The Ubiquity of Smooth Hilbert Schemes

TL;DR

This paper explores the structure and properties of Hilbert schemes, revealing regularities and probabilistic results about their irreducibility and smoothness within a combinatorial and geometric framework.

Contribution

It classifies certain Hilbert schemes with unique Borel-fixed points and uncovers probabilistic regularities in their geometric properties.

Findings

Hilbert schemes are irreducible with probability > 0.5 under natural distributions.

Classification of Hilbert schemes with unique Borel-fixed points.

Hilbert schemes form vertices of an infinite binary tree with regular geometric patterns.

Abstract

We investigate the geography of Hilbert schemes parametrizing closed subschemes of projective space with specified Hilbert polynomials. We classify Hilbert schemes with unique Borel-fixed points via combinatorial expressions for their Hilbert polynomials. These expressions naturally lead to an arrangement of nonempty Hilbert schemes as the vertices of an infinite full binary tree. Here we discover regularities in the geometry of Hilbert schemes. Specifically, under natural probability distributions on the tree, we prove that Hilbert schemes are irreducible and nonsingular with probability greater than $0.5$.

Tables

Table 1. Table 1. Summary of Basic Operations

$$ Hilbert polynomial $𝗉$ with partitions $(b_1,b_2,\dots ,b_r)$ and $(e_0,e_1,\dots ,e_d)$	$A\left(𝗉\right)=1+𝗉$ with partitions $(b_1,b_2,\dots ,b_r,0)$ and $(e_0+1,e_1,e_2,\dots ,e_d)$
	$\Phi \left(𝗉\right)$ with partitions $(b_1+1,b_2+1,\dots ,b_r+1)$ and $(e_0,e_0,e_1,\dots ,e_d)$
lex ideal $L_n^{𝗉}=L(a_0,a_1,\dots ,a_{n-1})$	$A\left(L_n^{𝗉}\right)=L_n^{A\left(𝗉\right)}=L(a_0+1,a_1,a_2,\dots ,a_{n-1})$
	$\Phi \left(L_n^{𝗉}\right)=L_{n+1}^{\Phi \left(𝗉\right)}=L(0,a_0,a_1,\dots ,a_{n-1})$
saturated str. st. ideal $I$ with ${𝗉}_I=𝗉$	expansion $I^{\prime }$ of $I$ with ${𝗉}_{I^{\prime }}=A\left(𝗉\right)$
	extension $\Phi \left(I\right)=I\cdot 𝕂[x_0,x_1,\dots ,x_{n+1}]$

Equations

$$\mathsf{p}(t)=\sum _{j=1}^{e_d}\left(\genfrac{}{}{0ex}{}{t+d-j+1}{d}\right)+{\left[\sum _{i=0}^{d-1}\left(\genfrac{}{}{0ex}{}{s+i}{i+1}\right)-\left(\genfrac{}{}{0ex}{}{s+i-e_i+e_d}{i+1}\right)\right]}_{s=t-e_d}$$

$$L^{\mathsf{p}_{I}}_{n}:=\bigl{(}L^{\mathsf{h}_{I}}_{n}:\langle x_{0},x_{1},\dotsc,x_{n}\rangle^{\infty}\bigr{)}=\bigcup_{j\geq 1}\bigl{\{}f\in\mathbb{K}[x_{0},x_{1},\dotsc,x_{n}]\mid f\langle x_{0},x_{1},\dotsc,x_{n}\rangle^{j}\subseteq L^{\mathsf{h}_{I}}_{n}\bigr{\}}.$$

$$⟨x_0^{a_{n-1}+1},x_0^{a_{n-1}}{x}_1^{a_{n-2}+1},\dots ,x_0^{a_{n-1}}{x}_1^{a_{n-2}}\cdots x_{n-3}^{a_2}{x}_{n-2}^{a_1+1},x_0^{a_{n-1}}{x}_1^{a_{n-2}}\cdots x_{n-2}^{a_1}{x}_{n-1}^{a_0}⟩.$$

$$L_n^{\mathsf{p}}$$

$$=⟨x_0,x_1,\dots ,x_{n-d-2},x_{n-d-1}^{a_d+1},$$

$$x_{n-d-1}^{a_d}{x}_{n-d}^{a_{d-1}+1},\dots ,x_{n-d-1}^{a_d}{x}_{n-d}^{a_{d-1}}\cdots x_{n-3}^{a_2}{x}_{n-2}^{a_1+1},x_{n-d-1}^{a_d}{x}_{n-d}^{a_{d-1}}\cdots x_{n-2}^{a_1}{x}_{n-1}^{a_0}⟩.$$

$$m_i$$

$$m_{n-d+k}$$

$$m_{n-\ell -1}$$

$$0\to \left(\mathbb{K}[x_0,x_1,\dots ,x_n]/L^{\prime \prime }\right)(-a_d)\to \mathbb{K}[x_0,x_1,\dots ,x_n]/L\to \mathbb{K}[x_0,x_1,\dots ,x_n]/L^{\prime }\to 0,$$

$$x_{n-d-1}^{a_d}{x}_{n-d}^{a_{d-1}}\cdots x_{n-\ell -3}^{a_{\ell +2}}{x}_{n-\ell -2}^{a_{\ell +1}+1},x_{n-d-1}^{a_d}{x}_{n-d}^{a_{d-1}}\cdots x_{n-\ell -2}^{a_{\ell +1}}{x}_{n-\ell -1}^{a_{\ell }+1},\dots ,$$

$$\text{hfill}{x}_{n-d-1}^{a_d}{x}_{n-d}^{a_{d-1}}\cdots x_{n-3}^{a_2}{x}_{n-2}^{a_1+1}$$

$$I^{\prime }=⟨I\setminus ⟨g⟩⟩+⟨g{x}_j\mid \max g\le j\le n-1⟩\subset \mathbb{K}[x_0,x_1,\dots ,x_n];$$

$$\{m_1,m_2,\dots ,m_{n-\ell -2},m_{n-\ell -1}{x}_{n-\ell -2},m_{n-\ell -1}{x}_{n-\ell -1},\dots ,m_{n-\ell -1}{x}_{n-1}\}.$$

$$0⟶\left(S/I\right)(-1)⟶S/I⟶\mathbb{K}[x_0,x_1,\dots ,x_{n-1}]/J⟶0$$

$${\mathsf{H}}_I(T)=\sum _{i\in \mathbb{N}}\left(c_0\left(\genfrac{}{}{0ex}{}{n+i}{n}\right)+c_1\left(\genfrac{}{}{0ex}{}{n+i-1}{n}\right)+\cdots +c_d\left(\genfrac{}{}{0ex}{}{n+i-d}{n}\right)\right)T^i.$$

$${\mathsf{p}}_I(i)$$

$$=\sum _{j=0}^{d-1}{c}_j{\left(\genfrac{}{}{0ex}{}{t+n-j}{n})\right|}_{t=i}+c_d(-1{)}^n$$

$$G((L^{\mathsf{p}}_{n})^{\prime})=\bigl{(}\{m_{1},m_{2},\dotsc,m_{n-\ell-1}\}\setminus\{m\}\bigr{)}\cup\bigl{\{}mx_{\max m},mx_{\max m+1},\dotsc,mx_{n-1}\bigr{\}},$$

$$g\in G(I)$$

$$\mathsf{h}(i)={\dim }_{\mathbb{K}}\mathbb{K}[x_0,x_1,\dots ,x_n{]}_i/(L_n^{\mathsf{h}}{)}_i\ge {\dim }_{\mathbb{K}}\mathbb{K}[x_0,x_1,\dots ,x_n{]}_i/(L_n^{\mathsf{p}}{)}_i,$$

$$\bigl{(}\Phi(I)\bigr{)}_{i}=\bigoplus_{j\in\mathbb{N},j\leq i}I_{j}\cdot x_{n+1}^{i-j}\subset\bigoplus_{j\in\mathbb{N},j\leq i}\mathbb{K}[x_{0},x_{1},\dotsc,x_{n}]_{j}\cdot x_{n+1}^{i-j}=\mathbb{K}[x_{0},x_{1},\dotsc,x_{n+1}]_{i}$$

$${\mathsf{h}}_I(i)$$

$${\mathsf{h}}_{L_{n+1}^{\mathsf{p}}}(i)$$

$$⟨x_0^{s+1},x_0^s{x}_1,\dots ,x_0^s{x}_{b_s-b_{s+1}-1},x_0^s{x}_{b_s-b_{s+1}}^2,x_0^s{x}_{b_s-b_{s+1}}{x}_{b_s-b_{s+1}+1},\dots ,x_0^s{x}_{b_s-b_{s+1}}{x}_{b_s-b_{s+2}}⟩.$$

$$L:=⟨x_0^{d_0+1},x_0^{d_0}{x}_1^{d_1+1},x_0^{d_0}{x}_1^{d_1}{x}_2^{d_2+1},\dots ,x_0^{d_0}{x}_1^{d_1}\cdots x_{k-1}^{d_{k-1}+1},x_0^{d_0}{x}_1^{d_1}\cdots x_k^{d_k}⟩.$$

$$\mathrm{Pr}(\mathcal{N})=\sum _{H\in \mathcal{N}}\mathrm{pr}\left(H\right)$$

$$=\sum _{c>0}\left(\frac{f(c)f_c(0)}{2}+\sum _{k\in \mathbb{N}}\frac{f(c)f_c(k)}{2}\right)$$

$$=\sum _{c>0}\left(\frac{f(c)f_c(0)}{2}+\frac{f(c)}{2}\right)=\sum _{c>0}\left(\frac{f(c)f_c(0)}{2}\right)+\frac{1}{2},$$

Full text

1. Introduction

Hilbert schemes parametrizing closed subschemes with a fixed Hilbert polynomial in projective space are fundamental moduli spaces. However, with the exception of Hilbert schemes parametrizing hypersurfaces [ACG11, Example 2.3] and points in the plane [Fog68], the geometric features of typical Hilbert schemes are still poorly understood. Techniques for producing pathological Hilbert schemes are known, generating Hilbert schemes with many irreducible components [Iar72, FP96], with generically nonreduced components [Mum62], and with arbitrary singularity types [Vak06]. This raises the questions: What should we expect from a random Hilbert scheme? Can we understand the geography of Hilbert schemes? Our answer is that the set of nonempty Hilbert schemes forms a collection of trees and a discrete probability space, and that irreducible, nonsingular Hilbert schemes are unexpectedly common.

Let $\operatorname{Hilb}^{\mathsf{p}}(\operatorname{\mathbb{P}}^{n})$ be the Hilbert scheme parametrizing closed subschemes of $\operatorname{\mathbb{P}}^{n}_{\mathbb{K}}$ with Hilbert polynomial $\mathsf{p}$, where $\mathbb{K}$ is a field. Macaulay classified Hilbert polynomials of homogeneous ideals in [Mac27]. Any such admissible Hilbert polynomial $\mathsf{p}(t)$ has a unique combinatorial expression of the form $\sum_{j=1}^{r}\binom{t+b_{j}-j+1}{b_{j}}$, for integers $b_{1}\geq b_{2}\geq\dotsb\geq b_{r}\geq 0$. Our main result is the following theorem.

