Newton-Okounkov bodies and Segre classes

TL;DR

This paper introduces a method to compute Segre classes of subschemes in projective space using Newton-Okounkov bodies, generalizing previous results for monomial schemes through convex geometric constructions.

Contribution

It adapts Newton-Okounkov bodies to calculate Segre classes, extending the approach from monomial schemes to arbitrary subschemes of projective space.

Findings

Provides a convex geometric framework for Segre class computation.

Generalizes known results from monomial schemes to all subschemes.

Connects algebraic geometry invariants with convex geometry.

Abstract

Given a homogeneous ideal in a polynomial ring over C, we adapt the construction of Newton-Okounkov bodies to obtain a convex subset of Euclidean space such that a suitable integral over this set computes the Segre zeta function of the ideal. That is, we extract the numerical information of the Segre class of a subscheme of projective space from an associated (unbounded) Newton-Okounkov convex set. The result generalizes to arbitrary subschemes of projective space the numerical form of a previously known result for monomial schemes.