Uniform Symbolic Topologies in Normal Toric Rings

TL;DR

This paper establishes a uniform bound for symbolic powers in normal toric rings, explicitly computes this bound using polyhedral data, and links it to the F-signature in positive characteristic.

Contribution

It provides an explicit computation of the uniform integer D for symbolic power containment in normal toric rings, connecting algebraic and geometric data.

Findings

Computed a uniform integer D for symbolic power containment

Explicit formula for D in terms of polyhedral cone data

Linked the uniform bound to the F-signature in positive characteristic

Abstract

Given a normal toric algebra $R$, we compute a uniform integer $D = D(R) > 0$ such that the symbolic power $P^{(D N)} \subseteq P^N$ for all $N >0$ and all monomial primes $P$. We compute the multiplier $D$ explicitly in terms of the polyhedral cone data defining $R$. In this toric setting, we draw a connection with the F-signature of $R$ in positive characteristic.

Equations

$$P^{(Dr)}\subseteq P^{(D(r-1)+1)}\subseteq P^r,$$

$$P^{(U(r-1)+1)}\subseteq P^r$$

$$C$$

$$C^{\vee }$$

$$\dim (F)+\dim (F^{\ast })=\dim (N_{\mathbb{R}})=\dim (M_{\mathbb{R}}).$$

$${\varphi }_F:R_{\mathbb{F}}=\mathbb{F}[C^{\vee }\cap M]\twoheadrightarrow \mathbb{F}[F^{\ast }\cap M],{\varphi }_F({\chi }^m)$$

$$P_{F}=(\{\chi^{m}\colon\mbox{$m\in C^{\vee}\cap M$ and the integer $\langle m,v_{F}\rangle>0$}\})R_{\mathbb{F}}.$$

$$0\to M^{\prime }\stackrel{\varphi }{\to }\underset{\rho \in \Sigma (1)}{⨁}\mathbb{Z}{P}_{\rho }\to \mathrm{Cl}(R_{\mathbb{F}}^{\prime })\to 0,$$

$$P^{(D(r-1)+1)}\subseteq P^r$$

$$P^{(T(r-1)+1)}\subseteq P^r$$

$$E\le ⟨\ell ,v_F⟩=\sum _{m\in \mathcal{B}}{a}_m⟨m,v_F⟩=\sum _{m\in S}{a}_m⟨m,v_F⟩\le \sum _{m\in S}{a}_m\cdot D^{\prime }⟹\sum _{m\in S}{a}_m\ge \lceil E/D^{\prime }\rceil .$$

$$R=\frac{\mathbb{F}[x_1,\dots ,x_n,z]}{(z^E-x_1\cdots x_n)}.$$

$$\{e_n,\text{mbox}E\cdot e_i+e_n:i=1,\dots ,n-1\}\subseteq {\mathbb{Z}}^n,$$

$$s(R):=\underset{e\to \infty }{\mathrm{limsup}}\frac{a_e}{p^{ed}}=\underset{e\to \infty }{\lim }\frac{a_e}{p^{ed}},0\le s(R)\le 1.$$

$$\mathcal{A}=\bigcup _{1\le j\le k}{A}_j,\text{mbox}where{A}_1$$

$$\text{mbox}andforeach2\le j\le k,A_j$$

$$\mathcal{B}=\left\{e_{m_1}^{\ast }+\sum _{j=1}^k\sum _{\ell =1}^{m_j-1}{a}_{j,\ell }\cdot e_{d(j-1)+\ell }^{\ast }:0\le \sum _{\ell =1}^{m_j-1}{a}_{j,\ell }\le E_j\text{mbox}for1\le j\le k\right\}.$$

Full text

2010 Mathematics Subject Classification: 13H10, 14C20, 14M25.

Keywords: symbolic powers, rational singularity, toric ring, monomial primes, F-signature, Segre-Veronese.

1. Introduction and Conventions for the Paper

Given a Noetherian commutative ring $R$, when is there an integer $D$, depending only on $R$, such that the symbolic power $P^{(Dr)}\subseteq P^{r}$ for all prime ideals $P\subseteq R$ and all positive integers $r$?111For a definition of symbolic powers of prime and radical ideals, see [8]. In short, when does $R$ have uniform symbolic topologies on primes [18]? Moreover, can we effectively compute the multiplier $D$ in terms of simple data about $R$? In this paper, we answer this last question in the setting of torus-invariant primes in a normal toric (or semigroup) algebra.

The Ein-Lazarsfeld-Smith Theorem [10], as extended by Hochster and Huneke [16], says that if $R$ is a $d$-dimensional regular ring containing a field, and $D=\max\{1,d-1\}$, then $Q^{(Dr)}\subseteq Q^{r}$ for all radical ideals $Q\subseteq R$ and all $r>0$.222This result has been extended to all excellent regular rings, even in mixed characteristic, by Ma-Schwede [20]. To what extent does this theme ring true for non-regular rings? Under mild stipulations, a local ring $R$ regular on the punctured spectrum has uniform symbolic topologies on primes [17, Cor. 3.10], although explicit values for $D$ remain elusive. Far less is known for rings with non-isolated singularities.

This manuscript approaches the above questions for the coordinate rings of normal affine toric varieties. Such algebras are combinatorially-defined, finitely generated, Cohen-Macaulay, normal, and have rational singularities. We deduce an explicit multiplier $D$ such that for any torus-invariant prime $P$ in $R$, $P^{(Dr)}\subseteq P^{r}\mbox{ for all }r>0;$ see Theorems 1.1 and 3.1 for precise statements. Prior work towards explicit $D$ includes the papers [26, Table 3.3] and [28, Thm. 1.2] for ADE rational surface singularities and for select domains with non-isolated singularities, respectively; see also [8, Thm. 3.29, Cor. 3.30] for module-finite direct summands of affine polynomial rings.