Theorem 1.1**.**

The lexicographic ideal is the unique saturated strongly stable ideal of codimension $c$ with Hilbert polynomial $\mathsf{p}$ if and only if at least one of the following holds:

1. (i)

$b_{r}>0$, 2. (ii)

$c\geq 2$* and $r\leq 2$,* 3. (iii)

$c=1$* and $b_{1}=b_{r}$, or* 4. (iv)

$c=1$* and $r-s\leq 2$, where $b_{1}=b_{2}=\dotsb=b_{s}>b_{s+1}\geq\dotsb\geq b_{r}$.*

If $\mathbb{K}$ is algebraically closed, then the lexicographic ideal is the unique saturated Borel-fixed ideal of codimension $c$ with Hilbert polynomial $\mathsf{p}$ if and only if at least one of (i)–(iv) holds.

Strongly stable ideals, including lexicographic ideals, are Borel-fixed. In characteristic [math], the converse also holds. Many fundamental properties of Hilbert schemes have been understood through these ideals. Hartshorne and later Peeva–Stillman found rational curves linking Borel-fixed points, proving connectedness [Har66, PS05]. Bayer used them to derive defining equations and proposed studying their tangent cones [Bay82]. Reeves, followed by Pardue, studied their combinatorial properties to give bounds for radii of Hilbert schemes [Ree95, Par94] and Reeves–Stillman proved that lexicographic points are nonsingular [RS97]. Gotzmann also discovered some irreducible Hilbert schemes in [Got89]. Theorem 1.1 advances this line of inquiry, identifies a large collection of well-behaved Hilbert schemes that includes Gotzmann’s examples, and improves our understanding of the geography of Hilbert schemes.

To make a quantitative statement about all Hilbert schemes, we interpret Macaulay’s classification as follows: First, identify any admissible Hilbert polynomial $\mathsf{p}$ with its sequence $b=(b_{1},b_{2},\ldots,b_{r})$. These sequences are generated by two operations, namely “integrating” $\mathsf{p}$ to $\Phi(\mathsf{p}):=b+(1,1,\ldots,1)$ and “adding one” to $\mathsf{p}$ to get $A(\mathsf{p}):=1+\mathsf{p}=(b,0)$. The set of all such sequences forms an infinite full binary tree. There is an associated tree, which we denote $\operatorname{\mathscr{H}}_{c}$, whose vertices are the Hilbert schemes $\operatorname{Hilb}^{\mathsf{p}}(\operatorname{\mathbb{P}}^{n})$ parametrizing codimension $c=n-\deg\mathsf{p}$ subschemes, for each positive $c\in\mathbb{Z}$. Geometrically, $\Phi$ corresponds to coning over parametrized schemes and $A$ to adding a point. We endow $\operatorname{\mathscr{H}}_{c}$ with a natural probability distribution, in which the vertices at a fixed height are equally likely. This leads to our second main result.

Theorem 1.2**.**

Let $\mathbb{K}$ be algebraically closed or have characteristic [math]. The probability that a random Hilbert scheme is irreducible and nonsingular is greater than $0.5$.

This theorem counterintuitively suggests that the geometry of the majority of Hilbert schemes is understandable. To prove Theorems 1.1 and 1.2, we study the algorithm generating saturated strongly stable ideals first described by Reeves [Ree92] and later generalized in [Moo12, CLMR11]. We obtain precise information about Hilbert series and $K$-polynomials of saturated strongly stable ideals. The primary technical result we need is the following.

Theorem 1.3**.**

Let $I\subset\mathbb{K}[x_{0},x_{1},\dotsc,x_{n}]$ be a saturated strongly stable ideal with Hilbert polynomial $\mathsf{p}$, let $L^{\mathsf{p}}_{n}$ be the corresponding lexicographic ideal in $\mathbb{K}[x_{0},x_{1},\dotsc,x_{n}]$, and let $\mathsf{K}_{I}$ be the numerator of the Hilbert series of $I$. If $I\neq L^{\mathsf{p}}_{n}$, then we have $\deg\mathsf{K}_{I}<\deg\mathsf{K}_{L^{\mathsf{p}}_{n}}$.

The structure of the paper is as follows. In Section 2.1, we introduce two binary relations on the set of admissible Hilbert polynomials and show that they generate all such polynomials. The set of lexicographic ideals is then partitioned by codimension into infinitely many binary trees in Section 2.2. Geometrically, these are trees of Hilbert schemes, as every Hilbert scheme contains a unique lexicographic ideal. To identify well-behaved Hilbert schemes, we review saturated strongly stable ideals in Section 3 and we examine their $K$-polynomials in Section 4. The main results are in Section 5.

Conventions.

Throughout, $\mathbb{K}$ is a field, $\mathbb{N}$ is the nonnegative integers, and $\mathbb{K}[x_{0},x_{1},\dotsc,x_{n}]$ is the standard $\mathbb{Z}$-graded polynomial ring. The Hilbert function, polynomial, series, and $K$-polynomial of the quotient $\mathbb{K}[x_{0},x_{1},\dotsc,x_{n}]/I$ by a homogeneous ideal $I$ are denoted $\mathsf{h}_{I},\mathsf{p}_{I},\mathsf{H}_{I}$, and $\mathsf{K}_{I}$, respectively.

Acknowledgments

We especially thank Gregory G. Smith for his guidance in this research. We thank Mike Roth, Ivan Dimitrov, Tony Geramita, Chris Dionne, Ilia Smirnov, Nathan Grieve, Andrew Fiori, Simon Rose, and Alex Duncan for many discussions. We also thank the anonymous referee for helpful remarks that improved the paper. This research was supported by an E.G. Bauman Fellowship in 2011-12, by Ontario Graduate Scholarships in 2012-15, and by Gregory G. Smith’s NSERC Discovery Grant in 2015-16.

2. Binary Trees and Hilbert Schemes

In 2.1 we observe that a tree structure exists on the set of numerical polynomials determining nonempty Hilbert schemes. Macaulay’s pioneering work [Mac27] classifies these polynomials and two mappings turn this set into an infinite binary tree. In 2.2 we find related binary trees in the sets of lexicographic ideals and Hilbert schemes.

2.1. The Macaulay Tree

Let $\mathbb{K}$ be a field and let $\mathbb{K}[x_{0},x_{1},\dotsc,x_{n}]$ denote the homogeneous (standard $\mathbb{Z}$-graded) coordinate ring of $n$-dimensional projective space $\operatorname{\mathbb{P}}^{n}_{\mathbb{K}}$. Let $M$ be a finitely generated graded $\mathbb{K}[x_{0},x_{1},\dotsc,x_{n}]$-module. The Hilbert function $\mathsf{h}_{M}\colon\mathbb{Z}\to\mathbb{Z}$ of $M$ is defined by $\mathsf{h}_{M}(i):=\dim_{\mathbb{K}}(M_{i})$ for all $i\in\mathbb{Z}$. Every such $M$ has a Hilbert polynomial $\mathsf{p}_{M}$, that is, a polynomial $\mathsf{p}_{M}(t)\in\mathbb{Q}[t]$ such that $\mathsf{h}_{M}(i)=\mathsf{p}_{M}(i)$ for $i\gg 0$; see [BH93, Theorem 4.1.3]. For a homogeneous ideal $I\subset\mathbb{K}[x_{0},x_{1},\dotsc,x_{n}]$, let $\mathsf{h}_{I}$ and $\mathsf{p}_{I}$ denote the Hilbert function and Hilbert polynomial of the quotient module $\mathbb{K}[x_{0},x_{1},\dotsc,x_{n}]/I$, respectively.

We begin with a basic example and make a notational convention, for later use.

Example 2.1**.**

Fix a nonnegative integer $n$. By the classic stars-and-bars argument [Sta12, Section 1.2], we have $\mathsf{h}_{S}(i)=\binom{i+n}{n}$ for all $i\in\mathbb{Z}$, where $S=\mathbb{K}[x_{0},x_{1},\dotsc,x_{n}]$. The equality $\mathsf{h}_{S}(i)=\mathsf{p}_{S}(i)$ is only valid for $i\geq-n$, because the polynomial $\mathsf{p}_{S}(t)=\binom{t+n}{n}\in\mathbb{Q}[t]$ only has roots $-n$, $-n+1,\dotsc,-1$, whereas $\mathsf{h}_{S}(i)=0$ for all $i<0$.

Remark 2.2**.**

For integers $j,k$ we set $\binom{j}{k}=\frac{j!}{k!(j-k)!}$ if $j\geq k\geq 0$ and $\binom{j}{k}=0$ otherwise. For a variable $t$ and $a,b\in\mathbb{Z}$, we define $\binom{t+a}{b}=\frac{(t+a)(t+a-1)\dotsb(t+a-b+1)}{b!}\in\mathbb{Q}[t]$ if $b\geq 0$, and $\binom{t+a}{b}=0$ otherwise. When $b\geq 0$, the polynomial $\binom{t+a}{b}$ has degree $b$ with zeros $-a,-a+1,\dotsc,-a+b-1$. Importantly, we have $\binom{t+a}{b}\rvert_{t=j}\neq\binom{j+a}{b}=0$ when $j<-a$. Interestingly, [Mac27, p.533] uses distinct notation for polynomial and integer binomial coefficients.

A polynomial is an admissible Hilbert polynomial if it is the Hilbert polynomial of some closed subscheme in some $\operatorname{\mathbb{P}}^{n}$. Admissible Hilbert polynomials correspond to nonempty Hilbert schemes. We use the well-known classification first discovered by Macaulay.

Proposition 2.3**.**

The following conditions are equivalent:

1. (i)

The polynomial $\mathsf{p}(t)\in\mathbb{Q}[t]$ is a nonzero admissible Hilbert polynomial. 2. (ii)

There exist integers $e_{0}\geq e_{1}\geq\dotsb\geq e_{d}>0$ such that $\mathsf{p}(t)=\sum_{i=0}^{d}\binom{t+i}{i+1}-\binom{t+i-e_{i}}{i+1}$. 3. (iii)

There exist integers $b_{1}\geq b_{2}\geq\dotsb\geq b_{r}\geq 0$ such that $\mathsf{p}(t)=\sum_{j=1}^{r}\binom{t+b_{j}-j+1}{b_{j}}$.

Moreover, these correspondences are bijective.

Proof.

1. (i) $\Leftrightarrow$ (ii)

This is proved in [Mac27]; see the formula for “$\chi(\ell)$” at the bottom of p.536. For a geometric account, see [Har66, Corollary 3.3 and Corollary 5.7]. 2. (i) $\Leftrightarrow$ (iii)

This follows from [Got78, Erinnerung 2.4]; see also [BH93, Exercise 4.2.17].

Uniqueness of the sequences of integers also follows. ∎

For simplicity, we always work with nonzero admissible Hilbert polynomials. Let the Macaulay–Hartshorne expression of an admissible Hilbert polynomial $\mathsf{p}$ be its expression $\mathsf{p}(t)=\sum_{i=0}^{d}\binom{t+i}{i+1}-\binom{t+i-e_{i}}{i+1}$, for $e_{0}\geq e_{1}\geq\dotsb\geq e_{d}>0$, and the Gotzmann expression of $\mathsf{p}$ be its expression $\mathsf{p}(t)=\sum_{j=1}^{r}\binom{t+b_{j}-j+1}{b_{j}}$, for $b_{1}\geq b_{2}\geq\dotsb\geq b_{r}\geq 0$. From these, we find the degree $d=b_{1}$, the leading coefficient $e_{d}/d!$, and the Gotzmann number $r$ of $\mathsf{p}$, which bounds the Castelnuovo–Mumford regularity of saturated ideals with Hilbert polynomial $\mathsf{p}$. In particular, such ideals are generated in degree $r$; see [IK99, p. 300-301].

