Relative mixed multiplicities and mixed Buchsbaum-Rim multiplicities

TL;DR

This paper introduces relative mixed multiplicities as a multigraded extension of existing invariants, demonstrating their stability, relation to mixed Buchsbaum-Rim multiplicities, and their use in detecting integral dependence and birationality.

Contribution

It defines relative mixed multiplicities, proves their stability and equality with mixed Buchsbaum-Rim multiplicities, and shows their application in algebraic geometry.

Findings

Relative mixed multiplicities have a stable value equal to mixed Buchsbaum-Rim multiplicities.

Vanishing of these invariants detects integral dependence.

They can also identify birationality.

Abstract

We define and study the natural multigraded extension of the relative multiplicities introduced by Simis, Ulrich and Vasconcelos. We call these new invariants relative mixed multiplicities. We show that they have a stable value equal to the mixed Buchsbaum-Rim multiplicity of Kleiman and Thorup. Furthermore, we prove that integral dependence and birationality can be detected via the vanishing of relative mixed multiplicities.