an ${\mathfrak{m}}$-primary monomial ideal, and $P_{1},\ldots,P_{s}$ be monomial prime ideals. Then in $S=R/I$ we have

[TABLE]

Proof.

If we consider the sets S_{j}\coloneqq\{\text{monomials in R not contained in }(I,P_{j})\}, then

[TABLE]

The conclusion now follows from the inclusion-exclusion principle in Proposition 6.1.1. ∎

We now derive a sharper version of Lech’s inequality for a power series ring over a Lech-stable ring.

Theorem 6.1.3**.**

Let $(S,\mathfrak{m})$ be a Lech-stable local ring (i.e., $\operatorname{c_{LM}}(S)=1$) of dimension at least one, $R=S[[x_{1},\ldots,x_{n}]]$ for some $n\geq 0$, and $I$ be an $(\mathfrak{m},x_{1},\ldots,x_{n})$-primary ideal in $R$. Suppose also that $I$ is generated by monomials (i.e., $s\prod x_{i}^{a_{i}}$ for some $s\in S$). Then the inequality

[TABLE]

holds. Moreover, the inequality is strict if $\dim(R)\geq 3$.

Proof.

We use induction on $n$. In the base case $n=0$, $R=S$ is Lech-stable and the conclusion is just a reformulation of $\operatorname{c_{LM}}(R)=1$.

We next consider the inductive step. Suppose that $x_{n}^{c+1}\in I$ but $x_{n}^{c}\notin I$. Denote

[TABLE]

Note that for $0\leq k\leq c$, each $I_{k}$ is an $(\mathfrak{m},x_{1},\ldots,x_{n-1})$-primary monomial ideal in $R^{\prime}\coloneqq S[[x_{1},\ldots,x_{n-1}]]$. By Proposition 2.1.12 and the Minkowski inequality for mixed multiplicities ([Tei77, RS78]), for $d\coloneqq\dim(S)+n-1=\dim(R)-1$ we have

[TABLE]

Interpreting the right-hand side as a geometric mean, we further obtain that

[TABLE]

where the second inequality is strict when $d\geq 2$ (i.e., $\dim(R)\geq 3$). Since $I$ is generated by monomials, it is easy to see that $\ell(R^{\prime}/I_{0})=\ell(R/(I,x_{n}))$. By the inductive hypothesis

[TABLE]

At this point, it remains to note that, since our ideals are monomial, we have the formula

[TABLE]

The conclusion of the theorem follows directly from (6.1) by recombining the sums. ∎

Corollary 6.1.4**.**

Let $(R,{\mathfrak{m}})\coloneqq K[[x_{0},\ldots,x_{d}]]$ where $K$ is a field and let $I\subseteq R$ be an ${\mathfrak{m}}$-primary monomial ideal. Then we have the inequality

[TABLE]

and it is strict if $d\geq 2$.

Proof.

By using the sets corresponding to $P_{i}=(x_{i})$, $1\leq i\leq d$, in the proof of Corollary 6.1.2, it follows from Proposition 6.1.1 that the estimate in Theorem 6.1.3 (applied to $S=K[[x_{0}]]$) is stronger than the conclusion. ∎

In dimension two, Corollary 6.1.4 recovers a result of Mumford.

Corollary 6.1.5** ([Mum77, Lemma 3.15]).**

Let $(R,{\mathfrak{m}})\coloneqq K[[x,y]]$ where $K$ is a field and let $I\subseteq R$ be an ${\mathfrak{m}}$-primary ideal. Then we have

[TABLE]

Proof.

First note that if $I$ is a monomial ideal, then the conclusion is the $d=1$ case of Corollary 6.1.4. In general, let $\prec$ be the lexicographic order with $x\prec y$. We see that

[TABLE]

Since $\operatorname{e}(I)\leq\operatorname{e}(\operatorname{in}_{\prec}(I)$ and $\ell(R/I)=\ell(R/\operatorname{in}_{\prec}(I))$ as discussed in Proposition 2.1.6, the conclusion follows from the monomial case established above. ∎

6.2. Simple normal crossings and determinantal rings

We will now prove main results of this section. We start with the case of simple normal crossing singularities.

Theorem 6.2.1**.**

Let $R\coloneqq K[[x_{1},\ldots,x_{d}]]/(x_{1}\cdots x_{d})$ where $K$ is a field. Then

• •

if $d=1$, then $R$ is Lech-stable;

• •

if $d=2$, then $R$ is semistable but not stable;

• •

if $d\geq 3$, then $R$ is stable but not Lech-stable.

Proof.

If $d=1$, then $R=K$ is a field and the conclusion is clear (see Proposition 2.2.5). We next show that when $d\geq 2$ (resp., $d\geq 3$), then $R$ is semistable (resp., stable). By Proposition 2.1.6, it is enough to prove that for any $\mathfrak{m}$-primary monomial ideal $I\subseteq S\coloneqq K[[x_{0},\ldots,x_{d}]]$

[TABLE]

and the inequality is strict if $d\geq 3$.

At this point, we note that $\operatorname{e}(I,S/(x_{1}\cdots x_{d}))=\sum_{i=1}^{d}\operatorname{e}(I,S/(x_{i}))$, so we may bound

[TABLE]

where the first inequality follows from Corollary 6.1.4 (and this inequality is strict when $d\geq 3$) and the last equality follows from Corollary 6.1.2 applied to $P_{i}=(x_{i})$ for $1\leq i\leq d$ (also note that for the equality in the second line above, the coefficient disappeared since each $\ell(R/I+(x_{j_{1}},\dots,x_{j_{k}}))$ appeared in the original summation $k$ times: one for each $S/(x_{j_{i}})$).

It remains to prove that $R$ is not stable when $d=2$ by Proposition 2.2.6 and and is not Lech-stable when $d\geq 3$ by Corollary 4.3.1. ∎

As an immediate consequence, we see that generic determinantal hypersurfaces are Lech-stable (we will later show a stronger result that generic determinantal rings of maximal minors are Lech-stable).

Corollary 6.2.2**.**

Let $X=\{x_{ij}\}_{1\leq i,j\leq n}$ be an $n\times n$ matrix of indeterminates. Then $R\coloneqq K[[X]]/\det(X)$ is Lech-stable.

Proof.