Macaulay–Hartshorne and Gotzmann expressions are conjugate. Recall that the conjugate partition to a partition $\lambda=\left(\lambda_{1},\lambda_{2},\dotsc,\lambda_{k}\right)$ of an integer $\ell=\sum_{i=1}^{k}\lambda_{i}$ is the partition of $\ell$ obtained from the Ferrers diagram of $\lambda$ by interchanging rows and columns, having $\lambda_{i}-\lambda_{i+1}$ parts equal to $i$; see [Sta12, Section 1.8].

Lemma 2.4**.**

If $\mathsf{p}(t)\in\mathbb{Q}[t]$ is an admissible Hilbert polynomial with Macaulay–Hartshorne expression $\sum_{i=0}^{d}\binom{t+i}{i+1}-\binom{t+i-e_{i}}{i+1}$ for $e_{0}\geq e_{1}\geq\dotsb\geq e_{d}>0$ and Gotzmann expression $\sum_{j=1}^{r}\binom{t+b_{j}-j+1}{b_{j}}$ for $b_{1}\geq b_{2}\geq\dotsb\geq b_{r}\geq 0$, then $r=e_{0}$ and the nonnegative partition $(b_{1},b_{2},\dotsc,b_{r})$ is conjugate to the partition $(e_{1},e_{2},\dotsc,e_{d})$.

Proof.

The key step is to rewrite $\mathsf{p}$ as $\sum_{i=0}^{d}\binom{t+i}{i+1}-\binom{t+i-e_{d}}{i+1}+\sum_{i=0}^{d-1}\binom{t+i-e_{d}}{i+1}-\binom{t+i-e_{i}}{i+1}$ and to prove $\sum_{i=0}^{d}\binom{t+i}{i+1}-\binom{t+i-e_{d}}{i+1}=\sum_{j=1}^{e_{d}}\binom{t+d-j+1}{d}$, by induction on $d$. This gives the expression

[TABLE]

and one can iterate on the second part. So $r=e_{0}$ and the partition $(b_{1},b_{2},\ldots,b_{r})$ has $e_{i}-e_{i+1}$ parts equal to $i$, for all $0\leq i\leq d$. The equalities $\sum_{j=1}^{r}b_{j}=\sum_{i=0}^{d}(e_{i}-e_{i+1})i=\sum_{i=1}^{d}e_{i}$ then show that $(b_{1},b_{2},\dotsc,b_{r})$ is conjugate to $(e_{1},e_{2},\dotsc,e_{d})$. ∎

For $\mathsf{p}(t)$ as in Lemma 2.4, it is convenient to refer to $(e_{0},e_{1},\dotsc,e_{d})$ as its Macaulay–Hartshorne partition and $(b_{1},b_{2},\dotsc,b_{r})$ as its (nonnegative) Gotzmann partition.

We now describe two fundamental binary relations on admissible Hilbert polynomials. The first takes the polynomial $\mathsf{p}$ with partitions $e=(e_{0},e_{1},\dotsc,e_{d})$ and $b=(b_{1},b_{2},\dotsc,b_{r})$ to the polynomial $\Phi(\mathsf{p})$ with partitions $(e_{0},e)$ and $b+(1,1,\ldots,1)$ (add one to each entry). The second takes $\mathsf{p}$ to $A(\mathsf{p})=1+\mathsf{p}$, with partitions $e+(1,0,\ldots,0)$ and $(b,0)$. Both $\Phi(\mathsf{p})$ and $A(\mathsf{p})$ are admissible by Proposition 2.3.

The backwards difference operator $\nabla$ maps any $\mathsf{q}\in\mathbb{Q}[t]$ to $\mathsf{q}(t)-\mathsf{q}(t-1)$. Backwards differences are discrete derivatives—in Lemma 2.5, (ii) says that $\Phi$ is the indefinite integral and (iii) is a well-known discrete analogue of the Fundamental Theorem of Calculus.

Lemma 2.5**.**

If $\mathsf{p}(t)$ is an admissible Hilbert polynomial with Macaulay–Hartshorne partition $(e_{0},e_{1},\dotsc,e_{d})$ and Gotzmann partition $(b_{1},b_{2},\dotsc,b_{r})$, then the following hold:

1. (i)

$\left[\nabla(\mathsf{p})\right](t)=\sum_{j=1}^{r}\binom{t+b_{j}-1-j+1}{b_{j}-1}=\sum_{i=0}^{d-1}\binom{t+i}{i+1}-\binom{t+i-e_{i+1}}{i+1}$; 2. (ii)

$\nabla A^{a}\Phi(\mathsf{p})=\mathsf{p}$, for all $a\in\mathbb{N}$; and 3. (iii)

if $\deg\mathsf{p}>0$ and $k\in\{1,2,\dotsc,r\}$ is the largest index such that $b_{k}\neq 0$, then we have $\mathsf{p}-\Phi\nabla(\mathsf{p})=r-k$, but if $\deg\mathsf{p}=0$, then $\nabla(\mathsf{p})=0$.

Proof.

These follow by linearity of $\nabla$ and the binomial addition formula. ∎

We now observe that the set of admissible Hilbert polynomials forms a tree.

Proposition 2.6**.**

The tree with vertices corresponding to admissible Hilbert polynomials and edges corresponding to pairs of the form \bigl{(}\mathsf{p},A(\mathsf{p})\bigr{)} and \bigl{(}\mathsf{p},\Phi(\mathsf{p})\bigr{)}, for all admissible Hilbert polynomials $\mathsf{p}$, forms an infinite full binary tree. The root of the tree corresponds to $1$.

We call this the Macaulay tree $\operatorname{\mathscr{M}}$. It has $2^{j}$ vertices at height $j$, for all $j\in\mathbb{N}$.

Proof.

By induction on $r$, $\mathsf{p}(t)=\sum_{j=1}^{r}\binom{t+b_{j}-j+1}{b_{j}}=\Phi^{b_{r}}A\Phi^{b_{r-1}-b_{r}}A\dotsb A\Phi^{b_{2}-b_{3}}A\Phi^{b_{1}-b_{2}}(1)$ holds. ∎

A portion of $\operatorname{\mathscr{M}}$ is displayed in Figure 1, in terms of Gotzmann expressions.

Remark 2.7**.**

The path from the root $1$ of the tree $\operatorname{\mathscr{M}}$ to $\mathsf{p}(t)=\sum_{i=0}^{d}\binom{t+i}{i+1}-\binom{t+i-e_{i}}{i+1}$ can also be expressed as $\mathsf{p}=A^{e_{0}-e_{1}}\Phi A^{e_{1}-e_{2}}\Phi\dotsb\Phi A^{e_{d-1}-e_{d}}\Phi A^{e_{d}-1}(1)$.

Example 2.8**.**

The Hilbert polynomial $3t+1$ of the twisted cubic curve $X\subset\operatorname{\mathbb{P}}^{3}$ has partitions $(b_{1},b_{2},b_{3},b_{4})=(1,1,1,0)$ and $(e_{0},e_{1})=(4,3)$. The path in $\operatorname{\mathscr{M}}$ from $1$ to $3t+1$ can be written as $\Phi^{0}A\Phi^{1-0}A\Phi^{1-1}A\Phi^{1-1}(1)=A^{4-3}\Phi A^{3-1}(1)=A\Phi A^{2}(1)$. This path is shown in Figure 2.

2.2. Lexicographic and Hilbert Trees

We now connect lexicographic ideals and Hilbert schemes to the Macaulay tree $\operatorname{\mathscr{M}}$. Specifically, $\operatorname{\mathscr{M}}$ reappears infinitely many times in the set of saturated lexicographic ideals and the set of Hilbert schemes, with one tree for each positive codimension. Two mappings on lexicographic ideals analogous to $\Phi$ and $A$ are essential.

For any vector $u=(u_{0},u_{1},\dotsc,u_{n})\in\mathbb{N}^{n+1}$, let $x^{u}=x_{0}^{u_{0}}x_{1}^{u_{1}}\dotsb x_{n}^{u_{n}}$. The lexicographic ordering is the relation $\operatorname{\succ}$ on the monomials in $\mathbb{K}[x_{0},x_{1},\dotsc,x_{n}]$ defined by $x^{u}\operatorname{\succ}x^{v}$ if the first nonzero coordinate of $u-v\in\mathbb{Z}^{n+1}$ is positive, where $u,v\in\mathbb{N}^{n+1}$.

Example 2.9**.**

We have $x_{0}\operatorname{\succ}x_{1}\operatorname{\succ}\dotsb\operatorname{\succ}x_{n}$ in lexicographic order on $\mathbb{K}[x_{0},x_{1},\dotsc,x_{n}]$. If $n\geq 2$, then $x_{0}x_{2}^{2}\operatorname{\succ}x_{1}^{4}\operatorname{\succ}x_{1}^{3}$.

Lexicographic ideals are monomial ideals whose homogeneous pieces are spanned by maximal monomials in lexicographic order. For a homogeneous ideal $I$ in $\mathbb{K}[x_{0},x_{1},\dotsc,x_{n}]$, lexicographic order gives rise to two monomial ideals associated to $I$. First, the lexicographic ideal for the Hilbert function $\mathsf{h}_{I}$ in $\mathbb{K}[x_{0},x_{1},\dotsc,x_{n}]$ is the monomial ideal $L^{\mathsf{h}_{I}}_{n}$ whose $i$th graded piece is spanned by the $\dim_{\mathbb{K}}I_{i}=\mathsf{h}_{\mathbb{K}[x_{0},x_{1},\dotsc,x_{n}]}(i)-\mathsf{h}_{I}(i)$ largest monomials in $\mathbb{K}[x_{0},x_{1},\dotsc,x_{n}]_{i}$, for all $i\in\mathbb{Z}$. The equality $\mathsf{h}_{I}=\mathsf{h}_{L^{\mathsf{h}_{I}}_{n}}$ holds by definition and $L^{\mathsf{h}_{I}}_{n}$ is a homogeneous ideal of $\mathbb{K}[x_{0},x_{1},\dotsc,x_{n}]$; see [Mac27, § II] or [MS05b, Proposition 2.21]. More importantly, the (saturated) lexicographic ideal for the Hilbert polynomial $\mathsf{p}_{I}$ is the monomial ideal

[TABLE]

Saturation with respect to the irrelevant ideal $\langle x_{0},x_{1},\dotsc,x_{n}\rangle\subset\mathbb{K}[x_{0},x_{1},\dotsc,x_{n}]$ does not affect the Hilbert function in large degrees, so $L^{\mathsf{p}_{I}}_{n}$ also has Hilbert polynomial $\mathsf{p}_{I}$. From here on, we essentially always work with saturated lexicographic ideals and point out when this is not the case.

Given a finite sequence of nonnegative integers $a_{0},a_{1},\dotsc,a_{n-1}\in\mathbb{N}$, consider the monomial ideal $L(a_{0},a_{1},\dots,a_{n-1})\subset\mathbb{K}[x_{0},x_{1},\dotsc,x_{n}]$ from [RS97, Notation 1.2] with generators

[TABLE]

Lemma 2.10(i) appears in [Moo12, Theorem 2.23].

Lemma 2.10**.**

Let $\mathsf{p}(t)=\sum_{i=0}^{d}\binom{t+i}{i+1}-\binom{t+i-e_{i}}{i+1}$, for integers $e_{0}\geq e_{1}\geq\dotsb\geq e_{d}>0$, and let $n\in\mathbb{N}$ satisfy $n>d=\deg\mathsf{p}$.