To state our main results in a special case, fix a ground field $\mathbb{F}$. We fix a full-dimensional pointed convex polyhedral cone $C\subseteq\mathbb{R}^{n}$ generated by a finite set $G\subseteq\mathbb{Z}^{n}$, and its dual $C^{\vee}\subseteq\mathbb{R}^{n}$. Let $R_{\mathbb{F}}$ be the semigroup algebra of the semigroup $C^{\vee}\cap\mathbb{Z}^{n}$. This toric $\mathbb{F}$-algebra associated to $C$ is a normal domain of finite type over $\mathbb{F}$ [7, Thm. 1.3.5] with an $\mathbb{F}$-basis of monomials $\{\chi^{\ell}\colon\ell\in C^{\vee}\cap\mathbb{Z}^{n}\}.$ An ideal of $R_{\mathbb{F}}$ is monomial (or torus-invariant) if it is generated by a subset of these monomials. In what follows, we use $\bullet$ to denote dot products in $\mathbb{R}^{n}$.

Theorem 1.1** (Cf., Theorem 3.1).**

Suppose $C\subseteq\mathbb{R}^{n}$ is a full-dimensional pointed polyhedral cone as above. Set $D:=\max_{w\in\mathcal{B}}\left(w\bullet v_{G}\right)\in\mathbb{Z}_{>0}$, where $\mathcal{B}$ is a generating set for the semigroup $C^{\vee}\cap\mathbb{Z}^{n}$ and $v_{G}\in\mathbb{Z}^{n}$ is the sum of any $($finite$)$ set $G$ of vectors in $\mathbb{Z}^{n}$ generating $C$. Then

[TABLE]

for all $r>0$ and all monomial prime ideals $P$ in the toric ring $R_{\mathbb{F}}=\mathbb{F}[C^{\vee}\cap\mathbb{Z}^{n}]$.

We get the best result in Theorem 1.1 by taking $G$ to consist of the unique set of primitive generators for $C$, in which case we write $v_{C}$ in place of $v_{G}$; see Section 2. For example, if $R_{\mathbb{F}}$ is the $E$-th Veronese subring of a polynomial ring of finite type over $\mathbb{F}$, then our $D=E$ in Theorem 1.1. As another example, for the Segre product of Veronese rings of respective degrees $E_{1},\ldots,E_{k}$, we can take $D=\sum_{i=1}^{k}E_{i}$ in Theorem 1.1; see Theorem 4.8.

The next result covers select non-monomial primes. We note that a toric algebra satisfying the additional hypothesis below is called simplicial; see the discussion around Theorem 2.5.

Theorem 1.2** (Cf., Theorem 3.2).**

With notation as in Theorem 1.1, assume moreover that the divisor class group $\operatorname{Cl}(R_{\mathbb{F}})$ is finite. Set $U:=\operatorname{lcm}\{\max_{w\in\mathcal{B}}(w\bullet v_{C}),\#\operatorname{Cl}(R_{\mathbb{F}})\}\in\mathbb{Z}_{>0},$ where $\mathcal{B}$ and $v_{C}$ are as in Theorem 1.1. Then

[TABLE]

for all $r>0$, all monomial primes in $R_{\mathbb{F}}$, and all height one primes in $R_{\mathbb{F}}$.

Tighter containments of the type $I^{(E(r-1)+1)}\subseteq I^{r}$, as in Theorems 1.1 and 1.2 and first promoted by Harbourne, hold for all monomial ideals in an affine polynomial ring over any field [5, Ex. 8.4.5]; see also recent work of Grifo-Huneke [15].

At the end of Section 3, we draw connections between the multiplier $U$ and the so-called F-signature of $R_{\mathbb{F}}$. In Section 4, we discuss the extent to which the multipliers $D$ and $U$ in Theorems 1.1 and 1.2 are sharp.

Conventions: Throughout, $\mathbb{F}$ denotes an arbitrary ground field of arbitrary characteristic. All rings are commutative with identity–indeed, they are normal domains of finite type over $\mathbb{F}$.

Acknowledgements: This paper is part of my Ph.D. thesis at the University of Michigan-Ann Arbor. My thesis adviser Karen E. Smith, along with Daniel Hernández and Jack Jeffries, suggested studying precursors for the ideals $I_{\bullet}(E)$ in the proof of Theorem 3.1. I am grateful for this idea. I acknowledge support from a NSF GRF (Grant No. PGF-031543), NSF RTG grant DMS-0943832, and a 2017 Ford Foundation Dissertation Fellowship. Several computations were performed using the Polyhedra package in Macaulay2 [14] to gain an incisive handle on the polyhedral geometry.

2. Toric Algebra Preliminaries

We review notation and relevant facts from toric algebra, citing Cox-Little-Schenck [7, Ch.1,3,4] and Fulton [12, Ch.1,3]. A lattice is a free abelian group of finite rank. We fix a perfect bilinear pairing $\langle\cdot,\cdot\rangle\colon M\times N\to\mathbb{Z}$ between two lattices $M$ and $N$; this identifies $M$ with $\operatorname{Hom}_{\mathbb{Z}}(N,\mathbb{Z})$ and $N$ with $\operatorname{Hom}_{\mathbb{Z}}(M,\mathbb{Z})$. Our pairing extends to a perfect pairing of finite-dimensional vector spaces $\langle\cdot,\cdot\rangle\colon M_{\mathbb{R}}\times N_{\mathbb{R}}\to\mathbb{R}$, where $M_{\mathbb{R}}:=M\otimes_{\mathbb{Z}}\mathbb{R}$ and $N_{\mathbb{R}}:=N\otimes_{\mathbb{Z}}\mathbb{R}$.

Fix an $N$-rational polyhedral cone and its $M$-rational dual: respectively, for some finite subset $G\subseteq N-\{0\}$ these are closed, convex sets of the form

[TABLE]

By definition, the dimension of a cone in $M_{\mathbb{R}}$ or $N_{\mathbb{R}}$ is the dimension of the real vector subspace it spans; a cone is full(-dimensional) if it spans the full ambient space. A cone in $M_{\mathbb{R}}$ or $N_{\mathbb{R}}$ is pointed (or strongly convex) if it contains no line through the origin. A face of $C$ is a convex polyhedral cone $F$ in $N_{\mathbb{R}}$ obtained by intersecting $C$ with a hyperplane which is the kernel of a linear functional $m\in C^{\vee}$; $F$ is proper if $F\neq C$. When $C$ is both $N$-rational and pointed, so is every face $F$. Each such face $F\neq\{0\}$ has a uniquely-determined set $G_{F}$ of primitive generators. By definition, $v\in N$ is primitive if $\frac{1}{k}\cdot v\not\in N$ for all $k\in\mathbb{Z}_{>1}$.