By Corollary 2.4.8 and Proposition 2.1.6 (applied to the lexicographic term order of the variables), we know that

[TABLE]

where the second equality follows from Theorem 6.2.1 and ($\dagger$ ‣ 2.2). Therefore $\operatorname{c_{LM}}(R)=1$ and thus $R$ is Lech-stable. ∎

Our approach also allows us to compute the Lech–Mumford constant for simple normal crossing in dimension two.

Proposition 6.2.3**.**

Let $R=K[[x,y,z]]/(xyz)$. Then we have $\operatorname{c_{LM}}(R)=3/2$.

Proof.

First of all, it is clear that $\operatorname{e}({\mathfrak{m}})/2!\ell(R/{\mathfrak{m}})=3/2$. Thus all we need to show is that $\operatorname{e}(I)\leq 3\ell(R/I)$ for every $\mathfrak{m}$-primary ideal $I\subseteq R$. By Proposition 2.1.6, we may assume that $I$ is a monomial ideal. By Corollary 6.1.5, we have that

[TABLE]

Thus it suffices to show that

[TABLE]

Now, by Corollary 6.1.2, we have that

[TABLE]

and then we use Corollary 6.1.2: $I(R/(x))$ is a monomial ideal in $K[[y,z]]$, so

[TABLE]

and similarly,

[TABLE]

[TABLE]

Combining these (6.2.3.1) follows. ∎

Remark 6.2.4*.*

A similar approach can be taken for higher-dimensional simple normal crossings. Namely, following the proof of Theorem 6.2.1 with the inequality from Theorem 6.1.3, we obtain that for an $\mathfrak{m}$-primary ideal $I$ in $R=K[[x_{1},\ldots,x_{d}]]/(x_{1}\cdots x_{d})$,

[TABLE]

It remains to compare the right-hand side with $\ell(R/I)$. One may obtain crude estimates using the approach in the proof of Proposition 6.1.1. We expect that $\operatorname{c_{LM}}(R)$ should be attained by a power of the maximal ideal in this case, but we do not see how to show this.

We end this subsection by proving that generic determinantal rings of maximal minors are Lech-stable.

Theorem 6.2.5**.**

Let $X=\{x_{ij}\}_{1\leq i\leq n,1\leq j\leq m}$ be an $n\times m$ matrix of indeterminates with $n\leq m$. Let $I_{n}(X)$ be the ideal generated by the $n\times n$ minors of $X$. Then $R\coloneqq K[[X]]/I_{n}(X)$ is Lech-stable.

Proof.

If $n=1$, then $R=K$ is a field and the conclusion is obvious. In what follows, we will assume $n\geq 2$. Under the standard lexicographic monomial order, the maximal minors form a Gröbner basis of $I_{n}(X)$ (see [Stu90]), the initial ideal of $I_{n}(X)$ under this term order is

[TABLE]

By Corollary 2.4.8, it is enough to show that $K[[X]]/J$ is Lech-stable. Now since $n\geq 2$, it is easy to see that $x_{n1}$ is a free variable in $K[[X]]/J$ and it is enough to show that the local ring $K[[x_{j{i_{j}}}\mid 1\leq i_{1}<\cdots<i_{n}\leq m]]/J$ is semistable. Furthermore, by Proposition 2.1.6, it is enough to consider only monomial ideals.

We will now use induction on $n$ and $m-n$. For the base case we will use $n=m$ from Corollary 6.2.2. To set the induction we observe that $J=(x_{11},J_{1})\cap J_{2}$, where $J_{1}$ is the initial ideal of the maximal minors of the $n\times(m-1)$ matrix obtained by removing the first column, and $J_{2}$ is the initial ideal of the maximal minors of the $(n-1)\times(m-1)$ matrix obtained by removing the first row and column. Set $S\coloneqq K[[x_{j{i_{j}}}\mid 1\leq i_{1}<\cdots<i_{n}\leq m]]$ and note that $\operatorname{ht}(x_{11},J_{1})=m-n=\operatorname{ht}J_{2}$. It follows that $\dim(S/(x_{11},J_{1}))=\dim(S/J_{2})=:d$. From the short exact sequence

[TABLE]

we thus obtain that

[TABLE]

and that

[TABLE]

We will now analyze $S[[t]]/(x_{11},J_{1})$ and $S[[t]]/(J_{2}))$ separately.

We start with $S[[t]]/(x_{11},J_{1})$. Observe that $x_{22},\ldots,x_{nn}$ are free variables in $S/(x_{11},J_{1})$, i.e., we can write $S/(x_{11},J_{1})\cong(S^{\prime}/J_{1})[[x_{22},\dots,x_{nn}]]$ where $S^{\prime}=K[[x_{j{i_{j}}}\mid 2\leq i_{1}<\cdots<i_{n}\leq m]]$. Thus, by the induction hypothesis, we know that $S^{\prime}/J_{1}$ is semistable. Hence by Theorem 6.1.3 applied to the Lech-stable ring $(S^{\prime}/J_{1})[[t]]$, we see that for any monomial ideal $I\subseteq S[[t]]$ of finite colength, we have

[TABLE]

By Corollary 6.1.2 (applied to the prime ideals $(x_{22}),\dots,(x_{nn})$), we have

[TABLE]

Hence, using the lower bound from Proposition 6.1.1, we obtain from 6.2 that

[TABLE]

Now for $S/J_{2}$, we note that $x_{11}$ is a free variable. Thus by a similar (and easier) analysis using the inductive hypothesis and Theorem 6.1.3, we have that

[TABLE]

By combining (6.2.5.1), (6.2.5.2), (6.2.5.4), and (6.2.5.5) we obtain the bound

[TABLE]

Finally, since $J_{1}+(x_{22}\cdots x_{nn})\subseteq J_{2}$, we have

[TABLE]

and the desired inequality follows. ∎

Given Theorem 6.2.5, it is natural to ask the following question.

Question 6.2.6*.*

Let $X=\{x_{ij}\}_{1\leq i\leq n,1\leq j\leq m}$ be an $n\times m$ matrix of indeterminates and let $I_{t}(X)$ be the ideal generated by the $t\times t$ minors of $X$. Then is $R\coloneqq K[[X]]/I_{t}(X)$ Lech-stable (or at least semistable)?