1. (i)

Define $e_{i}=0$, for $d+1\leq i\leq n$, and $a_{j}=e_{j}-e_{j+1}$, for all $0\leq j\leq n-1$. We have

[TABLE] 2. (ii)

If there is an integer $0\leq\ell\leq d-1$ such that $a_{j}=0$ for all $j\leq\ell$, and $a_{\ell+1}>0$, then the minimal monomial generators of $L^{\mathsf{p}}_{n}$ are given by $m_{1},m_{2},\dotsc,m_{n-\ell-1}$, where

[TABLE]

If $a_{0}\neq 0$, then the minimal monomial generators are those listed in (i).

Proof.

1. (i)

It is straightforward to check that the ideal $L=L(a_{0},a_{1},\dotsc,a_{n-1})$ is saturated and lexicographic. To see that $L$ has the correct Hilbert polynomial, one can first prove that $L^{\prime}=L(0,\dotsc,0,a_{d},0,\dotsc,0)$ has Hilbert polynomial $\sum_{i=0}^{d}\binom{t+i}{i+1}-\binom{t+i-a_{d}}{i+1}$. The general case then follows by induction on $d=\deg\mathsf{p}$ and the short exact sequence

[TABLE]

where $L^{\prime\prime}=L(a_{0},a_{1},\dotsc,a_{d-1},0,0,\dotsc,0)$ and the injection sends $1\mapsto x_{n-d-1}^{a_{d}}$. 2. (ii)

We know that $x_{n-d-1}^{a_{d}}x_{n-d}^{a_{d-1}}\dotsb x_{n-2}^{a_{1}}x_{n-1}^{a_{0}}=x_{n-d-1}^{a_{d}}x_{n-d}^{a_{d-1}}\dotsb x_{n-\ell-3}^{a_{\ell+2}}x_{n-\ell-2}^{a_{\ell+1}}$, because either $a_{0}=a_{1}=\dotsb=a_{\ell}=0$, or $a_{0}\neq 0$ and $\ell=-1$. If $\ell\geq 0$, then the monomial generators

[TABLE]

from (i) are redundant, as they are multiples of the last monomial generator. Removing these gives the monomial generators $m_{1},m_{2},\dotsc,m_{n-\ell-1}$. For all $1<i<n-\ell$ and all $0<j<i$ there exists $x_{k}$ dividing $m_{j}$ to higher order than the order to which it divides $m_{i}$ and minimality follows. ∎

The nonminimal list of generators in Lemma 2.10(i) is useful for describing operations on lexicographic ideals in terms of Macaulay–Hartshorne and Gotzmann partitions. Importantly, Lemma 2.10 shows that all sequences $(a_{0},a_{1},\dotsc,a_{n-1})$ of nonnegative integers determine a lexicographic ideal.

The next example uses Lemma 2.10 to identify minimal monomial generators.

Example 2.11**.**

The twisted cubic $X\subset\operatorname{\mathbb{P}}^{3}$ has $\mathsf{p}_{X}(t)=\left[\binom{t+0}{0+1}-\binom{t+0-4}{0+1}\right]+\left[\binom{t+1}{1+1}-\binom{t+1-3}{1+1}\right]$, with ideal $L^{3t+1}_{3}\subset\mathbb{K}[x_{0},x_{1},x_{2},x_{3}]$. We have $d=1$ and $(e_{0},e_{1},e_{2},e_{3})=(4,3,0,0)$, so that $(a_{0},a_{1},a_{2})=(1,3,0)$. Applying Lemma 2.10 yields $L^{3t+1}_{3}=L(1,3,0)=\langle x_{0},x_{1}^{4},x_{1}^{3}x_{2}\rangle$.

By analogy with $A$, we define the lex-expansion of $L^{\mathsf{p}}_{n}=L(a_{0},a_{1},\dotsc,a_{n-1})$ to be the lexicographic ideal A\bigl{(}L^{\mathsf{p}}_{n}\bigr{)}:=L(a_{0}+1,a_{1},a_{2},\dotsc,a_{n-1}).

Lemma 2.12**.**

Let $\mathsf{p}$ be an admissible Hilbert polynomial and $n>\deg\mathsf{p}$ a positive integer. We have A\bigl{(}L^{\mathsf{p}}_{n}\bigr{)}=L^{A(\mathsf{p})}_{n} and the mapping $A$ on lexicographic ideals preserves codimension.

Proof.

See Lemma 2.10(i). Note that $n-\deg A(\mathsf{p})=n-\deg\mathsf{p}$. ∎

In analogy with $\Phi$, for any ideal $I\subseteq\mathbb{K}[x_{0},x_{1},\dotsc,x_{n}]$, we denote the extension ideal by $\Phi(I)=I\cdot\mathbb{K}[x_{0},x_{1},\dotsc,x_{n+1}]$. The following is similar to Lemma 2.12.

Proposition 2.13**.**

Let $\mathsf{p}$ be an admissible Hilbert polynomial and $n>\deg\mathsf{p}$ an integer. We have \Phi\bigl{(}L^{\mathsf{p}}_{n}\bigr{)}=L^{\Phi(\mathsf{p})}_{n+1}. Equivalently \Phi\bigl{(}L(a_{0},a_{1},\dotsc,a_{n-1})\bigr{)}=L(0,a_{0},a_{1},\dotsc,a_{n-1}) holds, for all $a_{0},a_{1},\dotsc,a_{n-1}\in\mathbb{N}$, and extension preserves codimension.

Proof.

This follows by relabelling $a_{0}^{\prime}=0$ and $a_{i}^{\prime}=a_{i-1}$, for $0<i\leq d+1$, on generators. ∎

The lexicographic point of $\operatorname{Hilb}^{\mathsf{p}}(\operatorname{\mathbb{P}}^{n})$ is the point \big{[}X_{L^{\mathsf{p}}_{n}}\big{]}, where $X_{I}\subseteq\operatorname{\mathbb{P}}^{n}$ denotes the closed subscheme with saturated ideal $I$. The lexicographic point is nonsingular and lies on a unique irreducible component $\operatorname{Hilb}^{\mathsf{p}}(\operatorname{\mathbb{P}}^{n})$ called the lexicographic component [RS97]. We can now describe a tree structure on the set of Hilbert schemes. We regard this as a rough chart of the geography of Hilbert schemes, developed further in Section 5.

Theorem 2.14**.**

For each positive codimension $c\in\mathbb{Z}$, the graph $\operatorname{\mathscr{H}}_{c}$ whose vertex set consists of all nonempty Hilbert schemes $\operatorname{Hilb}^{\mathsf{p}}(\operatorname{\mathbb{P}}^{n})$ parametrizing codimension $c$ subschemes and whose edges are all pairs $\left(\operatorname{Hilb}^{\mathsf{p}}(\operatorname{\mathbb{P}}^{n}),\operatorname{Hilb}^{A(\mathsf{p})}(\operatorname{\mathbb{P}}^{n})\right)$ and $\left(\operatorname{Hilb}^{\mathsf{p}}(\operatorname{\mathbb{P}}^{n}),\operatorname{Hilb}^{\Phi(\mathsf{p})}(\operatorname{\mathbb{P}}^{n+1})\right)$, where $\mathsf{p}$ is an admissible Hilbert polynomial and $n=c+\deg\mathsf{p}$, is an infinite full binary tree. The root of the tree $\operatorname{\mathscr{H}}_{c}$ is the Hilbert scheme $\operatorname{Hilb}^{1}(\operatorname{\mathbb{P}}^{c})=\operatorname{\mathbb{P}}^{c}$.

Proof.

Each pair of an admissible Hilbert polynomial $\mathsf{p}$ and positive $c\in\mathbb{Z}$ uniquely determines $L=L^{\mathsf{p}}_{c+\deg\mathsf{p}}$ and the Hilbert scheme $\operatorname{Hilb}^{\mathsf{p}}(\operatorname{\mathbb{P}}^{c+\deg\mathsf{p}})$ containing $[X_{L}]$. Lemma 2.12 and Proposition 2.13 combined with Lemma 2.10 show that the mapping $\mathsf{p}\mapsto\operatorname{Hilb}^{\mathsf{p}}(\operatorname{\mathbb{P}}^{c+\deg(\mathsf{p})})$ is a graph isomorphism. The root is then $\operatorname{Hilb}^{1}(\operatorname{\mathbb{P}}^{c})=\operatorname{\mathbb{P}}^{c}$. ∎

For positive $c\in\mathbb{Z}$, we call the tree of Theorem 2.14 the Hilbert tree of codimension $c$, denoted $\operatorname{\mathscr{H}}_{c}$. We call the disjoint union $\operatorname{\mathscr{H}}=\bigsqcup_{c\in\mathbb{N},c>0}\operatorname{\mathscr{H}}_{c}$ the Hilbert forest.

3. Strongly Stable Ideals

This section reviews a well-known algorithm that generates saturated strongly stable ideals and then examines analogues of $\Phi$ and $A$ for strongly stable ideals.

A monomial ideal $I\subseteq\mathbb{K}[x_{0},x_{1},\dotsc,x_{n}]$ is strongly stable, or [math]-Borel, if, for all monomials $m\in I$, for all $x_{j}$ dividing $m$, and for all $x_{i}\operatorname{\succ}x_{j}$, we have $x_{j}^{-1}mx_{i}\in I$. In characteristic [math], this is equivalent to being Borel-fixed, i.e. fixed by the action $\gamma\cdot x_{j}=\sum_{i=0}^{n}\gamma_{ij}x_{i}$ of upper triangular matrices $\gamma\in\operatorname{GL}_{n+1}(\mathbb{K})$ [BS87a, Proposition 2.7]. For any monomial $m$, let $\max m$ be the maximum index $j$ such that $x_{j}$ divides $m$, and $\min m$ be the minimum such index.

Example 3.1**.**

The monomial ideal $I=\langle x_{0}^{2},x_{0}x_{1},x_{1}^{2}\rangle\subset\mathbb{K}[x_{0},x_{1},x_{2}]$ is strongly stable. The monomial $m=x_{1}^{5}x_{2}x_{7}^{2}\in\mathbb{K}[x_{0},x_{1},\dotsc,x_{13}]$ satisfies $\max m=7$ and $\min m=1$.

For a monomial ideal $I$, let $G(I)$ denote its minimal set of monomial generators. We gather some useful properties of strongly stable ideals in the following lemma.

Lemma 3.2**.**

Let $I\subseteq\mathbb{K}[x_{0},x_{1},\dotsc,x_{n}]$ be a monomial ideal.

1. (i)

The ideal $I$ is strongly stable if and only if, for all $g\in G(I)$, for all $x_{j}$ dividing $g$, and for all $x_{i}\operatorname{\succ}x_{j}$, we have $gx_{i}x_{j}^{-1}\in I$. 2. (ii)

If, for all $g^{\prime}\in G(I)$, for all $x_{j}$ dividing $g^{\prime}$, and for all $x_{i}\operatorname{\succ}x_{j}$, we have $g^{\prime}x_{i}x_{j}^{-1}\in I$, then, for all monomials $m\in I$, there exists a unique $g\in G(I)$ and unique monomial $m^{\prime}\in\mathbb{K}[x_{0},x_{1},\dotsc,x_{n}]$ such that $m=gm^{\prime}$ and $\max g\leq\min m^{\prime}$. 3. (iii)

If $I$ is a strongly stable ideal, then $I$ is saturated with respect to the irrelevant ideal $\langle x_{0},x_{1},\dotsc,x_{n}\rangle$ if and only if no minimal monomial generators of $I$ are divisible by $x_{n}$. 4. (iv)

If $I$ is strongly stable with constant Hilbert polynomial $\mathsf{p}_{I}\in\mathbb{N}$, then there exists an integer $k\in\mathbb{N}$ such that $x_{n-1}^{k}\in I$.

Proof.