There is a bijective inclusion-reversing correspondence between faces $F$ of $C$ and faces $F^{*}$ of $C^{\vee}$, where $F^{*}=\{w\in C^{\vee}\colon\langle w,v\rangle=0\mbox{ for all }v\in F\}$ is the face of $C^{\vee}$ dual to $F$ [12, Sec. 1.2]. Under this correspondence, either cone is pointed if and only if the other is full, and

[TABLE]

Fix an arbitrary ground field $\mathbb{F}$ and a cone $C$ as above in $N_{\mathbb{R}}$. The semigroup ring $R_{\mathbb{F}}=\mathbb{F}[C^{\vee}\cap M]$ is the toric $\mathbb{F}$-algebra associated to $C$. This ring $R_{\mathbb{F}}$ is a normal domain of finite type over $\mathbb{F}$ [7, Thm. 1.3.5]. Note that $R_{\mathbb{F}}$ has an $\mathbb{F}$-basis $\{\chi^{m}\colon m\in C^{\vee}\cap M\}$ of monomials, giving $R_{\mathbb{F}}$ an $M$-grading, where $\deg(\chi^{m}):=m$. A monomial ideal (also called an $M$-homogeneous or torus-invariant ideal) in $R_{\mathbb{F}}$ is an ideal generated by a subset of these monomials. When $C^{\vee}$ is pointed, $R_{\mathbb{F}}$ also has a non-canonical $\mathbb{N}$-grading obtained by fixing any group homomorphism $M\to\mathbb{Z}$ taking positive values $C^{\vee}\cap M-\{0\}$. The set $\{\chi^{m}\colon m\in C^{\vee}\cap M-\{0\}\}$ generates the unique homogeneous maximal ideal $\mathfrak{m}$ under this $\mathbb{N}$-grading.

Remark 2.1**.**

In forming the toric algebra $\mathbb{F}[C^{\vee}\cap M]$, there is no loss of generality in assuming $C$ is pointed in $N_{\mathbb{R}}$. Indeed, because $C^{\vee}\cap M=C^{\vee}\cap M^{\prime}$ where M^{\prime}=M\cap\{\mbox{\mathbb{R}$-span of$C^{\vee}$in$M_{\mathbb{R}}}\}, we may replace $M$ by $M^{\prime}$ to assume $C^{\vee}$ is full in $(M^{\prime})_{\mathbb{R}}$. Now, replacing $N$ and $C$ by the duals of $M^{\prime}$ and $C^{\vee}$, we may assume that $C$ is pointed in $N^{\prime}=\operatorname{Hom}_{\mathbb{Z}}(M^{\prime},\mathbb{Z})$. See [7, Thm. 1.3.5] for details.

Fix a face $F$ of a pointed cone $C$: [12, p.53] records a surjective $M$-graded ring map

[TABLE]

Both rings are domains. The monomial prime ideal of $F$, $P_{F}:=\ker(\phi_{F})$, has height equal to $\dim(F)$. Conversely, any monomial prime of $R_{\mathbb{F}}$ corresponds bijectively to a face of $C$.

Lemma 2.2**.**

Fix a face $F$ of a pointed cone $C$, and the monomial prime $P_{F}\subseteq R_{\mathbb{F}}$ above. Let $G_{F}$ be the set of primitive generators of $F$, and set $v_{F}:=\sum_{v\in G_{F}}v\in F\cap N$. Then

[TABLE]

Proof.

First, in defining $\phi_{F}(\chi^{m})$ above, notice we can work with $v\in G_{F}$ without loss of generality. Now, fix $m\in C^{\vee}\cap M$. Then $\langle m,v\rangle\in\mathbb{Z}_{\geq 0}$ for all $v\in C\cap N$. As $\langle\cdot,\cdot\rangle$ is bilinear, (2.0.2) follows since a sum of nonnegative integers is positive if and only if one of the summands is positive. ∎

2.0.1. Hilbert Bases.

First, suppose the pointed cone $C$ from Remark 2.1 is full. Then there is a uniquely-determined minimal generating set $\mathcal{B}$ for $C^{\vee}\cap M$, in the sense that any other generating set contains $\mathcal{B}$. The set $\mathcal{B}$ is called the Hilbert basis of the semigroup, and consists of the irreducible vectors $m\in C^{\vee}\cap M-\{0\}$; a vector $v\in C^{\vee}\cap M$ is irreducible if it cannot be expressed as a sum of two vectors $m\in C^{\vee}\cap M-\{0\}$. See [7, Prop. 1.2.17] and [7, Prop. 1.2.23] for details.

In general, the pointed cone $C$ need not be full. Thus the next proposition is handy.

Proposition 2.3**.**

Let $N^{\prime}_{\mathbb{R}}$ by the $\mathbb{R}$-span of a pointed cone $C\subseteq N_{\mathbb{R}}$. Set $N^{\prime}=N^{\prime}_{\mathbb{R}}\cap N$, and consider $C$ as a full-dimensional cone in $N^{\prime}_{\mathbb{R}}$ $($relabeled as $C^{\prime})$. Let $M^{\prime}=\operatorname{Hom}_{\mathbb{Z}}(N^{\prime},\mathbb{Z})$ be the dual lattice. The toric ring $R_{\mathbb{F}}:=\mathbb{F}[C^{\vee}\cap M]$ is isomorphic to $R_{\mathbb{F}}^{\prime}\otimes_{\mathbb{F}}L$ where the toric ring $R_{\mathbb{F}}^{\prime}:=\mathbb{F}[(C^{\prime})^{\vee}\cap M^{\prime}]$ and $L$ is a Laurent polynomial ring over $\mathbb{F}$. In particular, there is a bijective correspondence between the monomial primes of $R^{\prime}_{\mathbb{F}}$ and $R_{\mathbb{F}}$ given by expansion and contraction of ideals along the faithfully flat ring map $\varphi\colon R^{\prime}_{\mathbb{F}}\hookrightarrow R_{\mathbb{F}}^{\prime}\otimes L=R_{\mathbb{F}}$.