6.3. Rational triple points

In [Sha81], Shah established a list of candidates for semistable surface singularities among triple points of embedding dimension four. They all happen to have rational singularities. Shah’s list contains several infinite families and several exceptional cases. We record here that our methods can be used to prove semistability for at least one of the infinite families. Following Tjurina’s terminology ([Tju68])

[TABLE]

i.e., the singularity $A_{l,m,n}$ is defined in $k[[x,y,z,w]]$ by the $2\times 2$ minors of the above matrix. Up to change of coordinates, one can check that this is symmetric on $l,m,n$ and thus we may assume that $l\leq m\leq n$. Note that $A_{0,0,0}$ is isomorphic to the (completion of the) cone of the twisted cubic curve.

The following proposition contains the key computation for showing semistability of these singularities. The proposition is a special case of Proposition 7.2.3 in Section 7, however we present it here since it follows directly from Theorem 6.1.3.

Proposition 6.3.1**.**

Let $R=k[[a,b,c,d]]/(ab,ac,cd)$. Then $R$ is stable. As a consequence, the (completion of the) cone of the twisted cubic $k[[s^{3},s^{2}t,st^{2},t^{3}]]$ is stable.

Proof.

We need to show that $\operatorname{c_{LM}}(R[[T]])=1$ and that the supremum is not attained, that is, $\operatorname{e}(I,R[[T]])<6\ell(R[[T]]/I)$ for every $(a,b,c,d,T)$-primary ideal $I\subseteq R[[T]]$. By Proposition 2.1.6, we may assume that $I$ is generated by monomials. By Corollary 6.1.2 applied to $P_{1}=(a,c)$, $P_{2}=(a,d)$ and $P_{3}=(b,c)$ in $k[[a,b,c,d,T]]$, we have

[TABLE]

Furthermore, by applying Theorem 6.1.3 to each $R[[T]]/(a,c),R[[T]]/(a,d),R[[T]]/(b,c)$, we estimate that

[TABLE]

This completes the proof of the first result. For the second claim, note that

[TABLE]

Consider the term order $a>c>b>d$, the initial ideal of the defining ideal of $S$ under this term order is $(ab,ac,cd)$ which is the defining ideal of $R$. Thus by Corollary 2.4.8, we have $\operatorname{c_{LM}}(S[[T]])\leq\operatorname{c_{LM}}(R[[T]])=1$ and the supremum in $\operatorname{c_{LM}}(S[[T]])$ is not attained. Thus $S$ is stable. ∎

Proposition 6.3.2**.**

For all $0\leq m,n\leq\infty$, the singularities $A_{0,m,n}$ are stable (thus by symmetry, $A_{m,0,n}$ and $A_{m,n,0}$ are stable).

Proof.

If $m,n>0$, then we may pass to the associated graded ring via Proposition 2.1.5 and reduce to the case where $m=n=\infty$. In the latter case, we have

[TABLE]

whose defining ideal is $(xy-yw,xz,yz)$. Under the monomial order $x<y<z<w$, the initial ideal is $(yw,xz,yz)$ and thus the singularity is stable by Corollary 2.4.8, Proposition 2.1.6, and Proposition 6.3.1.

It remains to study the case where more than one of $l,m,n$ is [math]. The defining ideal of $A_{0,0,\infty}$ is $(x(y+w)-yw,xz,yz)$ and under the monomial order $x<y<z<w$ its initial ideal is $(yw,xz,yz)$. The defining ideal of $A_{0,0,0}$ is $(x(y+w)-yw,yz-(y+w)w,xz-w^{2})$ and under the monomial order $x<y<w<z$ its initial ideal is $(yw,yz,xz)$. Thus in both cases, the singularities are stable by Corollary 2.4.8, Proposition 2.1.6, and Proposition 6.3.1. ∎

Remark 6.3.3*.*

In [Sha81], Shah proved that $A_{l,m,n}$ is unstable if $\frac{1}{l+1}+\frac{1}{m+1}+\frac{1}{n+1}<1$. Thus, to fully understand this family, it remains to determine the (semi)stability of $A_{1,1,n}$ for $1\leq n\leq\infty$ along with the exceptional cases $A_{1,2,2},A_{1,2,3},A_{1,3,3},A_{2,2,2}$.

We also want to highlight a peculiar connection between the non-normal limit $A_{1,1,\infty}$ and the $A_{3}$-singularity. The defining ideal of $A_{1,1,\infty}$ is

[TABLE]

which becomes isomorphic to $(x+w^{2},y+w^{2})\cap(z,xy+w^{4})$ after a change of variables.

7. Lech’s inequality revisited

In this section we investigate optimal versions of Lech’s inequality (see [HSV19] for closely related developments). We establish a sharp version of Lech’s inequality for regular local rings of dimension three: equality is attained for powers of the maximal ideal. This result can be also viewed as an extension of the sharp two-dimensional inequality in [Mum77, Lemma 3.15]. As an application, we show that elliptic and rational polygonal $n$-cones for $n\leq 6$ are semistable – results which were stated by Mumford in [Mum77, Propositions 3.18 and 3.19] without proofs. We then propose a general conjecture for the optimal form of Lech’s inequality in higher dimensions, provide partial evidence, and explore its connections with the results and conjectures in [HSV19].

7.1. Optimal Lech’s inequality in dimension three

The main ingredient of the proof of the sharp Lech-type inequality in dimension three is an extension of Mumford’s two-dimensional Lech’s inequality (see Corollary 6.1.5) to mixed multiplicities.

Lemma 7.1.1**.**

Let $(R,{\mathfrak{m}})=k[[x,y]]$ and let $I\subseteq J\subseteq R$ be two ${\mathfrak{m}}$-primary monomial ideals. Then we have

[TABLE]

Proof.

Write $I=(y^{r_{0}},y^{r_{1}}x,\ldots,y^{r_{c}}x^{c},x^{c+1})$, where $r_{0}\geq r_{1}\geq\cdots\geq r_{c}>0$. Since $I\subseteq J$, we may write $J=(y^{s_{0}},y^{s_{1}}x,\ldots,y^{s_{c}}x^{c},x^{c+1})$ with $r_{j}\geq s_{j}\geq 0$. For convenience, we set $r_{j}=s_{j}=0$ if $j\geq c+1$. We will use the formula $\operatorname{e}(I|J)=\ell(R/IJ)-\ell(R/I)-\ell(R/J)$ given in [HS06, Exercise 17.8]. Thus it suffices to show that

[TABLE]

By our way of writing $I$ and $J$, we know that $\ell(R/I)=\sum^{c}_{i=0}r_{i}$, $\ell(R/I)=\sum^{c}_{j=0}s_{j}$, and $\ell(R/I+(x))=r_{0}$. It is now a straightforward computation that

[TABLE]

In the sum above, each $s_{j}$ appears twice and each $r_{i}$ for $i\geq 1$ appears twice, while $r_{0}$ appears only once. Therefore we have

[TABLE]

which is precisely what we want to show. ∎

We now present the Lech-type inequality for power series rings of dimension three which is sharp for powers of the maximal ideal.