1. (i)

If $I$ is strongly stable, then this holds for all $g\in I$. Conversely, let $m\in I$ be any monomial. By (ii), there is a unique factorization $m=gm^{\prime}$, where $g\in G(I)$ and $m^{\prime}\in\mathbb{K}[x_{0},x_{1},\dotsc,x_{n}]$ is a monomial such that $\max g\leq\min m^{\prime}$. Let $x_{j}$ divide $m$ and let $x_{i}\operatorname{\succ}x_{j}$. Either $x_{j}$ divides $g$, in which case $gx_{i}x_{j}^{-1}\in I$ and $mx_{i}x_{j}^{-1}\in I$, or $x_{j}$ divides $m^{\prime}$, in which case $mx_{i}x_{j}^{-1}$ is a multiple of $g$. 2. (ii)

See the proof of [MS05b, Lemma 2.11]. 3. (iii)

If $x_{n}$ divides a minimal generator $g\in G(I)$, then for all $x_{j}$ we have $(gx_{n}^{-1})x_{j}\in I$, while $gx_{n}^{-1}\notin I$. Conversely, any monomial $m\in(I:\langle x_{0},x_{1},\dotsc,x_{n}\rangle)\setminus I$ yields a minimal monomial generator $mx_{n}\in G(I)$, by (ii). 4. (iv)

See the proof of [Moo12, Lemma 3.17]. ∎

Strongly stable ideals are generated by expansions. Let $I\subseteq\mathbb{K}[x_{0},x_{1},\dotsc,x_{n}]$ be a saturated strongly stable ideal. A generator $g\in G(I)$ is expandable if there are no minimal monomial generators of $I$ in the set $\left\{x_{i}^{-1}gx_{i+1}\mid x_{i}\text{ divides }g\text{ and }0\leq i<n-1\right\}$. The expansion of $I$ at an expandable generator $g$ is the monomial ideal

[TABLE]

see [Moo12, Definition 3.4]. The monomial $1\in\langle 1\rangle$ is vacuously expandable with expansion $\langle x_{0},x_{1},\dotsc,x_{n-1}\rangle\subset\mathbb{K}[x_{0},x_{1},\dotsc,x_{n}]$. Parts (i) and (iii) of Lemma 3.2 ensure that the expansion of a saturated strongly stable ideal is again saturated and strongly stable.

Example 3.3**.**

Let $L^{\mathsf{p}}_{n}=L(a_{0},a_{1},\dotsc,a_{n-1})=\langle m_{1},m_{2},\dotsc,m_{n-\ell-1}\rangle$ be lexicographic, as in Lemma 2.10. The last generator is $m_{n-\ell-1}=x_{n-d-1}^{a_{d}}x_{n-d}^{a_{d-1}}\dotsb x_{n-\ell-3}^{a_{\ell+2}}x_{n-\ell-2}^{a_{\ell+1}}$. If $a_{i}>0$, then $x_{n-i-1}$ divides $m_{n-\ell-1}x_{n-i}x_{n-i-1}^{-1}$ to order $a_{i}-1$, which is not the case for any $m_{j}$. Therefore, the expansion at $m_{n-\ell-1}$ has generators

[TABLE]

These are easily verified to be minimal generators of L(a_{0}+1,a_{1},a_{2},\dotsc,a_{n-1})=A\bigl{(}L^{\mathsf{p}}_{n}\bigr{)}.

For a saturated strongly stable ideal $I\subseteq\mathbb{K}[x_{0},x_{1},\dotsc,x_{n}]$, let $\nabla\left(I\right)\subseteq\mathbb{K}[x_{0},x_{1},\dotsc,x_{n-1}]$ be the image of $I$ under the mapping $\mathbb{K}[x_{0},x_{1},\dotsc,x_{n}]\to\mathbb{K}[x_{0},x_{1},\dotsc,x_{n-1}]$, defined by $x_{j}\mapsto x_{j}$ for $0\leq j\leq n-2$ and $x_{j}\mapsto 1$ for $n-1\leq j\leq n$. This is the saturation of $I\cap\mathbb{K}[x_{0},x_{1},\dotsc,x_{n-1}]$.

The following lemma generalizes Lemma 2.12 and Proposition 2.13 to strongly stable ideals.

Lemma 3.4**.**

Let $I$ be a saturated strongly stable ideal.

1. (i)

If $I^{\prime}$ is any expansion of $I$, then we have $\mathsf{p}_{I^{\prime}}=A(\mathsf{p}_{I})$. 2. (ii)

We have $\mathsf{p}_{\nabla\left(I\right)}=\nabla(\mathsf{p}_{I})$. 3. (iii)

There exists $j\in\mathbb{N}$ such that $\mathsf{p}_{\Phi(I)}=A^{j}\Phi(\mathsf{p}_{I})$.

Proof.

1. (i)

Let $I^{\prime}$ be the expansion of $I$ at $g$. For all $d\geq\deg g$, Lemma 3.2(ii) shows that the only monomial in $I_{d}\setminus I^{\prime}_{d}$ is $gx_{n}^{d-\deg g}$, so that $\mathsf{h}_{I^{\prime}}(d)=1+\mathsf{h}_{I}(d)$. Hence, we have $\mathsf{p}_{I^{\prime}}=1+\mathsf{p}_{I}$. 2. (ii)

Let $S=\mathbb{K}[x_{0},x_{1},\dotsc,x_{n}]$ and $J=I\cap\mathbb{K}[x_{0},x_{1},\dotsc,x_{n-1}]$. Because $I$ is saturated, $x_{n}$ is not a zero-divisor and

[TABLE]

is a short exact sequence. The Hilbert function of $J$ now satisfies $\mathsf{h}_{J}(i)=\mathsf{h}_{I}(i)-\mathsf{h}_{I}(i-1)$, for all $i\in\mathbb{Z}$. Hence, by saturating $J$ with respect to $\langle x_{0},x_{1},\dotsc,x_{n-1}\rangle$, we find that $\mathsf{p}_{\nabla\left(I\right)}(t)=\mathsf{p}_{I}(t)-\mathsf{p}_{I}(t-1)=\left[\nabla(\mathsf{p}_{I})\right](t)$. 3. (iii)

The ideal $\Phi(I)$ is saturated strongly stable by Lemma 3.2(i),(iii) with no minimal monomial generators divisible by $x_{n}$. Thus, \nabla\bigl{(}\Phi(I)\bigr{)}=I. Part (ii) shows that $\nabla(\mathsf{p}_{\Phi(I)})=\mathsf{p}_{I}$, so that $\Phi\nabla(\mathsf{p}_{\Phi(I)})=\Phi(\mathsf{p}_{I})$. Lemma 2.5(iii) then shows $\mathsf{p}_{\Phi(I)}-\Phi\nabla(\mathsf{p}_{\Phi(I)})\in\mathbb{N}$. ∎

The heart of Reeves’ algorithm [Ree92] is the following.

Theorem 3.5**.**

If $I$ is a saturated strongly stable ideal of codimension $c$, then there is a finite sequence $I_{(0)},I_{(1)},\ldots,I_{(i)}$ such that $I_{(0)}=\langle 1\rangle=\mathbb{K}[x_{0},x_{1},\dotsc,x_{c}]$, $I_{(i)}=I$, and $I_{(j)}$ is an expansion or extension of $I_{(j-1)}$, for all $1\leq j\leq i$.

Proof.

See [Moo12, Theorem 3.20] or [MN14, Theorem 4.4]. ∎

When $I\neq\langle 1\rangle$, the first step is always to expand $\langle 1\rangle$ to $\langle x_{0},x_{1},\ldots,x_{c-1}\rangle$. We will start after this step, as we assume $\mathsf{p}\neq 0$. The sequences of expansions and extensions are not generally unique, but Theorem 2.14 shows that they are for lexicographic ideals. Theorem 3.5 leads to the following algorithm. The original is in [Ree92, Appendix A], but we follow [Moo12, MN14]; see also [CLMR11, Section 5].

Algorithm 3.6**.**

[TABLE]

Proof.

See [Moo12, Algorithm 3.22]. Here, $\mathcal{S}$ is reset at the $j$th step to Moore’s $\mathcal{S}^{(d-j)}$. ∎

Example 3.7**.**

To compute all codimension $2$ saturated strongly stable ideals with Hilbert polynomial $\mathsf{p}(t)=3t+1$, we first compute $\nabla(\mathsf{p})=3$. We produce all length $2$ sequences of expansions beginning at $\langle x_{0},x_{1}\rangle\subset\mathbb{K}[x_{0},x_{1},x_{2}]$. The only expandable generator of $\langle x_{0},x_{1}\rangle$ is $x_{1}$, with expansion $\langle x_{0},x_{1}^{2}\rangle$. Both $x_{0}$ and $x_{1}^{2}$ are expandable in $\langle x_{0},x_{1}^{2}\rangle$, with expansions $\langle x_{0}^{2},x_{0}x_{1},x_{1}^{2}\rangle,\langle x_{0},x_{1}^{3}\rangle\subset\mathbb{K}[x_{0},x_{1},x_{2}]$. Extending each of these to $\mathbb{K}[x_{0},x_{1},x_{2},x_{3}]$, their Hilbert polynomials are $3t+1$ and $3t$, respectively. Thus, we make all possible expansions of $\langle x_{0},x_{1}^{3}\rangle$; expansion at $x_{0}$ gives $\langle x_{0}^{2},x_{0}x_{1},x_{0}x_{2},x_{1}^{3}\rangle$, and expansion at $x_{1}^{3}$ gives $\langle x_{0},x_{1}^{4},x_{1}^{3}x_{2}\rangle$. Hence, there are three codimension $2$ saturated strongly stable ideals with Hilbert polynomial $3t+1$ namely, $\langle x_{0}^{2},x_{0}x_{1},x_{1}^{2}\rangle$, $\langle x_{0}^{2},x_{0}x_{1},x_{0}x_{2},x_{1}^{3}\rangle$, and $\langle x_{0},x_{1}^{4},x_{1}^{3}x_{2}\rangle$ in $\mathbb{K}[x_{0},x_{1},x_{2},x_{3}]$.

4. K-Polynomials and Climbing Trees

The goal of this section is to understand where Hilbert functions and Hilbert polynomials of saturated strongly stable ideals coincide. Theorem 4.6 states that among the saturated strongly stable ideals with a fixed codimension and Hilbert polynomial, the degree of the $K$-polynomial of the lexicographic ideal is strictly the largest. The proof tracks the genesis of minimal monomial generators as Algorithm 3.6 traces the path from $1$ to $\mathsf{p}$ in $\operatorname{\mathscr{M}}$. Proposition 4.4 identifies where the inequality first occurs and Proposition 4.5 shows that it persists.

The Hilbert series of a finitely generated graded $\mathbb{K}[x_{0},x_{1},\dotsc,x_{n}]$-module $M$ is the formal power series $\mathsf{H}_{M}(T)=\sum_{i\in\mathbb{Z}}\mathsf{h}_{M}(i)\,T^{i}\in\mathbb{Z}[T^{-1}][\![T]\!]$. The Hilbert series of $M$ is a rational function $\mathsf{H}_{M}(T)=(1-T)^{-n-1}\mathsf{K}_{M}(T)$ and the $K$-polynomial of $M$ is the numerator $\mathsf{K}_{M}$, possibly divisible by $1-T$, of $\mathsf{H}_{M}$; see [MS05b, Theorem 8.20]. For a quotient by a homogeneous ideal $I$, we use the notation $\mathsf{H}_{I}=\mathsf{H}_{\mathbb{K}[x_{0},x_{1},\dotsc,x_{n}]/I}$ and $\mathsf{K}_{I}=\mathsf{K}_{\mathbb{K}[x_{0},x_{1},\dotsc,x_{n}]/I}$.