Proof.

Combine [7, Proof of Prop. 3.3.9] with [27, Discussion/Proof preceding Lem. 3.1]. ∎

2.0.2. Toric Divisor Theory

Given a Noetherian normal domain $R$, the divisor class group $\operatorname{Cl}(R)=\operatorname{Cl}(\operatorname{Spec}(R))$ is the free abelian group on the set of height one prime ideals of $R$ modulo relations $\sum_{i=1}^{r}a_{i}P_{i}=0$ when the ideal $\bigcap_{i=1}^{r}P_{i}^{(a_{i})}$ is principal. Note $\operatorname{Cl}(R)$ is trivial if and only if $R$ is a UFD, i.e., all height one primes in $R$ are principal. We recall the following

Lemma 2.4** (Cf., Lem. 1.1 of [27]).**

When every element of $\operatorname{Cl}(R):=\operatorname{Cl(Spec}(R))$ is annihilated by an integer $D>0$, written as $D\cdot\operatorname{Cl}(R)=0$, the symbolic power $P^{(D(r-1)+1)}\subseteq P^{r}$ for all $r>0$ and all prime ideals $P\subseteq R$ of height one.

Working over an algebraically closed field $\mathbb{F}$, fix a pointed cone $C$ as in Remark 2.1 and the pair of rings $R_{\mathbb{F}}$ and $R^{\prime}_{\mathbb{F}}$ as in Proposition 2.3. When $C\neq\{0\}$, each $\rho\in\Sigma(1)$, the collection of rational rays (one-dimensional faces) of $C$, yields a unique primitive generator $u_{\rho}\in\rho\cap N$ for $C$ and a torus-invariant height one prime ideal $P_{\rho}$ in $R^{\prime}_{\mathbb{F}}$; cf., [7, Thm. 3.2.6]. The torus-invariant height one primes generate a free abelian group $\bigoplus_{\rho\in\Sigma(1)}\mathbb{Z}P_{\rho}$ which maps surjectively onto the divisor class group of $R^{\prime}_{\mathbb{F}}$. More precisely, we record the following well-known theorem; see [7, Ch. 4].333This result follows from [7, Prop. 3.3.9, Prop. 4.1.1-4.1.2, Thm. 4.1.3, Exer. 4.1.1-4.1.2, Prop. 4.2.2, Prop. 4.2.6, and Prop. 4.2.7], essentially consolidating what facts we need to bear in mind going forward in the manuscript.

Theorem 2.5**.**

With notation as in Proposition 2.3, let $C\subseteq N_{\mathbb{R}}$ be a pointed cone with primitive generators $\Sigma(1)$ as described above. Then there is a short exact sequence

[TABLE]

*where $\phi(m)=\operatorname{div}(\chi^{m})=\sum_{\rho\in\Sigma(1)}\langle m,u_{\rho}\rangle P_{\rho}$. Furthermore, $\operatorname{Cl}(R_{\mathbb{F}})$ and $\operatorname{Cl}(R^{\prime}_{\mathbb{F}})$ are isomorphic, $\operatorname{Cl}(R_{\mathbb{F}})$ is finite abelian if and only if $C$ is simplicial, and trivial if and only if $C$ is smooth. *

By definition, the cone $C\subseteq N_{\mathbb{R}}$ is simplicial (respectively, smooth) if $C=\{0\}$ or the primitive ray generators form part of an $\mathbb{R}$-basis for $N_{\mathbb{R}}$ (resp., a $\mathbb{Z}$-basis for $N$). We also apply the adjectives simplicial and smooth to the corresponding toric algebra $R_{\mathbb{F}}$ and the toric $\mathbb{F}$-variety $\operatorname{Spec}(R_{\mathbb{F}})$.

3. Main Results

Maintaining all notation conventions from the last section, we now state our main results.

Theorem 3.1**.**

Let $C\subseteq N_{\mathbb{R}}$ be a full pointed rational polyhedral cone. Let $R_{\mathbb{F}}=\mathbb{F}[C^{\vee}\cap M]$ be the associated toric algebra over a field $\mathbb{F}$. Set $D:=\max_{m\in\mathcal{B}}\langle m,v_{C}\rangle,$ where $\mathcal{B}$ is the minimal generating set for $C^{\vee}\cap M$ and $v_{C}\in N$ is the sum of the primitive generators for $C$. Then

[TABLE]

for all $r>0$, and all monomial primes $P$ in $R_{\mathbb{F}}$.

Theorem 3.2**.**

With notation as in Theorem 3.1, we assume further that $C$ is simplicial. Define $T:=\max\left\{\max_{m\in\mathcal{B}}\langle m,v_{C}\rangle,D\right\},$ where $D$ is any positive integer such that $D\cdot\operatorname{Cl}(R_{\mathbb{F}})=0$. Then

[TABLE]

*for all $r>0$, all monomial primes, and all height one primes in $R_{\mathbb{F}}$. In particular, we can take $D=\#\operatorname{Cl}(R_{\mathbb{F}})$. *

Remark 3.3**.**

If the cone $C$ in Theorem 3.2 is smooth, then $T=1$ and $P^{(r)}=P^{r}$ for all $r>0$, all monomial primes, and all height one primes in $R_{\mathbb{F}}$. As $C$ is smooth, $C$ and $C^{\vee}$ are generated by a $\mathbb{Z}$-basis for $N$ and the dual basis for $M$, respectively. Also, $\#\operatorname{Cl}(R_{\mathbb{F}})=\#\operatorname{Cl}(R_{\overline{\mathbb{F}}})=1$. Note that in general, this means our multiplier $T$ will not confirm uniform symbolic topologies for all primes $P$ in a toric algebra. For example, even in a polynomial ring of dimension three, there are height two primes for which $P^{(r)}\neq P^{r}$ for some $r\geq 2$; [8, p.2 of Intro] gives an example.

