When are multidegrees positive?

Abstract

Let $\mathbb{k}$ be an arbitrary field, $\mathbb{P}={\mathbb{P}}_{\mathbb{k}}^{m_1}{\times }_{\mathbb{k}}\cdots {\times }_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_p}$ be a multiprojective space over $\mathbb{k}$, and $X\subseteq \mathbb{P}$ be a closed subscheme of $\mathbb{P}$. We provide necessary and sufficient conditions for the positivity of the multidegrees of X. As a consequence of our methods, we show that when X is irreducible, the support of multidegrees forms a discrete algebraic polymatroid. In algebraic terms, we characterize the positivity of the mixed multiplicities of a standard multigraded algebra over an Artinian local ring, and we apply this to the positivity of mixed multiplicities of ideals. Furthermore, we use our results to recover several results in the literature in the context of combinatorial algebraic geometry.

Introduction

Let $\mathbb{k}$ be an arbitrary field, $\mathbb{P}={\mathbb{P}}_{\mathbb{k}}^{m_1}{\times }_{\mathbb{k}}\cdots {\times }_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_p}$ be a multiprojective space over $\mathbb{k}$, and $X\subseteq \mathbb{P}$ be a closed subscheme of $\mathbb{P}$. The multidegrees of X are fundamental invariants that describe algebraic and geometric properties of X. For each $\mathbf{n}=(n_1,\dots ,n_p)\in {\mathbb{N}}^p$ with $n_1+\cdots +n_p=\dim (X)$ one can define the multidegree of X of type n with respect to $\mathbb{P}$, denoted by ${\mathrm{deg}}_{\mathbb{P}}^{\mathbf{n}}(X)$, in different ways (see Definition 2.7, Remark 2.8 and Remark 2.9). In classical geometrical terms, when $\mathbb{k}$ is algebraically closed, ${\mathrm{deg}}_{\mathbb{P}}^{\mathbf{n}}(X)$ equals the number of points (counting multiplicity) in the intersection of X with the product $L_1{\times }_{\mathbb{k}}\cdots {\times }_{\mathbb{k}}{L}_p\subset \mathbb{P}$, where $L_i\subset {\mathbb{P}}_{\mathbb{k}}^{m_i}$ is a general linear subspace of dimension $m_i-n_i$ for each $1\le i\le p$.

The study of multidegrees goes back to pioneering work by van der Waerden [60]. From a more algebraic point of view, multidegrees receive the name of mixed multiplicities (see Definition 2.7). More recent papers where the notion of multidegree (or mixed multiplicity) is studied are, e.g., [1], [9], [11], [23], [33], [36], [41], [42], [57].

The main goal of this paper is to answer the following fundamental question considered by Trung [57] and by Huh [25] in the case $p=2$.

• For $\mathbf{n}\in {\mathbb{N}}^p$ with $n_1+\cdots +n_p=\dim (X)$, when do we have that ${\mathrm{deg}}_{\mathbb{P}}^{\mathbf{n}}(X)>0$?

Our main result says that the positivity of ${\mathrm{deg}}_{\mathbb{P}}^{\mathbf{n}}(X)$ is determined by the dimensions of the images of the natural projections from $\mathbb{P}$ restricted to the irreducible components of X. First, we set a basic notation: for each $\mathfrak{J}=\{j_1,\dots ,j_k\}\subseteq \{1,\dots ,p\}$, let ${\mathrm{\Pi }}_{\mathfrak{J}}$ be the natural projection

$${\mathrm{\Pi }}_{\mathfrak{J}}:\mathbb{P}={\mathbb{P}}_{\mathbb{k}}^{m_1}{\times }_{\mathbb{k}}\cdots {\times }_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_p}\to {\mathbb{P}}_{\mathbb{k}}^{m_{j_1}}{\times }_{\mathbb{k}}\cdots {\times }_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{j_k}}.$$

The following is the main theorem of this article. Here, we give necessary and sufficient conditions for the positivity of multidegrees.