We consider a fundamental example.

Example 4.1**.**

If $S=\mathbb{K}[x_{0},x_{1},\dotsc,x_{n}]$, then we have $\mathsf{K}_{S}(T)=1$, as $\mathsf{H}_{S}(T)=(1-T)^{-n-1}$. If $d\in\mathbb{N}$, then we have $\mathsf{H}_{S(-d)}(T)=(1-T)^{-n-1}T^{d}$ and $\mathsf{K}_{S(-d)}(T)=T^{d}$.

The following well-known lemma is useful. As before, $G(I)$ denotes the minimal set of monomial generators of a monomial ideal $I$.

Lemma 4.2**.**

Let $I\subseteq\mathbb{K}[x_{0},x_{1},\dotsc,x_{n}]$ be a strongly stable ideal.

1. (i)

We have $\mathsf{K}_{I}(T)=1-\sum_{g\in G(I)}T^{\,\deg g}(1-T)^{\max g}$. 2. (ii)

We have \deg\mathsf{K}_{I}\leq\max_{g\in G(I)}\bigl{\{}\deg g+\max g\bigr{\}}.

Proof.

1. (i)

This follows by Lemma 3.2(ii) and Example 4.1; also see [MS05b, Proposition 2.12]. 2. (ii)

This follows immediately from (i). ∎

Let $\deg\mathsf{H}_{I}:=\deg\mathsf{K}_{I}-n-1$. The next lemma establishes that $\deg\mathsf{H}_{I}$ is the maximal value where $\mathsf{h}_{I}$ and $\mathsf{p}_{I}$ differ.

Lemma 4.3**.**

Let $I\subseteq\mathbb{K}[x_{0},x_{1},\dotsc,x_{n}]$ be a homogeneous ideal with rational Hilbert series $\mathsf{H}_{I}(T)=\sum_{i\in\mathbb{Z}}\mathsf{h}_{I}(i)\,T^{i}=\mathsf{K}_{I}(T)(1-T)^{-n-1}$ and Hilbert polynomial $\mathsf{p}_{I}$. We have $\mathsf{h}_{I}(i)=\mathsf{p}_{I}(i)$ for all $i>\deg\mathsf{H}_{I}$, while $\mathsf{h}_{I}(i)\neq\mathsf{p}_{I}(i)$ for $i=\deg\mathsf{H}_{I}$.

Proof.

Let $\mathsf{K}_{I}(T)=\sum_{k=0}^{d}c_{k}T^{k}\in\mathbb{Z}[T]$. Expanding $(1-T)^{-n-1}$ and gathering terms yields

[TABLE]

Thus, the Hilbert function equals $\mathsf{h}_{I}(i)=c_{0}\binom{n+i}{n}+c_{1}\binom{n+i-1}{n}+\dotsb+c_{d}\binom{n+i-d}{n}$, for all $i\in\mathbb{Z}$, and the Hilbert polynomial is $\mathsf{p}_{I}(t)=c_{0}\binom{t+n}{n}+c_{1}\binom{t+n-1}{n}+\dotsb+c_{d}\binom{t+n-d}{n}$. Following our convention in Remark 2.2, the equality $\binom{n+i-j}{n}=\left.\binom{t+n-j}{n}\right\rvert_{t=i}$ holds if and only if $i\geq-n+j$. This implies that $\mathsf{h}_{I}(i)=\mathsf{p}_{I}(i)$ whenever $i\geq-n+d=1+\deg\mathsf{H}_{I}$, proving the first statement. To finish, set $i=d-n-1=\deg\mathsf{H}_{I}$ and compare the value

[TABLE]

with the value $\mathsf{h}_{I}(i)=\sum_{j=0}^{d-1}c_{j}\binom{i+n-j}{n}+c_{d}\cdot 0$. As $c_{d}\neq 0$, we are finished. ∎

In the next two propositions, let $L^{\mathsf{p}}_{n}=\langle m_{1},m_{2},\dotsc,m_{n-\ell-1}\rangle$, as in Lemma 2.10. These propositions examine the behaviour of $\deg\mathsf{K}_{I}$ for saturated strongly stable ideals $I$.

Proposition 4.4**.**

Let $L^{\mathsf{p}}_{n}\subset\mathbb{K}[x_{0},x_{1},\dotsc,x_{n}]$ be any lexicographic ideal.

1. (i)

If $m\in G(L^{\mathsf{p}}_{n})$ is a minimal monomial generator, then $m$ is expandable if and only if $m$ is the smallest minimal monomial generator of its degree. 2. (ii)

Let $m_{n-\ell-1}$ denote the last minimal monomial generator of $L^{\mathsf{p}}_{n}$. If $m\neq m_{n-\ell-1}$ is any other expandable generator of $L^{\mathsf{p}}_{n}$ and $(L^{\mathsf{p}}_{n})^{\prime}$ is the expansion of $L^{\mathsf{p}}_{n}$ at $m$, then every minimal monomial generator $g\in G((L^{\mathsf{p}}_{n})^{\prime})$ satisfies $\deg g<1+\deg m_{n-\ell-1}$. 3. (iii)

Moreover, in (ii), we have $\deg\mathsf{K}_{L^{A(\mathsf{p})}_{n}}>\deg\mathsf{K}_{(L^{\mathsf{p}}_{n})^{\prime}}$.

Proof.

1. (i)

By inspection, if $\deg m_{j}=\deg m_{j+1}$ holds, then $m_{j}$ is not expandable. 2. (ii)

By (i), we have $\deg m<\deg m_{n-\ell-1}$. But the minimal generators of $(L^{\mathsf{p}}_{n})^{\prime}$ are

[TABLE]

and $\deg m_{j}$ is maximized at $j=n-\ell-1$, which gives the inequality. 3. (iii)

We know $L^{A(\mathsf{p})}_{n}=\langle m_{1},m_{2},\dotsc,m_{n-\ell-2},m_{n-\ell-1}x_{n-\ell-2},m_{n-\ell-1}x_{n-\ell-1},\dotsc,m_{n-\ell-1}x_{n-1}\rangle$, by Example 3.3. As $\deg m_{j}$ is maximized at $m_{n-\ell-1}$, we get $\deg\mathsf{K}_{L^{A(\mathsf{p})}_{n}}=\deg m_{n-\ell-1}+n$. Then (ii) and Lemma 4.2(ii) yield the desired inequality. ∎

Proposition 4.5 explains how the $K$-polynomial inequality in Proposition 4.4 persists.

Proposition 4.5**.**

Let $I\subset\mathbb{K}[x_{0},x_{1},\dotsc,x_{n}]$ be a saturated strongly stable ideal, $\mathsf{p}=\mathsf{p}_{I}$, and $m_{n-\ell-1}$ be the last minimal generator of $L^{\mathsf{p}}_{n}$. Consider the following condition on $I$:

[TABLE]

If $I$ satisfies (4.5), then the following are true:

1. (i)

$\deg\mathsf{K}_{L^{\mathsf{p}}_{n}}>\deg\mathsf{K}_{I}$, or equivalently, $\deg\mathsf{H}_{L^{\mathsf{p}}_{n}}>\deg\mathsf{H}_{I}$; 2. (ii)

if $I^{\prime}$ denotes any expansion of $I$, then $I^{\prime}$ satisfies (4.5) with respect to $L^{A(\mathsf{p})}_{n}$; 3. (iii)

the extension $\Phi(I)$ satisfies (4.5) with respect to $L^{\mathsf{p}_{\Phi(I)}}_{n+1}$; and 4. (iv)

if $I_{(0)},I_{(1)},\dotsc,I_{(i)}$ is any finite sequence such that $I_{(0)}=I$ and $I_{(j)}$ is an expansion or extension of $I_{(j-1)}$, for all $0<j\leq i$, then $I_{(i)}$ satisfies (4.5).

Proof.

1. (i)

This follows immediately from Lemma 4.2(ii) and (4.5). 2. (ii)

The condition (4.5) for the expansion $I^{\prime}$ becomes: every generator $g^{\prime}\in G(I^{\prime})$ satisfies $\deg g^{\prime}<1+\deg m_{n-\ell-1}$ and $\max g^{\prime}\leq n-1$. Both inequalities hold, by definition of the minimal monomial generators of $I^{\prime}$ and because $I$ satisfies (4.5). 3. (iii)

An analogous condition to (4.5) holds between $\Phi(I)$ and $L^{\Phi(\mathsf{p})}_{n}$. Replacing $L^{\Phi(\mathsf{p})}_{n}$ by $L^{A^{j}\Phi(\mathsf{p})}_{n}$, with $j$ defined by Lemma 3.4(iii), results in higher degree and maximum index of the last minimal generator of $L^{A^{j}\Phi(\mathsf{p})}_{n}$; cf. Example 3.3. Hence, $\Phi(I)$ satisfies (4.5). 4. (iv)

We apply induction to $i$. The case $i=1$ is resolved by (ii) and (iii). If $i>1$, then (ii) and (iii) ensure that $I_{(1)}$ satisfies (4.5), and we apply the induction hypothesis. ∎

Combining Propositions 4.4 and 4.5 leads to the main result of this section.

Theorem 4.6**.**

Let $I\subset\mathbb{K}[x_{0},x_{1},\dotsc,x_{n}]$ be a saturated strongly stable ideal and denote $\mathsf{p}=\mathsf{p}_{I}$. If $I\neq L^{\mathsf{p}}_{n}$, then we have $\deg\mathsf{K}_{L^{\mathsf{p}}_{n}}>\deg\mathsf{K}_{I}$, or equivalently, $\deg\mathsf{H}_{L^{\mathsf{p}}_{n}}>\deg\mathsf{H}_{I}$.

Proof.

Both $I$ and $L^{\mathsf{p}}_{n}$ are generated by Algorithm 3.6. Let $c$ be their codimension and $I_{(1)},I_{(2)},\dotsc,I_{(i)}$ be a finite sequence such that $I_{(1)}=\langle x_{0},x_{1},\dotsc,x_{c-1}\rangle\subset\mathbb{K}[x_{0},x_{1},\dotsc,x_{c}]$, $I_{(i)}=I$, and $I_{(j)}$ is an expansion or extension of $I_{(j-1)}$, for all $1<j\leq i$. Theorem 2.14 implies that if $I\neq L^{\mathsf{p}}_{n}$, then there is some $2\leq k\leq i$ such that $I_{(j)}$ is lexicographic, for all $1\leq j\leq k-1$, but $I_{(k)}$ is not. By Proposition 4.4(i), $I_{(k)}$ is the expansion of $I_{(k-1)}$ at a minimal generator of nonmaximal degree. Proposition 4.4(ii) then shows that $I_{(k)}$ satisfies (4.5). Applying Proposition 4.5(iv) to the subsequence $I_{(k)},I_{(k+1)},\dotsc,I_{(i)}$ shows that $I_{(i)}=I$ satisfies (4.5), hence, applying Proposition 4.5(i) finishes the proof. ∎

Proof of Theorem 1.3.