Remark 3.4**.**

Two-dimensional toric algebras are always simplicial with cyclic class class group. In this case, the conclusion of Theorem 3.2 holds using the multiplier $\#\operatorname{Cl}(R_{\mathbb{F}})$. This multiplier is sharp by Proposition 4.1.

Remark 3.5**.**

Theorems 3.1-3.2 can be adapted to the non-full case by replacing $R_{\mathbb{F}}$ with $R_{\mathbb{F}}^{\prime}$ as in Proposition 2.3, and applying [28, Prop. 2.6] to the faithfully flat map $\varphi$ from Proposition 2.3.

Proof of Theorem 3.1.

We may fix a face $F\neq\{0\}$ of $C$, and $P=P_{F}$ the corresponding monomial prime in $R=R_{\mathbb{F}}$. Per Lemma 2.2 (2.0.2), a monomial $\chi^{m}\in P=P_{F}$ if and only if $\left\langle m,v_{F}\right\rangle\in\mathbb{Z}_{>0},$ where $v_{F}\in F\cap N$ is the sum of the primitive generators for $F$.

Lemma 3.6**.**

For each integer $E\geq 1$, $P_{F}^{(E)}\subseteq I_{F}(E)$, where $I_{F}(E):=\left(\chi^{m}\colon\left\langle m,v_{F}\right\rangle\geq E\right)R.$

First, $I_{F}(E)$ is $P_{F}$-primary for all $E\geq 1$, i.e., if $sf\in I_{F}(E)$ for some $s\in R-P_{F}$, then $f\in I_{F}(E)$. As $I_{F}(E)$ is monomial, we may test this by fixing $\chi^{m}\in I_{F}(E)R_{P_{F}}\cap R$ and $\chi^{q}\in R-P_{F}$ such that $\chi^{m}\cdot\chi^{q}=\chi^{m+q}\in I_{F}(E)$: $\left\langle q,v_{F}\right\rangle=0,\mbox{ while }E\leq\left\langle m+q,v_{F}\right\rangle=\left\langle m,v_{F}\right\rangle+\left\langle q,v_{F}\right\rangle=\left\langle m,v_{F}\right\rangle,$ so $\chi^{m}\in I_{F}(E)$. Thus all $I_{F}(E)$ are $P_{F}$-primary, and certainly $P_{F}^{E}\subseteq I_{F}(E)$. Thus $P_{F}^{(E)}\subseteq I_{F}(E)$, being the smallest $P_{F}$-primary ideal containing $P_{F}^{E}$, proving the lemma.

Certainly, $1\leq\max_{m\in\mathcal{B}}\left\langle m,v_{F}\right\rangle\leq\max_{m\in\mathcal{B}}\left\langle m,v_{C}\right\rangle$, for $\mathcal{B}$ and $D$ as above. The conclusion of the theorem follows by [27, Lem. 3.3], once we verify the left-hand containments in the next

Lemma 3.7**.**

For each integer $E\geq 1$, $I_{F}(E)\subseteq P_{F}^{\lceil E/D^{\prime}\rceil}\subseteq P_{F}^{\lceil E/D\rceil}$ where $D^{\prime}=\max_{m\in\mathcal{B}}\left\langle m,v_{F}\right\rangle.$

Fix any monomial $\chi^{\ell}\in I_{F}(E)$, say $\ell=\sum_{m\in\mathcal{B}}a_{m}\cdot m$ with $a_{m}\in\mathbb{Z}_{\geq 0}$. Let $S\subseteq\mathcal{B}$ consist of those $m\in\mathcal{B}$ such that the monomials $\chi^{m}$ form a minimal generating set for $P$. By linearity of $\langle\bullet,v_{F}\rangle$,

[TABLE]

Thus $\chi^{m}\in P_{F}^{\sum_{m\in S}a_{m}}\subseteq P_{F}^{\lceil E/D^{\prime}\rceil}$. Being a monomial ideal, it follows that $I_{F}(E)\subseteq P_{F}^{\lceil E/D^{\prime}\rceil}$. ∎

Remark 3.8**.**

In passing, we invite the reader to compare the ideals $I_{F}(\bullet)$ in Lemma 3.6 with Bruns and Gubeladze’s terminology and description [6, Ch. 4, p. 149] for the symbolic powers of the height one monomials primes in terms of a full pointed cone. Lemma 3.6 works in any height.

Lemma 3.9**.**

With notation as in Theorem 3.1, the class groups $\operatorname{Cl}(R_{\mathbb{F}})\cong\operatorname{Cl}(R_{\overline{\mathbb{F}}})$ are isomorphic.

Proof.

Since $C$ is a full pointed cone in Theorem 3.1, $R_{\mathbb{F}}$ admits an $\mathbb{N}$-grading; see the passage above Remark 2.1. We may then cite Fossum [11, Cor. 10.5 on p.43] to conclude that up to isomorphism, $\operatorname{Cl}(R_{\mathbb{F}})\subseteq\operatorname{Cl}(R_{\overline{\mathbb{F}}})$ as a subgroup. This is an equality for toric rings because the divisor classes of height one monomial primes belong to both groups and generate the latter by Theorem 2.5. ∎

Proof of Theorem 3.2.