Theorem 7.1.2**.**

Let $(S,{\mathfrak{m}})=k[[x_{0},x_{1},x_{2}]]$ and let $I\subseteq S$ be an $\mathfrak{m}$-primary ideal. Then

[TABLE]

and equality holds when $I={\mathfrak{m}}^{n}$.

Proof.

It is straightforward to verify the equality for $I={\mathfrak{m}}^{n}$. Therefore we only need to prove the inequality. We first handle the case when $I$ is a monomial ideal. The strategy will be similar to the proof of Theorem 6.1.3. We set $R\coloneqq k[[x_{0},x_{1}]]$ and $I_{j}\coloneqq\{m\in R\mid mx_{2}^{j}\in I\}$, a monomial ideal in $R$. By Proposition 2.1.12 (note that we are in dimension two), we can bound

[TABLE]

We now estimate the right-hand side in the following way. By Corollary 6.1.5, we bound

[TABLE]

where we are using that $\sum\ell(R/I_{j})=\ell(S/I)$ and $\sum\ell(R/(I_{j},x_{1}))=\ell(S/(I,x_{1}))$. Similarly, using Lemma 7.1.1 we obtain the bound

[TABLE]

where we have used that $\ell(R/I_{c})\geq\ell(R/I_{c}+(x_{1}))$ (note that the equality may happen, namely when $I_{c}$ contains $x_{1}$). By combining the two multiplicity estimates and noting that $\ell(R/I_{0})=\ell(S/(I,x_{2}))$ and $\ell(R/(I_{0},x_{1}))=\ell(S/(I,x_{1},x_{2}))$, we obtain that

[TABLE]

This completes the proof when $I\subseteq S$ is an ${\mathfrak{m}}$-primary monomial ideal.

We now prove the general case. We first note that $\operatorname{e}(I)\leq\operatorname{e}(\operatorname{in}_{\prec}(I))$, $\ell(S/I)=\ell(S/\operatorname{in}_{\prec}(I))$, and $\ell(S/(\operatorname{in}_{\prec}(I),x_{i}))\geq\ell(S/\operatorname{in}_{\prec}(I,x_{i}))=\ell(S/(I,x_{i}))$ for any monomial order $\prec$ since we have a containment $\operatorname{in}_{\prec}(I)\subseteq\operatorname{in}_{\prec}(I,x_{i})$. We next claim that if $\prec$ is the lexicographic order with $x_{2}\prec x_{1}\prec x_{0}$ then $\ell(S/(I,x_{1},x_{2}))=\ell(S/\operatorname{in}_{\prec}(I)+(x_{1},x_{2})).$ To see this, note that

[TABLE]

But since the $\prec$-leading term of $x_{0}^{n}+f(x_{0},x_{1},x_{2})$, for $f(x_{0},x_{1},x_{2})\in(x_{1},x_{2})$, is precisely $x_{0}^{n}$, we see that $\ell(S/(I,x_{1},x_{2}))=\ell(S/\operatorname{in}_{\prec}(I)+(x_{1},x_{2}))$ as wanted. Therefore, we bound

[TABLE]

where the second inequality follows from the already established monomial case. This completes the proof of the theorem. ∎

Remark 7.1.3*.*

If we compare Theorem 6.1.3 (for $S=k[[x_{0}]]$ and $n=2$) with Theorem 7.1.2, we see that there are two improvements: first, Theorem 6.1.3 only applies to monomial ${\mathfrak{m}}$-primary ideals while Theorem 7.1.2 applies to all ${\mathfrak{m}}$-primary ideals; secondly, the upper bound in Theorem 6.1.3 is

[TABLE]

while the bound in Theorem 7.1.2 is

[TABLE]

which is sharper. This latter improvement will be crucial in our applications later.

7.2. Elliptic and Rational polygonal cones

In this subsection, we provide purely algebraic proofs for [Mum77, Proposition 3.18 and Proposition 3.19] in arbitrary characteristic. We first introduce definitions following [Mum77, 3.17 and 3.18]. First, in $\mathbb{P}^{n-1}$ take a generic $n$-gon $\cup_{i=0}^{n}\overline{P_{i}P_{i+1}}$ where $P_{n+1}=P_{0}$, its affine cone in $\mathbb{A}^{n}$ is called an elliptic polygonal $n$-cone. Similarly, take $n$ generic line segments $\cup_{i=0}^{n-1}\overline{P_{i}P_{i+1}}$ in $\mathbb{P}^{n}$, its affine cone in $\mathbb{A}^{n+1}$ is called a rational polygonal $n$-cone.111111It seems to the authors that there are some notational inaccuracies in [Mum77, 3.18], when defining rational polygonal $n$-cones.

Proposition 7.2.1**.**

Elliptic polygonal $n$-cones are semistable if and only if $1\leq n\leq 6$. Specifically, these are the singularities:

[TABLE]

In fact, when $1\leq n\leq 5$ these singularities are stable, while if $n=6$, the singularity is semistable but not stable.

Remark 7.2.2*.*

In the degenerate cases $n=1$ corresponds to glueing a pair of transversal lines in a plane together and $n=2$ corresponds to glueing two planes to each other along a pair of transversal lines. For $n=1$, the equation is $x^{2}-xyz+y^{3}$, which, when $\operatorname{char}(k)\neq 2$, is equivalent to the equation $x^{2}-y^{3}-y^{2}z^{2}$ after a change of variables. Similarly, for $n=2$, the equation is $x^{2}-xyz$, which, when $\operatorname{char}(k)\neq 2$, is equivalent to the equation $x^{2}-y^{2}z^{2}$ after a change of variables. These latter equations are Mumford’s descriptions of these singularities over $\mathbb{C}$, see [Mum77, 3.17].

