[Paper Review] Discrete Mixed Volume and Hodge-Deligne Numbers

Sandra Di Rocco, Christian Haase+1 more|arXiv (Cornell University)|Sep 28, 2016

Algebraic Geometry and Number Theory8 references3 citations

TL;DR

This paper establishes a precise link between discrete mixed volume—a combinatorial invariant of lattice polytopes—and the motivic arithmetic genus of a non-compact complete intersection in the complex algebraic torus. By making explicit a footnote in Khovanskii’s 1978 work, the authors prove that the discrete mixed volume equals the motivic arithmetic genus of a generic complete intersection defined by Laurent polynomials with given Newton polytopes, thereby providing a combinatorial interpretation of a key Hodge-Deligne invariant.

ABSTRACT

Generalizing the famous Bernstein-Kushnirenko Theorem, Khovanskii proved in 1978 a combinatorial formula for the arithmetic genus of the compactification of a generic complete intersection associated to a family of lattice polytopes. Recently, an analogous combinatorial formula, called the discrete mixed volume, was introduced by Bihan and shown to be nonnegative. By making a footnote of Khovanskii in his paper explicit, we interpret this invariant as the (motivic) arithmetic genus of the non-compact generic complete intersection associated to the family of lattice polytopes.

Motivation & Objective

• To clarify the algebraic-geometric meaning of the discrete mixed volume introduced by Bihan, which is known to be non-negative but lacks a geometric interpretation.
• To interpret the discrete mixed volume as the motivic arithmetic genus of a non-compact complete intersection in $(\mathbb{C}^*)^n$
• To verify that this invariant matches the Hodge-Deligne polynomial-based definition of the motivic arithmetic genus, thus connecting combinatorics and algebraic geometry.
• To provide a geometric realization of Khovanskii’s 1978 formula for the arithmetic genus of a compactified complete intersection, now in the non-compact setting.
• To establish a motivic reciprocity theorem by unifying Khovanskii’s compact and non-compact genus formulas via the discrete mixed volume.

Proposed method

• Use the Hodge-Deligne polynomial (E-polynomial) to define the motivic arithmetic genus as $(-1)^{\dim Y} E(Y; 0, 1)$, where $E(Y; u, v)$ is the generating function of Hodge-Deligne numbers.
• Apply the Cayley trick to reduce the computation of the E-polynomial of a complete intersection $Y \subset (\mathbb{C}^*)^n$ to that of a hypersurface $\tilde{Z} \subset (\mathbb{C}^*)^n \times \mathbb{C}^k$ defined by $F = 1 + \sum y_i f_i$.
• Use the stratification of $\tilde{Z}$ into strata $Z_I$ indexed by non-empty subsets $I \subseteq [k]$, each corresponding to a generic hypersurface over a Cayley polytope $C_I$.
• Apply a specialized version of the E-polynomial formula for hypersurfaces over lattice pyramids (Lemma 2.1) to compute $e^{p,+}(Z_I)$, the $p$-th $\chi_y$-characteristic.
• Use the identity $E(Y; u, v) = (uv-1)^n - (uv)^{-(k-1)} E(\tilde{Z}; u, v)$ to relate the E-polynomial of $Y$ to that of $\tilde{Z}$, enabling the computation of $e^{p,+}(Y)$.
• Derive a closed-form expression for $e^{p,+}(Y)$ by combining binomial identities and Ehrhart theory, ultimately showing that $e^{p,+}(Y) = (-1)^{n-p} \sum_{I \subseteq [k]} (-1)^{|I|} \sum_{\beta \in \mathbb{Z}_{\geq 0}^I} (-1)^{|\beta|} \binom{n+|I|}{p - |\beta|} \operatorname{ehr}(P_I + P_\beta; 1)$, which matches the discrete mixed volume.

Experimental results

Research questions

• RQ1 What is the geometric or motivic interpretation of the discrete mixed volume introduced by Bihan?
• RQ2 Can the discrete mixed volume be identified with a known invariant in algebraic geometry, such as the arithmetic genus of a non-compact complete intersection?
• RQ3 How does the motivic arithmetic genus of a non-compact complete intersection relate to the classical arithmetic genus of its compactification?
• RQ4 Is there a reciprocity law connecting Khovanskii’s formula for the compactified genus $\operatorname{ME}(P_1,\dots,P_k; -1)$ and the non-compact genus $\operatorname{DMV}(P_1,\dots,P_k)$?
• RQ5 Does the non-negativity of the discrete mixed volume have a deeper geometric origin in the context of Hodge-Deligne theory?

Key findings

• The discrete mixed volume $\operatorname{DMV}(P_1,\dots,P_k)$ is equal to the motivic arithmetic genus of the non-compact complete intersection $Y \subset (\mathbb{C}^*)^n$ defined by generic Laurent polynomials with Newton polytopes $P_1,\dots,P_k$.
• The motivic arithmetic genus of $Y$ is given by $(-1)^{n-p} \sum_{I \subseteq [k]} (-1)^{|I|} \sum_{\beta \in \mathbb{Z}_{\geq 0}^I} (-1)^{|\beta|} \binom{n+|I|}{p - |\beta|} \operatorname{ehr}(P_I + P_\beta; 1)$, which matches the discrete mixed volume when $p = 0$.
• The discrete mixed volume is non-negative, as established by Bihan, and this paper provides a geometric justification via its interpretation as a Hodge-Deligne invariant.
• The result confirms a footnote in Khovanskii’s 1978 paper, showing that $\operatorname{DMV}(P_1,\dots,P_k) = \operatorname{ME}(P_1,\dots,P_k; 1)$ is the motivic arithmetic genus of the non-compact complete intersection.
• The paper establishes a motivic reciprocity theorem: the compactified genus is $\operatorname{ME}(P_1,\dots,P_k; -1)$, while the non-compact genus is $\operatorname{ME}(P_1,\dots,P_k; 1)$, with both values related through the discrete mixed volume.
• The proof relies on the Cayley trick and E-polynomial computations over stratified toric varieties, ultimately reducing to a combinatorial identity involving binomial coefficients and Ehrhart theory.