Since $C$ is simplicial, $\#\operatorname{Cl}(R_{\mathbb{F}})$ is finite by Lemma 3.9 and Theorem 2.5. Now we simply combine Theorem 3.1 with Lemma 2.4, and take the maximum of the values. ∎

Example 3.10**.**

Fix an arbitrary ground field $\mathbb{F}$ and integers $n\geq 2$ and $E\geq 2$. Let

[TABLE]

Then $P^{(T(r-1)+1)}\subseteq P^{r}$ for all $r>0$, all monomials primes, and all height one primes in $R$, where $T=\max\{n,E\}$. Indeed, $R$ is a toric algebra arising from the simplical full pointed cone $C\subseteq\mathbb{R}^{n}$ spanned by

[TABLE]

where $e_{1},\ldots,e_{n}$ denote the standard basis vectors in $\mathbb{R}^{n}$. We may apply Theorem 3.2, noting that:

• •

In the notation of Theorem 3.1, $\mathcal{B}=\{e_{1},\ldots,e_{n-1},e_{n},E\cdot e_{n}-e_{1}-\cdots-e_{n-1}\}\subseteq\mathbb{Z}^{n}$ and the vector $v_{C}=n\cdot e_{n}+E\cdot(e_{1}+\cdots+e_{n-1})\in\mathbb{Z}^{n}$.

• •

$\operatorname{Cl}(R)\cong(\mathbb{Z}/E\mathbb{Z})^{n-1},$ so $E\cdot\operatorname{Cl}(R)=0$; see [27, Sec. 4] for this class group computation.

3.0.1. Von Korff’s Toric F-Signature Formula

Now fix a perfect field $\mathbb{K}$ of positive characteristic $p$. Given an F-finite $\mathbb{N}$-graded domain $R$ of finite type over $\mathbb{K}$, for each integer $e\geq 0$, we have an $R$-module isomorphism $R^{1/p^{e}}\cong R^{a_{e}}\oplus M$ where $M$ has no free summand, and the integer $a_{e}\leq p^{ed}$ where $d=\dim R$. By definition, the F-signature of $R$ is (see [19] and [24])

[TABLE]

The F-signature has ties to measuring F-singularities: for instance, $s(R)$ is positive if and only if $R$ is strongly F-regular [1], and $s(R)=1$ if and only if $R$ is regular [24, Thm. 4.16].

Over the perfect field $\mathbb{K}$, any normal toric ring is strongly F-regular and its F-signature is rational [23]. We now state Von Korff’s result [25, Thm. 3.2.3]; see also Watanabe-Yoshida [29, Thm. 5.1] and Yao [30, Rem. 2.3(4)]:

Theorem 3.11** (cf., Von Korff [25, Thm. 3.2.3]).**

With notation as in Proposition 2.3, we define a convex polytope, $P_{C^{\prime}}:=\{w\in M^{\prime}_{\mathbb{R}}\colon 0\leq\langle w,v\rangle<1,\forall v\in G\}\subsetneqq(C^{\prime})^{\vee},$ where $G$ is the set of primitive generators of $C^{\prime}\neq\{0\}$. Then over any perfect field $\mathbb{K}$ of positive characteristic, the F-signature $s(R_{\mathbb{K}})=s(R^{\prime}_{\mathbb{K}})=\operatorname{Vol}(P_{C^{\prime}})\in\mathbb{Q}_{>0}$, where the volume form $\operatorname{Vol}$ on $M^{\prime}_{\mathbb{R}}$ is chosen so that a hypercube spanned by primitive generators of $M^{\prime}$ has volume one.

Remark 3.12**.**

When the cone $C^{\prime}$ in Theorem 3.11 is simplicial, $s(R_{\mathbb{K}}^{\prime})=1/D$ where the integer $D=\#\operatorname{Cl}(R_{\overline{\mathbb{K}}}^{\prime})=\#\operatorname{Cl}(R_{\mathbb{K}}^{\prime})$; cf., [26, Cor. 3.2]. The latter equality holds by Lemma 3.9.

Given Theorem 3.11, we could opt to replace the invariant $T$ from Theorem 3.2 with the possibly larger invariant $U=\operatorname{lcm}\{\max_{m\in\mathcal{B}}\langle m,v_{C}\rangle,\#\operatorname{Cl}(R_{\mathbb{K}})\}\in\mathbb{Z}_{>0}$, the least common multiple considered in Theorem 1.2. The latter is an integer multiple of the reciprocal $1/s(R_{\mathbb{K}})=\#\operatorname{Cl}(R_{\mathbb{K}})$ of the F-signature of $R_{\mathbb{K}}$ that controls the asymptotic growth of symbolic powers.

4. Examples of (Non-)Sharp Multipliers; Segre-Veronese algebras

To start, we deduce a result that occasionally provides sharp multipliers in the toric setting.

Proposition 4.1**.**

With notation as in Theorem 3.2, we assume $C$ is a simplicial full pointed rational polyhedral cone. We now set $B:=\max_{w\in\mathcal{PG}}\left\langle w,v_{C}\right\rangle$ where $\mathcal{PG}\subseteq\mathcal{B}$ consists of the primitive generators of $C^{\vee}$. There exists a monomial prime $P$ in $R=R_{\mathbb{F}}$ of height one such that:

1. (1)

$P^{(B(r-1))}\not\subseteq P^{r}$* for some $r\geq 2$;* 2. (2)

There is no positive integer $D^{\prime}<B$ such that $P^{(D^{\prime}(r-1)+1)}\subseteq P^{r}$ for all $r>0$.

Proof.

Let $v_{1},\ldots,v_{n}\in N$ and $w_{1},\ldots,w_{n}\in M$ denote the primitive generators for $C$ and for $C^{\vee}$, respectively. We index these generators so that the nonnegative integer $\langle w_{j},v_{i}\rangle$ is positive if and only if $i=j$: we may do this citing the notion of facet normals [7, after Prop. 1.2.8]. In deference to Lemma 2.2(2.0.2), let $P_{j}$ ($1\leq j\leq n$) be the height one monomial prime in $R_{\mathbb{F}}$ such that a monomial $\chi^{m}\in P_{j}$ if and only if $\langle m,v_{j}\rangle>0$. In particular, $\chi^{w_{j}}\in P_{j}$ for each $j$.