Theorem A Theorem 3.12, Corollary 3.13

Let $\mathbb{k}$ be an arbitrary field, $\mathbb{P}={\mathbb{P}}_{\mathbb{k}}^{m_1}{\times }_{\mathbb{k}}\cdots {\times }_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_p}$ be a multiprojective space over $\mathbb{k}$, and $X\subseteq \mathbb{P}$ be a closed subscheme of $\mathbb{P}$. Let $\mathbf{n}=(n_1,\dots ,n_p)\in {\mathbb{N}}^p$ be such that $n_1+\cdots +n_p=\dim (X)$. Then, ${\mathrm{deg}}_{\mathbb{P}}^{\mathbf{n}}(X)>0$ if and only if there is an irreducible component $Y\subseteq X$ of X that satisfies the following two conditions:

• $\dim (Y)=\dim (X)$.

• For each $\mathfrak{J}=\{j_1,\dots ,j_k\}\subseteq \{1,\dots ,p\}$ the inequality

  $$n_{j_1}+\cdots +n_{j_k}\le \dim ({\mathit{\Pi }}_{\mathfrak{J}}(Y))$$

  holds.

When $\mathbb{k}$ is the field of complex numbers Theorem A is essentially covered by the geometric results in [34, Theorems 2.14, 2.19],² however their methods do not extend to arbitrary fields. Here we follow an algebraic approach that allows us to prove the result for all fields, and hence a general version for algebras over Artinian local rings (see Theorem B). The main idea in the proof of Theorem A is the study of the dimensions of the images of the natural projections after cutting by a general hyperplane (see Theorem 3.7).

We note that if $p=2$ and X is arithmetically Cohen-Macaulay, the conclusion of Theorem A in the irreducible case also holds for X (see [57, Corollary 2.8]). In Example 5.2 we show that this is not necessarily true for $p>2$.

If X is irreducible, then the function $r:2^{\{1,\dots ,p\}}\to \mathbb{Z}$ defined by $r(\mathfrak{J}):=\dim ({\mathrm{\Pi }}_{\mathfrak{J}}(Y))$ is a submodular function, i.e., $r({\mathfrak{J}}_1\cap {\mathfrak{J}}_2)+r({\mathfrak{J}}_1\cup {\mathfrak{J}}_2)\le r({\mathfrak{J}}_1)+r({\mathfrak{J}}_2)$ for any two subsets ${\mathfrak{J}}_1,{\mathfrak{J}}_2\subseteq \{1,\dots ,p\}$, as proved in Proposition 5.1 (see also Definition 2.16). By the Submodular Theorem (see, e.g., [7, Theorem 3.11] or [44, Appendix B]) and the inequalities of Theorem A, the points $\mathbf{n}\in {\mathbb{N}}^p$ for which ${\mathrm{deg}}_{\mathbb{P}}^{\mathbf{n}}(X)>0$ are the lattice points of a generalized permutohedron. Defined by A. Postnikov in [48] generalized permutohedra are polytopes obtained by deforming usual permutohedra. In recent years this family of polytopes has been studied in relation to other fields such as probability, combinatorics, and representation theory (see [44], [45], [49]).

In a more algebraic flavor, we state the translation of Theorem A to the mixed multiplicities of a standard multigraded algebra over an Artinian local ring (see Definition 2.13).

Theorem B Corollary 3.14

Let A be an Artinian local ring and R be a finitely generated standard ${\mathbb{N}}^p$-graded A-algebra. For each $1\le j\le p$, let ${\mathfrak{m}}_j\subset R$ be the ideal generated by the elements of degree ${\mathbf{e}}_j$, where ${\mathbf{e}}_j\in {\mathbb{N}}^p$ denotes the j-th elementary vector. Let $\mathfrak{N}={\mathfrak{m}}_1\cap \cdots \cap {\mathfrak{m}}_p\subset R$. Let $\mathbf{n}=(n_1,\dots ,n_p)\in {\mathbb{N}}^p$ be such that $n_1+\cdots +n_p=\dim (R/\left(0{:}_R{\mathfrak{N}}^{\infty }\right))-p$. Then, $e(\mathbf{n};R)>0$ if and only if there is a minimal prime ideal $\mathfrak{P}\in \mathit{\text{Min}}\left(0{:}_R{\mathfrak{N}}^{\infty }\right)$ of $\left(0{:}_R{\mathfrak{N}}^{\infty }\right)$ that satisfies the following two conditions:

• $\dim (R/\mathfrak{P})=\dim (R/\left(0{:}_R{\mathfrak{N}}^{\infty }\right))$.