Theorem 4.6 proves the claim. ∎

Example 4.7**.**

Example 3.7 shows the saturated strongly stable ideals in $\mathbb{K}[x_{0},x_{1},x_{2},x_{3}]$ with Hilbert polynomial $3t+1$ are $\langle x_{0}^{2},x_{0}x_{1},x_{1}^{2}\rangle$, $\langle x_{0}^{2},x_{0}x_{1},x_{0}x_{2},x_{1}^{3}\rangle$, and $L^{3t+1}_{3}=\langle x_{0},x_{1}^{4},x_{1}^{3}x_{2}\rangle$. Lemma 4.2(i) yields $\deg\mathsf{K}_{L^{3t+1}_{3}}=6$, $\deg\mathsf{K}_{\langle x_{0}^{2},x_{0}x_{1},x_{0}x_{2},x_{1}^{3}\rangle}=3$, and $\deg\mathsf{K}_{\langle x_{0}^{2},x_{0}x_{1},x_{1}^{2}\rangle}=3$.

Corollary 4.8**.**

Let $I\subset\mathbb{K}[x_{0},x_{1},\dotsc,x_{n}]$ be a saturated strongly stable ideal and let $\mathsf{p}=\mathsf{p}_{I}$. If $I\neq L^{\mathsf{p}}_{n}$, then there exists $k\in\mathbb{Z}$ such that $\mathsf{h}_{I}(j)=\mathsf{p}(j)$, for all $j\geq k$, but $\mathsf{h}_{L^{\mathsf{p}}_{n}}(k)\neq\mathsf{p}(k)$.

Proof.

Lemma 4.3 and Theorem 4.6 show that this is the case for $k=\deg\mathsf{H}_{L^{\mathsf{p}}_{n}}$. ∎

5. The Ubiquity of Smooth Hilbert Schemes

The goal now is to investigate our proposed geography of Hilbert schemes, formally described by the collection of trees $\operatorname{\mathscr{H}}$, by applying the Hilbert series inequalities of Theorem 4.6. Surprisingly, we recover a not-so-well-known family of irreducible Hilbert schemes, first discovered by Gotzmann [Got89, Proposition 1]. Moreover, we observe that a complete classification of Hilbert schemes with unique strongly stable ideals can be given by examining how Reeves’ algorithm interacts with $\operatorname{\mathscr{H}}_{c}$. These Hilbert schemes are nonsingular and irreducible over algebraically closed or characteristic [math] fields, and under natural probability distributions on the trees $\operatorname{\mathscr{H}}_{c}$ occur with probability at least $0.5$.

The next two lemmas are used to prove the classification Theorem 5.3.

Lemma 5.1**.**

Let $I\subset\mathbb{K}[x_{0},x_{1},\dotsc,x_{n}]$ be a homogeneous ideal, and let $\mathsf{p}=\mathsf{p}_{I}$.

1. (i)

We have $\mathsf{h}_{I}(i)\geq\mathsf{h}_{L^{\mathsf{p}}_{n}}(i)$, for all $i\in\mathbb{Z}$, where $L^{\mathsf{p}}_{n}$ is the corresponding lexicographic ideal. 2. (ii)

The Hilbert function of $\Phi(I)=I\cdot\mathbb{K}[x_{0},x_{1},\dotsc,x_{n+1}]$ is given by $\mathsf{h}_{\Phi(I)}(i)=\sum_{0\leq j\leq i}\mathsf{h}_{I}(j)$.

Proof.

1. (i)

Section 2.2 defines the (possibly unsaturated) lexicographic ideal $L^{\mathsf{h}}_{n}\subset\mathbb{K}[x_{0},x_{1},\dotsc,x_{n}]$ for the Hilbert function $\mathsf{h}=\mathsf{h}_{I}$. We have

[TABLE]

for all $i\in\mathbb{Z}$, because $L^{\mathsf{p}}_{n}$ contains $L^{\mathsf{h}}_{n}$. It follows that $\mathsf{h}(i)\geq\mathsf{h}_{L^{\mathsf{p}}_{n}}(i)$. 2. (ii)

The homogeneous piece $\left(\Phi(I)\right)_{i}$ has decomposition

[TABLE]

and the desired equality follows directly. ∎

Lemma 5.2**.**

Let $c>0$, $\mathsf{p}$ be an admissible Hilbert polynomial, and $\Lambda$ be a finite sequence of $\Phi$’s and $A$’s. The number of expandable minimal monomial generators of $L^{\Lambda(\mathsf{p})}_{c+\deg\Lambda(\mathsf{p})}$ is greater than or equal to the corresponding number for $L^{\mathsf{p}}_{c+\deg\mathsf{p}}$.

Proof.

This follows from Proposition 4.4(i) and the definition of expandable. ∎

We can now prove our classification result.

Theorem 5.3**.**

Let $\mathsf{p}(t)=\sum_{j=1}^{r}\binom{t+b_{j}-j+1}{b_{j}}$, for $b_{1}\geq b_{2}\geq\dotsb\geq b_{r}\geq 0$. The lexicographic ideal is the unique saturated strongly stable ideal of codimension $c$ with Hilbert polynomial $\mathsf{p}$ if and only if at least one of the following holds: (i) $b_{r}>0$, (ii) $c\geq 2$ and $r\leq 2$, (iii) $c=1$ and $b_{1}=b_{r}$, or (iv) $c=1$ and $r-s\leq 2$, where $b_{1}=b_{2}=\dotsb=b_{s}>b_{s+1}\geq\dotsb\geq b_{r}$.

Proof.

Let $b_{r}>0$. Proposition 2.6 gives $\mathsf{p}=\Phi^{b_{r}}A\Phi^{b_{r-1}-b_{r}}\dotsb A\Phi^{b_{2}-b_{3}}A\Phi^{b_{1}-b_{2}}(1)$, so there exists $\mathsf{q}$ such that $\mathsf{p}=\Phi(\mathsf{q})$. Saturated strongly stable ideals are generated by Algorithm 3.6. The procedure is recursive and generates the codimension $c$ saturated strongly stable ideals with Hilbert polynomial $\mathsf{p}$ by extending all codimension $c$ saturated strongly stable ideals with Hilbert polynomial $\mathsf{q}=\nabla\Phi(\mathsf{q})$ and keeping those with Hilbert polynomial $\mathsf{p}$.

By Proposition 2.13, we have $\Phi(L^{\mathsf{q}}_{n})=L^{\mathsf{p}}_{n+1}\subset\mathbb{K}[x_{0},x_{1},\dotsc,x_{n+1}]$, where $n=c+\deg\mathsf{q}$. It suffices to prove the following statement:

If $J\subset\mathbb{K}[x_{0},\dotsc,x_{n}]$ is a saturated, strongly stable, nonlexicographic ideal, then $\mathsf{p}_{\Phi(J)}\neq\Phi(\mathsf{p}_{J})$.

Let $I=\Phi(J)$ be the extension of such an ideal, $\mathsf{q}=\mathsf{p}_{J}$, and $\mathsf{p}=\Phi(\mathsf{q})$. By Lemma 3.4(iii), we must show that $\mathsf{p}_{I}-\mathsf{p}>0$. Setting $d_{\mathsf{q}}=\deg\mathsf{H}_{L^{\mathsf{q}}_{n}}$, we show that $\mathsf{h}_{I}(i)>\mathsf{h}_{L^{\mathsf{p}}_{n+1}}(i)$, for all integers $i\geq d_{\mathsf{q}}$. Lemma 5.1(ii) implies that

[TABLE]

Theorem 4.6 implies $\deg\mathsf{H}_{J}<d_{\mathsf{q}}$, so that $\sum_{d_{\mathsf{q}}<j\leq i}\mathsf{h}_{J}(j)=\sum_{d_{\mathsf{q}}<j\leq i}\mathsf{q}(j)=\sum_{d_{\mathsf{q}}<j\leq i}\mathsf{h}_{L^{\mathsf{q}}_{n}}(j)$, by Lemma 4.3. We must prove that $\sum_{0\leq j\leq d_{\mathsf{q}}}\mathsf{h}_{J}(j)>\sum_{0\leq j\leq d_{\mathsf{q}}}\mathsf{h}_{L^{\mathsf{q}}_{n}}(j)$. Lemma 5.1(i) gives $\sum_{0\leq j\leq d_{\mathsf{q}}}\mathsf{h}_{J}(j)\geq\sum_{0\leq j\leq d_{\mathsf{q}}}\mathsf{h}_{L^{\mathsf{q}}_{n}}(j)$ and strict inequality fails if and only if $\mathsf{h}_{J}(j)=\mathsf{h}_{L^{\mathsf{q}}_{n}}(j)$, for all $0\leq j\leq d_{\mathsf{q}}$. But this contradicts Corollary 4.8, so strict inequality holds, and $\mathsf{p}_{I}-\mathsf{p}>0$. This settles the case $b_{r}>0$. To prove the remaining cases, we examine Algorithm 3.6.

Let $c\geq 2$ and $b_{r}=0$. If $r=1$, then $\mathsf{p}=1$ and uniqueness holds. If $r=2$, then $\mathsf{p}=A\Phi^{b_{1}}(1)$ and to generate saturated strongly stable ideals with codimension $c$ and Hilbert polynomial $\mathsf{p}$, we take $b_{1}$ extensions from $L^{1}_{c}=\langle x_{0},x_{1},\dotsc,x_{c-1}\rangle$ followed by one expansion. The only expandable generator is $x_{c-1}$, hence uniqueness holds. If $r\geq 3$, then consider $A\Phi^{b_{1}-b_{2}}(1)$ and its lexicographic ideal $\langle x_{0},x_{1},\dotsc,x_{c-2},x_{c-1}^{2},x_{c-1}x_{c},\dotsc,x_{c-1}x_{c+b_{1}-b_{2}-1}\rangle$ in $\mathbb{K}[x_{0},x_{1},\dotsc,x_{c+b_{1}-b_{2}}]$. As $c\geq 2$, this ideal has two expandable generators, $x_{c-2}$ and $x_{c-1}x_{c+b_{1}-b_{2}-1}$. Lemma 5.2 then implies that $L^{\Phi^{b_{r-1}}A\Phi^{b_{r-2}-b_{r-1}}\dotsb A\Phi^{b_{2}-b_{3}}A\Phi^{b_{1}-b_{2}}(1)}_{c+b_{1}}$ has at least two expandable generators, which give distinct saturated strongly stable ideals with Hilbert polynomial $\mathsf{p}$ and codimension $c$.

