Multidegrees, families, and integral dependence

TL;DR

This paper explores the behavior of multidegrees and introduces new invariants to detect integral dependence in families of ideals, providing criteria and examples that advance understanding of algebraic invariants.

Contribution

It introduces polar-Segre multiplicities and establishes new semicontinuity and integral dependence criteria for ideals and modules.

Findings

Mixed multiplicities are upper semicontinuous under fibers.

Projective degrees are lower semicontinuous under specialization.

Segre numbers uniquely detect integral dependence among studied invariants.

Abstract

We study the behavior of multidegrees in families and the existence of numerical criteria to detect integral dependence. We show that mixed multiplicities of modules are upper semicontinuous functions when taking fibers and that projective degrees of rational maps are lower semicontinuous under specialization. We investigate various aspects of the polar multiplicities and Segre numbers of an ideal and introduce a new invariant that we call polar-Segre multiplicities. In terms of polar multiplicities and our new invariants, we provide a new integral dependence criterion for certain families of ideals. By giving specific examples, we show that the Segre numbers are the only invariants among the ones we consider that can detect integral dependence. Finally, we generalize the result of Gaffney and Gassler regarding the lexicographic upper semicontinuity of Segre numbers.