Segre classes and integral dependence

TL;DR

This paper explores the relationship between Segre classes and integral dependence, showing that Segre classes encode integral dependence criteria and applying this to algebraic geometry and polynomial ideals.

Contribution

It demonstrates that Segre classes not only are invariant under birational transformations but also encode integral dependence information, offering new criteria for algebraic ideals.

Findings

Segre classes determine integral dependence of ideal sheaves.

Segre zeta function provides an integral dependence criterion for polynomial ideals.

Segre classes encode algebraic and geometric information about subschemes.

Abstract

A fundamental property of Segre classes is their birational invariance. This invariance implies that the Segre class of a closed subscheme only depends on the integral closure of the defining ideal sheaf. In this paper, we show that, conversely, the Segre class of a closed subscheme encodes an integral dependence criterion for its defining ideal sheaf. As an application, we prove that Aluffi's Segre zeta function provides an integral dependence criterion for homogeneous ideals in polynomial rings.