Proof of Proposition 7.2.1.

First of all, if $n\geq 7$, then $\operatorname{e}(R)=n\geq 7$ and thus $R$ is unstable by Remark 2.2.3. For $n=1$, we may assign weights $\deg(x)=4$, $\deg(y)=3$, $\deg(z)=1$, its initial ideal is $(x^{2}-xyz)$. Thus by Corollary 2.4.7, the $n=1$ case reduces to the $n=2$ case.

Now we suppose $n=2$. Let $I\subseteq R[[T]]$ be an $(x,y,z,T)$-primary ideal. By tensoring with $R[[T]]/I$ the short exact sequence

[TABLE]

we obtain the inequality

[TABLE]

For now on, let $\prec$ be any monomial order on $R[[T]]=k[[x,y,z,T]]$. By using the inclusion-exclusion formula of Corollary 6.1.2, we have that

[TABLE]

Next, note that $\operatorname{e}(I,R[[T]])=\operatorname{e}(I,R[[T]]/(x-yz))+\operatorname{e}(I,R[[T]]/(x))$. Since $R[[T]]/(x-yz)$ and $R[[T]]/(x)$ are both isomorphic to $k[[y,z,T]]$, by applying Theorem 7.1.2 to the initial ideal of the image of $I$ inside $R[[T]]/(x-yz)$ and $R[[T]]/(x)$ respectively, we bound

[TABLE]

[TABLE]

By collecting all these bounds, we find that

[TABLE]

Since $\operatorname{in}_{\prec}(I+(x-yz))\subseteq\operatorname{in}_{\prec}(I+(x,yz))$ and $\operatorname{in}_{\prec}(I+(x))\subseteq\operatorname{in}_{\prec}(I+(x,yz))$ for any monomial order $\prec$, we get that the terms paired in the first four lines of the formula are all nonnegative. Therefore we obtain that

[TABLE]

At this point, we take $\prec$ to be the lexicographic order with $x\prec y\prec z\prec T$. By the same argument as in the proof of Theorem 7.1.2, this monomial order forces that

[TABLE]

It follows that $6\ell(R[[T]]/I)-\operatorname{e}(I,R[[T]])\geq 4\ell\left(R[[T]]/(I+(x,y,z))\right)>0$, so $R$ is stable.

Finally, we suppose that $3\leq n\leq 6$. Since the defining ideal is a monomial ideal, by Proposition 2.1.6, in order to show $R$ is stable or semistable, it is enough to consider monomial ideals $I\subseteq R[[T]]$. Note that $P_{i}+P_{j}=(x_{1},\ldots,x_{n})$ if $j\neq i-1,i,i+1$, thus by Corollary 6.1.2, we obtain the formula

[TABLE]

where for convenience we denote $P_{n+1}\coloneqq P_{1}$.

On the other hand, each $R[[T]]/P_{i}$ is isomorphic to a power series ring over $k$ in three variables. Thus by Theorem 7.1.2, we bound

[TABLE]

Summing these up, we obtain the inequality

[TABLE]

After comparing with (7.2.2.1) we immediately see that $\operatorname{e}(I,R[[T]])<6\ell(R[[T]])$ when $n\leq 5$, so $R$ is stable in this case. On the other hand, when $n=6$ we only get that $\operatorname{e}(I,R[[T]])\leq 6\ell(R[[T]])$, showing that $R$ is semistable. Since equality holds for the maximal ideal, we see that $R$ is not stable for $n=6$. ∎

The result on rational polygonal $n$-cones is proved similarly.

Proposition 7.2.3**.**

Rational polygonal $n$-cones are semistable if and only if $2\leq n\leq 6$. Specifically, these are $R=k[[x_{1},\ldots,x_{n+1}]]/(P_{1}\cap\cdots\cap P_{n}),$ where $P_{1}=(x_{1},\ldots,x_{n-1}),P_{2}=(x_{2},\ldots,x_{n}),\ldots,P_{n}=(x_{n},x_{n+1},x_{1},\ldots,x_{n-3})$. Moreover, these singularities are stable if and only if $2\leq n\leq 5$.

Proof.

Since $\operatorname{e}(R)=n$, $R$ is unstable whenever $n\geq 7$ by Remark 2.2.3. It remains to prove that $R$ is stable for $2\leq n\leq 5$ and is semistable but not stable when $n=6$.

We now assume that $2\leq n\leq 6$. The strategy is similar to the proof of Proposition 7.2.1. First of all, since the defining ideal is a monomial ideal, by Proposition 2.1.6, in order to show $R$ is stable, it is enough to consider monomial ideals $I\subseteq R[[T]]$. Note that $P_{i}+P_{j}=(x_{1},\ldots,x_{n+1})$ if $j\neq i-1,i,i+1$, thus by Corollary 6.1.2, we have that

[TABLE]

since the terms equal to $\ell(R[[T]]/I+(x_{1},\ldots,x_{n+1}))$ cancel each other.

Next, each $R[[T]]/P_{i}$ is isomorphic to a power series ring over $k$ in three variables. Thus by Theorem 7.1.2, we have that

[TABLE]

Putting these together, we obtain that

[TABLE]

Immediately, we obtain that $\operatorname{e}(I,R[[T]])-6\ell(R[[T]]/I)<0$ if $n\leq 5$, showing that $R$ is stable in this case. On the other hand, for $n=6$ we only get that $\operatorname{e}(I,R[[T]])\leq 6\ell(R[[T]]/I)$. Indeed, equality holds for the maximal ideal of $R[[T]]$, so $R$ is semistable but not stable. ∎

Example 7.2.4**.**

Let $V_{n}\coloneqq k[x,y]^{(n)}$ be the $n$th Veronese subring of $k[x,y]$ and let $\widehat{V_{n}}$ denote the completion of $V_{n}$ at its unique homogeneous maximal ideal. Then

[TABLE]

Proof.