Let $c=1$ and $b_{r}=0$. If $b_{1}=b_{r}$, then $\mathsf{p}=r$ and to get codimension $1$ saturated strongly stable ideals with Hilbert polynomial $\mathsf{p}$, we take $r-1$ expansions from $L^{1}_{1}=\langle x_{0}\rangle\subset\mathbb{K}[x_{0},x_{1}]$; the possibilities are $\langle x_{0}^{2}\rangle,\langle x_{0}^{3}\rangle,\dotsc,\langle x_{0}^{r}\rangle\subset\mathbb{K}[x_{0},x_{1}]$. Let $b_{1}=b_{2}=\dotsb=b_{s}>b_{s+1}\geq\dotsb\geq b_{r}$. If $r-s=1$, then we have $\mathsf{p}=A\Phi^{b_{r-1}}A^{r-2}(1)$ and we take $b_{r-1}$ extensions of $\langle x_{0}^{r-1}\rangle\subset\mathbb{K}[x_{0},x_{1}]$, followed by the unique expansion of $\langle x_{0}^{r-1}\rangle\subset\mathbb{K}[x_{0},x_{1},\dotsc,x_{1+b_{r-1}}]$. If $r-s=2$, then we have $\mathsf{p}=A\Phi^{b_{r-1}}A\Phi^{b_{r-2}-b_{r-1}}A^{r-3}(1)$. We extend $b_{r-2}-b_{r-1}$ times from $\langle x_{0}^{r-2}\rangle\subset\mathbb{K}[x_{0},x_{1}]$, we expand to obtain $\langle x_{0}^{r-1},x_{0}^{r-2}x_{1},\dotsc,x_{0}^{r-2}x_{b_{r-2}-b_{r-1}}\rangle\subset\mathbb{K}[x_{0},x_{1},\dotsc,x_{1+b_{r-2}-b_{r-1}}]$, we take $b_{r-1}$ further extensions, and we expand at $x_{0}^{r-2}x_{b_{r-2}-b_{r-1}}$. Now suppose $r-s\geq 3$. Consider the polynomial $A\Phi^{b_{s+1}-b_{s+2}}A\Phi^{b_{s}-b_{s+1}}A^{s-1}(1)$ obtained from $\mathsf{p}$ by truncation, with lexicographic ideal

[TABLE]

As $b_{s}>b_{s+1}$, both $x_{0}^{s}x_{b_{s}-b_{s+1}-1}$ and $x_{0}^{s}x_{b_{s}-b_{s+1}}x_{b_{s}-b_{s+2}}$ are expandable and Lemma 5.2 shows that $L^{\Phi^{b_{r-1}}A\Phi^{b_{r-2}-b_{r-1}}A\dotsb A\Phi^{b_{2}-b_{3}}A\Phi^{b_{1}-b_{2}}(1)}_{1+b_{1}}$ has at least two distinct expansions. ∎

Remark 5.4**.**

The case $b_{r}>0$ is a consequence of Theorem 4.6. Another approach might exist using Stanley decompositions; see [MS05a, SW91, Sta82]. Indeed, Proposition 2.13 follows by considering a Stanley decomposition of the lexicographic ideal, while Lemma 3.2(ii) gives a Stanley decomposition of $I$. We thank D. Maclagan for pointing this out.

A non-standard Borel-fixed ideal is a Borel-fixed ideal that is not strongly stable. Such ideals exist only in positive characteristic. Pardue [Par94, Chapter 2] gives the following combinatorial criterion for $I\subset\mathbb{K}[x_{0},x_{1},\ldots,x_{n}]$ to be Borel-fixed when $\mathbb{K}$ is infinite of characteristic $p>0$: $I$ is monomial and for all monomials $m\in I$, if $x_{j}^{\ell}\|m$ and $x_{i}\operatorname{\succ}x_{j}$, then $x_{j}^{-k}x_{i}^{k}m\in I$ holds, for all $k\leq_{p}\ell$. Here, $x_{j}^{\ell}\|m$ means $x_{j}^{\ell}$ divides $m$ but $x_{j}^{\ell+1}$ does not; $k\leq_{p}\ell$ means that in the base-$p$ expansions $k=\sum_{i}k_{i}p^{i},\ell=\sum_{i}\ell_{i}p^{i}$, we have $k_{i}\leq\ell_{i}$, for all $i$; also see [Eis95, § 15.9.3].

When $\mathbb{K}$ has characteristic [math], Theorem 5.3 generalizes a result of Gotzmann. In fact, the Hilbert schemes with $b_{r}>0$ are the irreducible ones in [Got89]. These include Grassmannians and the Hilbert schemes of hypersurfaces studied in [Ådl85]. We extend our classification to positive characteristic.

Corollary 5.5**.**

Let $\mathbb{K}$ be an algebraically closed field and $\mathsf{p},c$ be as in Theorem 5.3. Then $\operatorname{Hilb}^{\mathsf{p}}(\operatorname{\mathbb{P}}^{n})$ has a unique Borel-fixed point, where $n=c+\deg\mathsf{p}$.

Proof.

Let $I$ be the saturated Borel-fixed ideal of a point on $\operatorname{Hilb}^{\mathsf{p}}(\operatorname{\mathbb{P}}^{n})$. So $I$ is monomial and no $g\in G(I)$ is divisible by $x_{n}$, by [Par94, II, Proposition 9]. Suppose $b_{r}>0$. As $x_{n}$ is not a zero-divisor of $\mathbb{K}[x_{0},x_{1},\ldots,x_{n}]/I$, the result [Got89, Proposition 2] applies directly, showing that $I=\langle f_{0}x_{0},f_{0}f_{1}x_{1},f_{0}f_{1}f_{2}x_{2},\ldots,f_{0}f_{1}\cdots f_{k-1}x_{k-1},f_{0}f_{1}\cdots f_{k}\rangle$, where $f_{i}\in\mathbb{K}[x_{i},\ldots,x_{n}]$ is homogeneous and $k\leq n-2$; cf. [RS97, Theorem 4.1]. We may assume each $f_{i}$ is monomial, of degree $d_{i}$. Then Pardue’s criterion shows that $I$ contains the saturated lexicographic ideal

[TABLE]

The degrees $d_{i}$ determine the Hilbert polynomials of $I$ and $L$, so we must have $I=L=L^{\mathsf{p}}_{n}$.

Let $c\geq 2$ and $b_{r}=0$. The cases $\mathsf{p}=1$ and $\mathsf{p}=2$ follow by inspection. When $r=2$ and $b_{1}>0$, consider the ideal $J:=I\cap\mathbb{K}[x_{0},x_{1},\ldots,x_{n-1}]$, which is Borel-fixed, has Hilbert polynomial $\nabla(\mathsf{p})$, and satisfies $\Phi(J)=I$ as $I$ is saturated. By the previous cases, we know that $(J:x_{n-1}^{\infty})=L^{\nabla(\mathsf{p})}_{n-1}=\langle x_{0},x_{1},\ldots,x_{c-1}\rangle$, which further implies $(I:x_{n-1}^{\infty})=L^{\mathsf{p}-1}_{n}$, by Lemma 3.4(iii), Proposition 2.13, and [Par94, II, Proposition 9]. So $I_{2}\subset(L^{\mathsf{p}-1}_{n})_{2}$ is a codimension $1$ vector subspace, the latter being spanned by $x_{0}^{2},x_{0}x_{1},\ldots,x_{c-1}^{2},x_{c-1}x_{c},\ldots,x_{c-1}x_{n}$. Now Pardue’s criterion implies that we can only obtain $I_{2}$ by removing $x_{c-1}x_{n}$, i.e. $I$ is the lex-expansion of $L^{\mathsf{p}-1}_{n}$, as $I$ is generated in degree $r=2$.

Now let $c=1$ and $b_{r}=0$. If $b_{1}=b_{r}$, then $I$ is lexicographic, as it is monomial and saturated. If $b_{1}>b_{r}$, then the argument is entirely analogous to the previous paragraph. ∎

Proof of Theorem 1.1.

Theorem 5.3 and Corollary 5.5 prove the claim. ∎

Geometrically, the Hilbert schemes corresponding to Theorem 5.3 are well-behaved.

Lemma 5.6**.**

Let $\mathbb{K}$ be an infinite field. If the lexicographic ideal is the unique saturated Borel-fixed ideal with Hilbert polynomial $\mathsf{p}$ and codimension $c$, then $\operatorname{Hilb}^{\mathsf{p}}(\operatorname{\mathbb{P}}^{n})$ is nonsingular and irreducible, where $n=c+\deg\mathsf{p}$.

Proof.

Every component and intersection of components of $\operatorname{Hilb}^{\mathsf{p}}(\operatorname{\mathbb{P}}^{n})$ contains a point $\left[X_{I}\right]$ defined by a saturated Borel-fixed ideal $I$; the Remarks in [Ree95, § 2] hold over infinite fields, by [BS87b, Proposition 1] and [Eis95, Theorem 15.17]. Lexicographic points are nonsingular by [RS97, Theorem 1.4], so $\left[X_{L^{\mathsf{p}}_{n}}\right]$ cannot lie on an intersection of components. Thus, $\operatorname{Hilb}^{\mathsf{p}}(\operatorname{\mathbb{P}}^{n})$ has a unique, generically nonsingular, irreducible component.

Suppose $\operatorname{Hilb}^{\mathsf{p}}(\operatorname{\mathbb{P}}^{n})$ has a singular point, given by $I\subset\mathbb{K}[x_{0},x_{1},\dotsc,x_{n}]$. For $\gamma\in\operatorname{GL}_{n+1}(\mathbb{K})$, the point $\left[X_{\gamma\cdot I}\right]$ is also singular, and for generic $\gamma\in\operatorname{GL}_{n+1}(\mathbb{K})$, the initial ideal of $\gamma\cdot I$ with respect to any monomial ordering is Borel-fixed, by Galligo’s Theorem [Gal74, BS87b]. Thus, a one-parameter family of singular points degenerating to the lexicographic ideal exists. By upper semicontinuity of cohomology of the normal sheaf, the lexicographic ideal is singular, a contradiction; see [Har77, III, Theorem 12.8], [Har10, Theorem 1.1(b)]. Hence, $\operatorname{Hilb}^{\mathsf{p}}(\operatorname{\mathbb{P}}^{n})$ is nonsingular and irreducible. ∎

Thus, our classification provides irreducible, nonsingular Hilbert schemes over algebraically closed or characteristic [math] fields. Moreover, these Hilbert schemes are rational, by [LR11, Theorem C].

We now wish to make a quantitative statement about the prevalence of this behaviour. To do so, we need a probability measure $\operatorname{Pr}\colon 2^{\operatorname{\mathscr{H}}}\to[0,1]$, which is determined by a normalized nonnegative function $\operatorname{pr}\colon\operatorname{\mathscr{H}}\to\mathbb{R}$, as in [Bil95, Examples 2.8–2.9]. That is, for every subset $\mathscr{N}\subseteq\operatorname{\mathscr{H}}$, we have $\operatorname{Pr}(\mathscr{N}):=\sum_{H\in\mathscr{N}}\operatorname{pr}(H)$. However, there is no known canonical distribution on $\operatorname{\mathscr{H}}$. A natural choice on $\operatorname{\mathscr{H}}_{c}$ is to mimic uniform distribution by making all vertices at a fixed height equally likely; given a mass function $f_{c}\colon\mathbb{N}\to[0,1]$, let $\operatorname{pr}_{c}(H)=2^{-k}f_{c}(k)$, for all $H\in\operatorname{\mathscr{H}}_{c}$ at height $k$. Distributions on $\operatorname{\mathscr{H}}$ are then specified via functions $f_{c}\colon\mathbb{N}\to[0,1]$, for all $\operatorname{\mathscr{H}}_{c}$, and a mass function $f\colon\mathbb{N}\setminus\{0\}\to[0,1]$, by setting $\operatorname{pr}(H)=2^{-k}f(c)f_{c}(k)$ (the probability of $H$), for all $H\in\operatorname{\mathscr{H}}_{c}$ at height $k$. Using $f_{c}(k):=2^{-k-1}$ and $f(c):=2^{-c}$ is sufficient for us, although other basic examples (geometric, Poisson, etc.) work similarly.

Theorem 5.7**.**

Let $\mathbb{K}$ be an algebraically closed or characteristic [math] field and endow $\operatorname{\mathscr{H}}$ with the structure of a probability space as in the preceding paragraph. The probability that a random Hilbert scheme is irreducible and nonsingular is greater than $0.5$.

Proof.

Let $\mathscr{N}$ be the set of nonsingular and irreducible Hilbert schemes. We compute

[TABLE]

because for all trees $\operatorname{\mathscr{H}}_{c}$ and heights $k\geq 1$, there are at least $2^{k-1}$ vertices corresponding to nonsingular and irreducible Hilbert schemes, by Theorem 5.3 and Lemma 5.6. The computation is similar in a fixed codimension. Hence, the probability that a Hilbert scheme is nonsingular and irreducible is greater than $0.5$. ∎

Proof of Theorem 1.2.

Theorem 5.7 proves the claim. ∎