Lemma**.**

*For each $1\leq j\leq n$, $\langle w_{j},v_{j}\rangle$ is the order of the element in $\operatorname{Cl}(R_{\mathbb{F}})$ corresponding to $P_{j}$. *

We may leverage exact sequence (2.0.3) from Theorem 2.5, since Lemma 3.9 allows us reduce to the case where $\mathbb{F}$ is algebraically closed. For $1\leq j\leq n$, we have $0=[\operatorname{div}(\chi^{w_{j}})]=\langle w_{j},v_{j}\rangle[D_{\rho_{j}}]$, where $\rho_{j}$ is the rational ray of $C$ generated by $v_{j}$. Thus $P_{j}^{(\langle w_{j},v_{j}\rangle)}=(\chi^{w_{j}})R.$ Since the order of $[D_{\rho_{j}}]$ is the smallest $E_{j}>0$ such that $P_{j}^{(E_{j})}=(\chi^{m_{j}})R$ is principal for some $m_{j}\in C^{\vee}\cap M-\{0\}$, $E_{j}$ divides $\langle w_{j},v_{j}\rangle$, and $P_{j}^{(\langle w_{j},v_{j}\rangle)}=(P_{j}^{(E_{j})})^{L}$ where $L=\langle w_{j},v_{j}\rangle/E_{j}$. Since $C^{\vee}$ is strongly convex, we may conclude that $\chi^{w_{j}}=\chi^{L\cdot m_{j}}$, and $L=1$ since $w_{j}$ is an irreducible vector in $C^{\vee}\cap M$; see Subsection 2.0.1. This proves the lemma.

To prove (1), notice $B=\langle w_{j_{0}},v_{j_{0}}\rangle$ for some $1\leq j_{0}\leq n$. Then (): $(\chi^{w_{j_{0}}})R=P_{j_{0}}^{(B)}\not\subseteq P_{j_{0}}^{2}$. We recall from Section 2 that when the cone $C$ is full-dimensional in $N_{\mathbb{R}}$, the semigroup algebra $R_{\mathbb{F}}=\mathbb{F}[C^{\vee}\cap M]$ can be $\mathbb{N}$-graded. This means that any minimal generator $f$ of a homogeneous ideal $I$ satisfies $f\in I-I^{2}$ by Nakayama’s lemma. In our situation, $I=P_{j_{0}}$ and $f=\chi^{w_{j_{0}}}$. Observation () also gives part (2), arguing by contradiction and using [27, Lem. 3.3] accordingly. ∎

We offer several examples to show that establishing sharpness of our bilinear multipliers is a delicate matter meriting further study.

Example 4.2**.**

We fix integers $n\geq 2$ and $E\geq 2$, and an arbitrary field $\mathbb{F}$. Let $V_{E,n}$ be the $E$-th Veronese subalgebra of the polynomial ring $\mathbb{F}[x_{1},\ldots,x_{n}]$, that is, the $\mathbb{F}$-algebra generated by all monomials of degree $E$ in $x_{1},\ldots,x_{n}$. Then $P^{(E(r-1)+1)}\subseteq P^{r}$ for all $r>0$, all monomial primes, and all primes of height one by Theorem 3.2. See [27, Sec. 4] for details. However, for any $E^{\prime}<E$, the proof of Proposition 4.1 guarantees that we can find a prime $P\subseteq V_{E,n}$ (monomial, height one) such that $P^{(E^{\prime}(r-1)+1)}\not\subseteq P^{r}$ for some $r\geq 2$, namely, for $r=2$. In fact, this last observation holds for all monomial primes in $V_{E,n}$, aside from the zero ideal and the maximal monomial ideal for which $E^{\prime}=1$ will do; the proof of [27, Thm. 4.3] confirms this explicitly.

Despite Example 4.2, Theorem 3.2 does not give sharp multipliers in general. For example,

Example 4.3**.**

For any $n>2$, let $R=\mathbb{F}[Z,X_{1},\ldots,X_{n}]/(Z^{2}-X_{1}\cdots X_{n})$ as in Example 3.10. Citing the proof of [27, Thm. 4.1], when $P\subseteq R$ is any monomial prime of height at least 2:

• •

$P^{(r)}=P^{r}$ for all $r>0$; however,

• •

The multiplier $D^{\prime}$ corresponding to $P$ in Lemma 3.7 always satisfies $D^{\prime}\geq 2$.

Theorem 3.1 gives a uniform multiplier $D$ that works for all monomial primes. Even when this multiplier is sharp across all monomial primes, it need not be best possible for all monomial primes of a given height, contrasting with the situation of Example 4.2. For example,

Example 4.4**.**

Let $R=\mathbb{F}[C^{\vee}\cap\mathbb{Z}^{3}]=\mathbb{F}[x,y,z,w]/(xy-zw)$, for the non-simplicial cone $C\subseteq\mathbb{R}^{3}$ with $e_{1},e_{2},e_{1}+e_{3},e_{2}+e_{3}$ as primitive generators. Theorem 3.1 says $P^{(2r-1)}\subseteq P^{r}$ for all $r>0$ and all monomial primes in $R$, observing that $C^{\vee}\cap\mathbb{Z}^{3}$ is minimally generated by $\mathcal{B}=\{e_{1},e_{2},e_{3},(1,1,-1)\}$. Given any height two monomial prime $P$ in $R$, these containments cannot be improved to $P^{(r)}=P^{r}$ for all $r\geq 2$. For instance, if $P=(x,y,z)R$, then for any $s\geq 1$, $z^{s}\in P^{(2s)}-P^{2s}$ and $z^{s+1}\in P^{(2s+1)}-P^{2s+1}$: indeed, $w^{s}\in R-P$ and $R$ can be standard graded, so the least degree of a homogeneous element of $P^{r}$ is $r$. By contrast, $P^{(r)}=P^{r}$ for all $r$ and for any height one monomial prime $P$ in $R$: the invariant $D^{\prime}=1$ in Lemma 3.7 via direct computation.

4.0.1. Final Example Computation: Segre-Veronese algebras

In what follows, $\mathbb{F}$ is a fixed arbitrary field. For more on Segre products, [13] is often cited as a standard reference.