First of all, if $n\leq 2$, then $\widehat{V_{n}}$ is a rational double point and thus Lech-stable by Corollary 3.1.2. Secondly, for $3\leq n\leq 6$, note that the defining ideal of $V_{n}$ is the two by two minors of the matrix

[TABLE]

It is easy to see that, under the lexicographic monomial order $z_{1}>z_{2}>\cdots>z_{n+1}$, the initial ideal defines the rational polygonal $n$-cone up to a change of variables. Therefore by Corollary 2.4.8 and Proposition 7.2.3, we know that $\widehat{V_{n}}$ is stable when $3\leq n\leq 5$ and is semistable but not stable when $n=6$. Finally, if $n\geq 7$ then $\operatorname{e}(\widehat{V_{n}})=n\geq 7$ so $\widehat{V_{n}}$ is not semistable by Remark 2.2.3, while if $n\geq 17$, then $\operatorname{e}(\widehat{V_{n}})=n\geq 17$ so $\widehat{V_{n}}$ is not lim-stable by Corollary 4.3.5. ∎

Remark 7.2.5*.*

We do not know whether $\widehat{V_{n}}$ is lim-stable when $7\leq n\leq 16$, nor do we know the values of $\operatorname{c^{\infty}_{LM}}(\widehat{V_{n}})$ for any $n\geq 7$. We do not know a good method to compute or estimate the (higher) Lech–Mumford constants for quotient singularities in general.

7.3. A more general conjecture

The discussed sharp Lech-type inequalities in dimension two and three suggest that in general there should be a sharp improvement, in terms of the colengths of the projections, of Lech’s inequality for monomial ideals in power series rings. The symmetry further implies that the coefficients for projections of the same dimension should be equal, leading us naturally to Conjecture 7.3.1.

Let us start with notation. Let $(x)_{d}\coloneqq x(x-1)\cdots(x-d+1)$. The unsigned Stirling numbers of the first kind $\genfrac{[}{]}{0.0pt}{}{d}{k}$ and the Stirling numbers of the second kind $\genfrac{\{}{\}}{0.0pt}{}{d}{k}$ arise from the decomposition of $(x)_{d}$ in the standard basis $x^{k}$ and vice versa:

[TABLE]

Conjecture 7.3.1**.**

Let $(R,{\mathfrak{m}})=k[[x_{0},\ldots,x_{d}]]$ and $I\subseteq R$ be an ${\mathfrak{m}}$-primary monomial ideal. Then

[TABLE]

Remark 7.3.2*.*

We note that in dimension three (i.e., $d=2$), Conjecture 7.3.1 holds, and this is exactly Theorem 7.1.2. We also point out that, when $d\geq 3$, the coefficients in Conjecture 7.3.1 are not necessarily integers. For example, if $d=3$, then the Stirling numbers are $1,7,6,1$. Thus the conjecture becomes

[TABLE]

Our conjecture can be compared to the refinement of Lech’s inequality that was conjectured in [HSV19].

Conjecture 7.3.3** ([HSV19, Conjecture 2.4]).**

Let $k$ be an infinite field and $(R,{\mathfrak{m}})=k[[x_{0},\dots,x_{d}]]$. Set $R_{i}=R/(\ell_{1},\ldots,\ell_{i})$ for general linear forms $\ell_{1},\ldots,\ell_{i}\in{\mathfrak{m}}$. Then for all ${\mathfrak{m}}$-primary ideals $I\subseteq R$, we have

[TABLE]

Remark 7.3.4*.*

In [HSV19], the conjecture was stated for all local rings, but it was shown in [HSV19, Section 3] that it reduces to the case of polynomial rings (or power series rings) over an infinite field.

We will next show that Theorem 7.1.2 implies Conjecture 7.3.3 in dimension three, thus recovering the main result of [HSV19].

Theorem 7.3.5**.**

Conjecture 7.3.1 implies Conjecture 7.3.3. In particular, Conjecture 7.3.3 holds in dimension three.

Proof.

Conjecture 7.3.3 was reduced to monomial ideals in [HSV19, Section 3]. Henceforth we assume that $I\subseteq R$ is an ${\mathfrak{m}}$-primary monomial ideal and we work with the left-hand side of Conjecture 7.3.3. Since $\ell_{1},\dots,\ell_{i}$ are general linear forms, we know that

[TABLE]

for any choice of $1\leq a_{1}<\cdots<a_{i}\leq d$. By Conjecture 7.3.1, we then have

[TABLE]

where $b_{j}$ take values in $\{1,\ldots,d\}\setminus\{a_{1},\ldots,a_{i}\}$. Averaging these inequalities over various choices of $a_{i}$ yields that

[TABLE]

where the equality above used the combinatorial identity $\binom{d}{i}\binom{d-i}{k}=\binom{d-k}{i}\binom{d}{k}$. Therefore

[TABLE]

We need to show that this last expression is equal to $(d+1)!\ell(R/I)$, which amounts to show that the coefficient in the parenthesis is zero for $k<d$. But by the definition of the Stirling numbers we have the identity

[TABLE]

Since $\{(x)_{n}\}_{n}$ are linearly independent, it follows that in the last expression above, the coefficient of $(x)_{k+1}$ is zero for $k<d$, which is exactly what we need. The last conclusion then follows from Theorem 7.1.2. ∎

An application of Theorem 6.1.3 gives us the following weaker version of Conjecture 7.3.3.

Proposition 7.3.6**.**

Let $k$ be an infinite field and $(R,{\mathfrak{m}})=k[[x_{0},\dots,x_{d}]]$. Let $R_{1}=R/(\ell_{1})$ for a general linear form $\ell_{1}\in{\mathfrak{m}}$. Then for all ${\mathfrak{m}}$-primary ideals $I\subseteq R$, we have

[TABLE]

Moreover, the inequality is strict if $d\geq 2$.

Proof.

By the same argument as in [HSV19, Section 3], we may assume that $I\subseteq R$ is an ${\mathfrak{m}}$-primary monomial ideal. Since $R_{1}=R/(\ell_{1})$ for a general linear form $\ell_{1}$, we know that $\operatorname{e}(IR_{1})\leq\operatorname{e}(I,R/(x_{i}))$ for every $i$. Thus, after applying Theorem 6.1.3 to $R/(x_{i})$ for $i=1,\ldots,d$, we obtain that

[TABLE]

Summing these up and noting that each $\ell(R/I+(x_{j_{1}},\ldots,x_{j_{k}}))$ will appear exactly $k$ times (one for each $R/(x_{j_{n}})$) we obtain the inequality

[TABLE]

Since $\genfrac{[}{]}{0.0pt}{}{d+1}{d}=\binom{d+1}{2}$, we may combine $(\ref{eqn: 7.3.6 bound})$ with the bound on $\operatorname{e}(I)$ given by Theorem 6.1.3 to derive the inequality

