Segre classes as 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’.