• For each $\mathfrak{J}=\{j_1,\dots ,j_k\}\subseteq \{1,\dots ,p\}$ the inequality

  $$n_{j_1}+\cdots +n_{j_k}\le \dim \left(\frac{R}{\mathfrak{P}+{\sum }_{j\notin \mathfrak{J}}{\mathfrak{m}}_j}\right)-k$$

  holds.

For a given finite set of ideals in a Noetherian local ring, such that one of them is zero-dimensional, we can define their mixed multiplicities by considering a certain associated standard multigraded algebra (see [58] for more information). These multiplicities have a long history of interconnecting problems from commutative algebra, algebraic geometry, and combinatorics, with applications to the topics of Milnor numbers, mixed volumes, and integral dependence (see, e.g., [25], [27], [55], [58]). As a direct consequence of Theorem B we are able to give a characterization for the positivity of mixed multiplicities of ideals (see Corollary 4.3). In another related result, we focus on homogeneous ideals generated in one degree; this case is of particular importance due to its relation with rational maps between projective varieties. In this setting, we provide more explicit conditions for positivity in terms of the analytic spread of products of these ideals (see Theorem 4.4).

Going back to the setting of Theorem A, we switch our attention to the following discrete set

$${\text{MSupp}}_{\mathbb{P}}(X)=\{\mathbf{n}\in {\mathbb{N}}^p|{\mathrm{deg}}_{\mathbb{P}}^{\mathbf{n}}(X)>0\},$$

which we call the support of X with respect to $\mathbb{P}$. When X is irreducible, we show that ${\text{MSupp}}_{\mathbb{P}}(X)$ is a (discrete) polymatroid (see §2.3, Proposition 5.1). The latter result was included in an earlier version of this paper when $\mathbb{k}$ is algebraically closed, and an alternative proof is given by Brändén and Huh in [3, Corollary 4.7] using the theory of Lorentzian polynomials. An advantage of our approach is that we can describe the corresponding rank submodular functions of the polymatroids, a fact that we exploit in the applications of Section 6. Additionally, our results are valid when X is just irreducible and not necessarily geometrically irreducible over $\mathbb{k}$ (i.e., we do not need to assume that $X{\times }_{\mathbb{k}}\bar{\mathbb{k}}$ is irreducible for an algebraic closure $\bar{\mathbb{k}}$ of $\mathbb{k}$); it should be noticed that this generality is not covered by the statements in [3] and [34].

Discrete polymatroids [24] have also been studied under the name of M-convex sets [46]. Polymatroids can also be described as the integer points in a generalized permutohedron [48], so they are closely related to submodular functions, which are well studied in optimization, see [38] and [52, Part IV] for comprehensive surveys on submodular functions, their applications, and their history. There are two distinguishable types of polymatroids, linear and algebraic polymatroids, whose main properties are inherited by their representation in terms of other algebraic structures. Theorem A allows us to define another type of polymatroids, that we call Chow polymatroids, and which interestingly lies in between the other two. In the following theorem we summarize our main results in this direction.

Theorem C Theorem 5.5

Over an arbitrary field $\mathbb{k}$, we have the following inclusions of families of polymatroids

Moreover, when $\mathbb{k}$ is a field of characteristic zero, the three families coincide.

If $\mathbb{k}$ has positive characteristic, then these types of polymatroids do not agree. In fact, there exist examples of polymatroids which are algebraic over any field of positive characteristic but never linear (see Remark 5.7).

Theorem A can be applied to particular examples of varieties coming from combinatorial algebraic geometry. In §6.1 we do so to matrix Schubert varieties; in this case the multidegrees are the coefficients of Schubert polynomials, thus our results allow us to give an alternative proof to a recent conjecture regarding the support of these polynomials (see Theorem 6.3). In §6.2 and §6.3 we study certain embeddings of flag varieties and of the moduli space ${\bar{M}}_{0,p+3}$, respectively (see Proposition 6.7 and Proposition 6.8). In §6.4 we recover a well-known characterization for the positivity of mixed volumes of convex bodies (see Theorem 6.9).

We now outline the contents of the article. In Section 2 we set up the notation used throughout the document. We also include key preliminary definitions and results, paying special attention to the connection between mixed multiplicities of standard multigraded graded algebras and multidegrees of their corresponding schemes. Section 3 is devoted to the proof of Theorem A and Theorem B. Our results for mixed multiplicities of ideals are included in Section 4. In Section 5 we relate our results to the theory of polymatroids. In particular, we show the proof of Theorem C. We finish the paper with Section 6 where the applications to combinatorial algebraic geometry are presented.