[TABLE]

Since this inequality is strict if $d\geq 2$ by Theorem 6.1.3, it remains to show that the right-hand side of the inequality is no greater than $\ell(R/I)$. We will use the method of Proposition 6.1.1. Since

[TABLE]

it is easy to deduce that

[TABLE]

By the corresponding inclusion-exclusion formula (see the proof of Proposition 6.1.1) we may now show that

[TABLE]

This completes the proof. ∎

7.4. Three points in $\mathbb{P}^{2}$

We believe that there exists a local ring $R$ such that

[TABLE]

As an evidence, we will show that the coordinate ring $R$ of three points in $\mathbb{P}^{2}$ is a one-dimensional local ring $(R,{\mathfrak{m}})$ such that $\operatorname{c_{LM}}(R)>\operatorname{c_{LM}}(R[[U]])>\operatorname{c_{LM}}(R[[U,V]])$. It turns out that even in this seemingly simple case, estimating $\operatorname{c_{LM}}(R[[U,V]])$ is not easy and requires nontrivial new estimates on the mixed multiplicities in a power series ring in two variables. We have not been able to progress this example further due to the lack of such estimates in higher dimensions.

We begin with the following variation of Lemma 7.1.1.

Lemma 7.4.1**.**

Let $(R,{\mathfrak{m}})=k[[x,y]]$ and let $I\subseteq J\subseteq R$ be two ${\mathfrak{m}}$-primary monomial ideals. Then we have

[TABLE]

Proof.

By [HS06, Exercise 17.8], we know that $\operatorname{e}(I|J)=\ell(R/IJ)-\ell(R/I)-\ell(R/J)$, so it suffices to show that $\ell(R/IJ)\leq\frac{3}{2}\ell(R/I)+4\ell(R/J)$. We now follow the notation of the proof of Lemma 7.1.1 and express

[TABLE]

Therefore it suffices to show that

[TABLE]

We demonstrate this inequality using the elementary inequality $\min\{a+b,c+d\}\leq(a+b+c+d)/2$ and grouping the terms in the following way:

[TABLE]

Direct inspection shows that each $r_{i}$ appears with coefficient at most $3/2$ and each $s_{i}$ appears with coefficient at most $3$. Thus $L\leq\frac{3}{2}\sum_{j=0}^{c}r_{j}+3\sum_{j=0}^{c}s_{j}$ as desired. ∎

We have the following variant of Theorem 7.1.2.

Proposition 7.4.2**.**

Let $(S,{\mathfrak{m}})=k[[x,y,z]]$ and let $I\subseteq S$ be an ${\mathfrak{m}}$-primary ideal. Then

[TABLE]

Proof.

The strategy is completely similar to the proof of Theorem 7.1.2. We first handle the case that $I$ is a monomial ideal. We set $R\coloneqq k[[x,y]]$ and consider monomial ideals $I_{j}\coloneqq\{m\in R\mid mz^{j}\in I\}.$ By Proposition 2.1.12, we have

[TABLE]

Using Lemma 7.4.1 for the mixed multiplicity terms and Lech’s inequality (Theorem I) for the usual multiplicity terms, we bound the right-hand side of (7.4.2.1) by

[TABLE]

This completes the proof when $I$ is a monomial ideal.

To prove the general case, we note that for any monomial order $\prec$ we have the containment $\operatorname{in}_{\prec}(I)\subseteq\operatorname{in}_{\prec}(I,z)$ and thus $\operatorname{e}(I)\leq\operatorname{e}(\operatorname{in}_{\prec}(I))$, $\ell(S/I)=\ell(S/\operatorname{in}_{\prec}(I))$, and $\ell(S/(\operatorname{in}_{\prec}(I),z))\geq\ell(S/\operatorname{in}_{\prec}(I,z))=\ell(S/(I,z))$. It follows that

[TABLE]

where the second inequality follows from the already established monomial case. ∎

We are now ready to study the (higher) Lech–Mumford constants of the coordinate ring of three points in $\mathbb{P}^{2}$.

Example 7.4.3**.**

Let $(R,{\mathfrak{m}})=k[[x,y,z]]/(x,y)\cap(y,z)\cap(x,z)$. Then we have

[TABLE]

Proof.

Since $\dim(R)=1$, $\operatorname{c_{LM}}(R)=\operatorname{e}(R)=3$ follows from Proposition 2.1.3(4). Next we verify that $\operatorname{c_{LM}}(R[[U]])=3/2$. Taking $I=(x,y,z,U)$ shows that $\operatorname{c_{LM}}(R[[U]])\geq 3/2$, so it suffices to verify that $\operatorname{e}(I)\leq 3\ell(R[[U]]/I)$ for all monomial ${\mathfrak{m}}+(U)$-primary ideals by Proposition 2.1.6. Now by Corollary 6.1.2 and Corollary 6.1.5, we have

[TABLE]

This completes the proof that $\operatorname{c_{LM}}(R[[U]])=3/2$.

Finally, we estimate $\operatorname{c_{LM}}(R[[U,V]])$. We set $S\coloneqq k[[x,y,z,U,V]]$. By Proposition 2.1.6 it suffices to consider monomial $({\mathfrak{m}}+(U,V))$-primary ideals $I\subseteq S$. By Corollary 6.1.2, we have

[TABLE]

where $P_{1}=(x,y)S$, $P_{2}=(y,z)S$, $P_{3}=(x,z)S$. Since $S/P_{i}$ is a power series ring in three variables, by Proposition 7.4.2 we have

[TABLE]

Since $\operatorname{e}(I)=\sum_{i=1}^{3}\operatorname{e}(I,S/P_{i})$, it follows that

[TABLE]

Therefore $\operatorname{c_{LM}}(R[[U,V]])\leq 3!\times 15/2=5/4$ as wanted. ∎

Remark 7.4.4*.*

One sees a drastic difference between the (higher) Lech–Mumford constants of $R=k[[x,y,z]]/(x,y)\cap(y,z)\cap(x,z)$ and $R^{\prime}=k[[x,y]]/(x^{3}+y^{3})$: while $\operatorname{c_{LM}}(R[[t_{1},t_{2}]])\leq 5/4$ (and thus $\operatorname{c^{\infty}_{LM}}(R)\leq 5/4$) by Example 7.4.3, by Remark 4.3.2 one has $\operatorname{c^{\infty}_{LM}}(R^{\prime})\geq 1.26>5/4$ (and thus $\operatorname{c_{LM}}(R^{\prime}[[t_{1},\dots,t_{n}]])>5/4$ for all $n$).

