Segre classes as integrals over polytopes

TL;DR

This paper presents a novel integral formula for Segre classes of monomial schemes, extending classical intersection theory results to a broader class of schemes using geometric integrals over associated polytopes.

Contribution

It introduces a new integral representation of Segre classes for monomial schemes, generalizing Bernstein-Kouchnirenko type theorems to refined intersection invariants.

Findings

Provides explicit integral formulas for Segre classes

Extends classical intersection theory to r.c. monomial schemes

Connects algebraic invariants with geometric integrals over polytopes

Abstract

We express the Segre class of a monomial scheme -- or, more generally, a scheme monomially supported on a set of divisors cutting out complete intersections -- in terms of an integral computed over an associated body in euclidean space. The formula is in the spirit of the classical Bernstein-Kouchnirenko theorem computing intersection numbers of equivariant divisors in a torus in terms of mixed volumes, but deals with the more refined intersection-theoretic invariants given by Segre classes, and holds in the less restrictive context of `r.c. monomial schemes'.