We conclude the Introduction with an illustrative example. The following example is constructed following the same ideas in Proposition 5.4.

Example 1.1

Consider the polynomial ring $S=\mathbb{k}[v_1,v_2,v_3][w_1,w_2,w_3]$ with the ${\mathbb{N}}^3$-grading $\mathrm{deg}(v_i)=(0,0,0)$, $\mathrm{deg}(w_i)={\mathbf{e}}_i$ for $1\le i\le 3$. Let T be the ${\mathbb{N}}^3$-graded polynomial ring $T=\mathbb{k}[x_0,\dots ,x_3][y_0,\dots ,y_3][z_0,\dots ,z_3]$ where $\mathrm{deg}(x_i)={\mathbf{e}}_1$, $\mathrm{deg}(y_i)={\mathbf{e}}_2$ and $\mathrm{deg}(z_i)={\mathbf{e}}_3$. Consider the ${\mathbb{N}}^3$-graded $\mathbb{k}$-algebra homomorphism

$$\phi =T\to S,\begin{array}{cccc}{x}_0\mapsto w_1,\hfill & x_1\mapsto v_1{w}_1,\hfill & x_2\mapsto v_1{w}_1,\hfill & x_3\mapsto v_1{w}_1,\hfill \\ y_0\mapsto w_2,\hfill & y_1\mapsto v_1{w}_2,\hfill & y_2\mapsto v_2{w}_2,\hfill & y_3\mapsto (v_1+v_2)w_2,\hfill \\ z_0\mapsto w_3,\hfill & z_1\mapsto v_1{w}_3,\hfill & z_2\mapsto v_2{w}_3,\hfill & z_3\mapsto v_3{w}_3.\hfill \end{array}$$

Note that $\mathfrak{P}=\text{Ker}(\phi )\subset T$ is an ${\mathbb{N}}^3$-graded prime ideal. Let $Y\subset \mathbb{P}={\mathbb{P}}_{\mathbb{k}}^3{\times }_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^3{\times }_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^3$ be the closed subscheme corresponding to $\mathfrak{P}$. In this case, one can easily compute the dimension of the projections ${\mathrm{\Pi }}_{\mathfrak{J}}(Y)$ for each $\mathfrak{J}\subseteq \{1,2,3\}$, and so Theorem A implies that ${\text{MSupp}}_{\mathbb{P}}(Y)$ is given by all $\mathbf{n}=(n_1,\dots ,n_3)\in {\mathbb{N}}^3$ satisfying the following conditions:

$$n_1+n_2+n_3=3=\dim (Y),n_1+n_2\le 2=\dim ({\mathrm{\Pi }}_{\{1,2\}}(Y)),n_1+n_3\le 3=\dim ({\mathrm{\Pi }}_{\{1,3\}}(Y)),n_2+n_3\le 3=\dim ({\mathrm{\Pi }}_{\{2,3\}}(Y)),n_1\le 1=\dim ({\mathrm{\Pi }}_{\{1\}}(Y)),n_2\le 2=\dim ({\mathrm{\Pi }}_{\{2\}}(Y)),n_3\le 3=\dim ({\mathrm{\Pi }}_{\{3\}}(Y)).$$

Hence ${\text{MSupp}}_{\mathbb{P}}(Y)=\{(0,0,3),(0,1,2),(0,2,1),(1,0,2),(1,1,1)\}\subset {\mathbb{N}}^3$. This set can also be represented graphically as follows:

Additionally, by using Macaulay2 [21] we can compute that its multidegree polynomial (see Definition 2.10) is equal to:

$${\mathrm{deg}}_{\mathbb{P}}(Y;t_1,t_2,t_3)=t_1^3{t}_2^3+t_1^3{t}_2^2{t}_3+t_1^3{t}_2{t}_3^2+t_1^2{t}_2^3{t}_3+t_1^2{t}_2^2{t}_3^2.$$

We note that here we are following the convention that ${\text{MSupp}}_{\mathbb{P}}(Y)$ is given by the complementary degrees of the polynomial ${\mathrm{deg}}_{\mathbb{P}}(Y;t_1,t_2,t_3)$; for instance, the term $t_1^3{t}_2^3$ corresponds to the point $(3,3,3)-(3,3,0)=(0,0,3)\in {\text{MSupp}}_{\mathbb{P}}(Y)$.