7.5. Semi-log canonical hypersurfaces in dimension two

Two-dimensional semi-log canonical hypersurface singularities over $\mathbb{C}$ are classified, see for example [LR12, Table 1]. Using this classification, we will show in this subsection that most of these singularities are semistable. In fact, we suspect that all of them are semistable, but our techniques cannot handle the case of simple elliptic singularities,121212In [Mum77, page 79, Case 3 a)], it was stated without proof that cones over nonsingular elliptic curves are semistable. We have not been able to verify this claim. A concrete example that we do not know how to show semistability is the singularity $\mathbb{C}[[x,y,z]]/(x^{3}+y^{3}+z^{3})$. see Conjecture 7.5.2.

We start by recalling the classification. In what follows, all singularities are hypersurfaces of dimension two over $\mathbb{C}$, i.e., (up to completion) they can be written as $\mathbb{C}[[x,y,z]]/(f)$.

1. (0)

Canonical singularities are either regular or ADE type singularities, in particular, they are Lech-stable by Theorem 3.1.5.

Next, log canonical but not canonical singularities consists of the next four families, the first three are simple elliptic singularities, and the fourth family is called cusp singularities:

1. (1)

$X_{1,0}\colon$ $f=x^{2}+y^{4}+z^{4}+\lambda xyz$, $\lambda^{4}\neq 64$, 2. (2)

$J_{2,0}\colon$ $f=x^{2}+y^{3}+z^{6}+\lambda xyz$, $\lambda^{6}\neq 432$, 3. (3)

$T_{3,3,3}\colon$ $f=x^{3}+y^{3}+z^{3}+\lambda xyz$, $\lambda^{3}\neq-27$, 4. (4)

$T_{p,q,r}$: $f=xyz+x^{p}+y^{q}+z^{r}$, $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}<1$.

Lastly, semi-log canonical but not log canonical singularities are classified as follows. First, we have normal crossings $A_{\infty}$ and pinched point $D_{\infty}$:

1. (5)

$A_{\infty}$: $f=x^{2}+y^{2}$, 2. (6)

$D_{\infty}$: $f=x^{2}+y^{2}z$.

The rest semi-log canonical singularities are degenerate cusps:

1. (7)

$T_{2,\infty,\infty}$: $f=x^{2}+y^{2}z^{2}$, 2. (8)

$T_{2,q,\infty}$: $f=x^{2}+y^{2}(z^{2}+y^{q-2})$, $q\geq 3$, 3. (9)

$T_{\infty,\infty,\infty}$: $f=xyz$, 4. (10)

$T_{p,\infty,\infty}$: $f=xyz+x^{p}$, $p\geq 3$, 5. (11)

$T_{p,q,\infty}$: $f=xyz+x^{p}+y^{q}$, $q\geq p\geq 3$.

Theorem 7.5.1**.**

All the singularities in the families (4)–(11) of the classification exhibited above are semistable.

Proof.

We first handle the families (5)–(11). $A_{\infty}$ and $D_{\infty}$ are semistable by Theorem 3.1.5 (they are even Lech-stable). $T_{2,\infty,\infty}$ is semistable by Proposition 7.2.1 $n=2$ case (see Remark 7.2.2). For $T_{2,q,\infty}$, by assigning $x,y,z$ weights $4,3,1$ respectively, the initial term is $x^{2}+y^{2}z^{2}$ which is semistable by the $T_{2,\infty,\infty}$ case. Thus $T_{2,q,\infty}$ is semistable by Corollary 2.4.7. $T_{\infty,\infty,\infty}$ is semistable by Theorem 6.2.1. For $T_{p,\infty,\infty}$ and $T_{p,q,\infty}$, we can assign $x,y,z$ weights $2,2,1$ so that the initial term is $xyz$, which is semistable by the $T_{\infty,\infty,\infty}$ case. Thus $T_{p,\infty,\infty}$ and $T_{p,q,\infty}$ are also semistable by Corollary 2.4.7.

We now handle the family (4), which follows from a degeneration very similar to above. We may assume $p\leq q\leq r$. If $p=2$ and $q=3$, then $r\geq 7$. So by assigning $x,y,z$ weights $3,2,1$, the initial term is $xyz+x^{2}+y^{3}$, which is semistable by Proposition 7.2.1 $n=1$ case. Thus $T_{2,3,r}$ is semistable by Corollary 2.4.7. If $p=2$ and $q=4$, then $r\geq 5$. By assigning $x,y,z$ weights $2,1,1$, the initial term is $xyz+x^{2}+y^{4}$, but then assigning $x,y,z$ weights $3,2,1$, the initial term is $xyz+x^{2}$, which is semistable by Proposition 7.2.1 $n=2$ case. Thus $T_{2,4,r}$ is semistable by Corollary 2.4.7 (applied twice). If $p=2$ and $q,r\geq 5$, then by assigning $x,y,z$ weights $2,1,1$, the initial term is $xyz+x^{2}$, which is semistable by Proposition 7.2.1 $n=2$ case. Thus $T_{2,q,r}$ is semistable by Corollary 2.4.7. Finally, if $p\geq 3$, then by assigning $x,y,z$ weights $1,1,1$, the initial term becomes either $T_{3,\infty,\infty}$ or $T_{3.3,\infty}$, which are semistable by the already established families (10) and (11). Thus they are semistable by Corollary 2.4.7 again. This completes the proof. ∎

Since our technique is essentially to degenerate the singularities to the monomial case where explicit computations are available, it does not seem to work for the simple elliptic singularities. However, we suspect that all two-dimensional semi-log canonical hypersurface singularities are semistable, see also [Mum77, page 79]. The following conjecture is all that remains to show.

Conjecture 7.5.2**.**

All the singularities in the families (1)–(3) above are semistable.

More generally, we do not know whether the following could be true.

Question 7.5.3*.*

Suppose $R$ is essentially of finite type over a field of characteristic zero that is ${\mathbb{Q}}$-Gorenstein and log canonical (or merely semi-log canonical), and $\operatorname{e}(R)\leq\dim(R)+1$. Then is $R$ semistable?