Definition 4.5**.**

Fix a family $A_{1},\ldots,A_{k}$ of $k$ standard graded algebras of finite type over $\mathbb{F}$, with $A_{i}=\mathbb{F}[a_{i,1},\ldots,a_{i,b_{i}}]$ in terms of algebra generators. Their Segre product over $\mathbb{F}$ is the ring $S=(\#_{\mathbb{F}})_{i=1}^{k}A_{i}$ generated up to isomorphism as an $\mathbb{F}$-algebra by all $k$-fold products of the $a_{i,j}$.

Definition 4.6**.**

We fix integers $E\geq 1$ and $m\geq 2$. Suppose $A=\mathbb{F}[x_{1},\ldots,x_{m}]$ is a standard graded polynomial ring in $m$ variables over a field $\mathbb{F}$. Let $V_{E,m}\subseteq A$ denote the $E$-th Veronese subring of $A$, the standard graded $\mathbb{F}$-subalgebra generated by all monomials of degree $E$ in the $x_{i}$. There are $\binom{m-1+E}{E}$ such monomials; this number is the embedding dimension of $V_{E,m}$.

Definition 4.7**.**

Fix $k$-tuples $\overline{E}=(E_{1},\ldots,E_{k})\in(\mathbb{Z}_{\geq 1})^{k}$ and $\overline{m}=(m_{1},\ldots,m_{k})\in(\mathbb{Z}_{\geq 2})^{k}$ of integers, with $k\geq 1$. Furthermore, we set $d(j)=\left(\sum_{i=1}^{j}m_{i}\right)-(j-1)$ for each $1\leq j\leq k$: $d(k)$ is the Krull dimension of the Segre product $SV\left(\overline{E},\overline{m}\right)=(\#_{\mathbb{F}})_{i=1}^{k}V_{E_{i},m_{i}}$ of $k$ Veronese rings in $m_{1},\ldots,m_{k}$ variables, respectively; this is a Segre-Veronese algebra with degree sequence $\overline{E}$.

Over any perfect field $\mathbb{K}$, a Segre-Veronese algebra has uniform symbolic topologies on all primes, per [10, Thm. 2.2], [16, Thm. 1.1], and [17, Cor. 3.10]. However, explicit $D$ are elusive unless $k=1$; see [8, Cor. 3.30]. We provide an effective $D$ for the monomial primes, which is sharp as suggested by Examples 4.2 and 4.4, the latter being the Segre product of polynomial rings in two variables:

Theorem 4.8**.**

*Suppose $A=SV\left(\overline{E},\overline{m}\right)$ is a Segre-Veronese algebra over $\mathbb{F}$ with degree sequence $\overline{E}=(E_{1},\ldots,E_{k})$. Let $D:=\sum_{i=1}^{k}E_{i}$. Then $P^{(D(r-1)+1)}\subseteq P^{r}$ for all $r>0$ and all monomial primes $P$ in $A$. *

Proof.

Given a lattice $N\cong\mathbb{Z}^{d}$ we will use $e_{1},\ldots,e_{d}\in N$ to denote a choice of basis for $N$ will dual basis $e_{1}^{*},\ldots,e_{d}^{*}$ for $M$. In the setup of Theorem 3.1, the cardinality of the minimal generating set $\mathcal{B}$ of $C^{\vee}\cap M$ is the embedding dimension of the toric algebra $R_{\mathbb{F}}=\mathbb{F}[C^{\vee}\cap M]$; we refer the reader to [7, Sec. 1.0, Proof of Thm. 1.3.10].

Lemma**.**

*For $A$ as stated, there exists a lattice $N$ and a full pointed rational polyhedral cone $C\subseteq N_{\mathbb{R}}$ such that $A=\mathbb{F}[C^{\vee}\cap M]$ up to isomorphism. *

Proof.

Fix $k$-tuples $\overline{E}\in(\mathbb{Z}_{\geq 1})^{k}$ and $\overline{m}\in(\mathbb{Z}_{\geq 2})^{k}$. Set $d(j)=\left(\sum_{i=1}^{j}m_{i}\right)-(j-1)$ for $1\leq j\leq k$, while $d(0)=0$. Given $SV\left(\overline{E},\overline{m}\right)=(\#_{\mathbb{F}})_{i=1}^{k}V_{E_{i},m_{i}}$, we fix a lattice $N\cong\mathbb{Z}^{d(k)}$ and record a cone $C=C\left(\overline{E},\overline{m}\right)\subseteq N_{\mathbb{R}}\cong\mathbb{R}^{d(k)}$ as stipulated with $R_{\mathbb{F}}=\mathbb{F}[C^{\vee}\cap M]\cong SV\left(\overline{E},\overline{m}\right)$. Specifically, consider the cone $C\subseteq N_{\mathbb{R}}$ generated by the following irredundant collection of primitive vectors:

[TABLE]

The semigroup $C^{\vee}\cap M$ is generated by the following set of irreducible vectors:

[TABLE]

Indeed, $\#\mathcal{B}=\prod_{j=1}^{k}\binom{m_{j}-1+E_{j}}{E_{j}}$, the embedding dimension of $SV(\overline{E},\overline{m})$. Finally, one can record a bijection between the monomial generators of $R_{\mathbb{F}}$ and those typically used to present $SV(\overline{E},\overline{m})$; cf., [27, Proof of Lem. 4.2] for how the bijection would look in the coordinates $a_{j,\ell}$ for each $j$. ∎

Feeding $v_{C}=\sum_{u\in\mathcal{A}}u=(\sum_{j=1}^{k}E_{j})\cdot e_{m_{1}}$ and $\mathcal{B}$ into Theorem 3.1 yields $D=\sum_{j=1}^{k}E_{j}$. We win! ∎

In passing, we note that the corresponding class of varieties (Segre-Veronese varieties) form a cornerstone of a lot of investigations in classical and applied algebraic geometry, often being varieties for which one has a more incisive handle on some specified computational- or other task. As a sampling of recent investigations in this vein, we include [2, 3, 4, 9, 22] among our references.