Mixed multiplicities and projective degrees of rational maps

TL;DR

This paper introduces a new approach to compute the projective degrees of rational maps using mixed multiplicities and Hilbert series, providing explicit formulas for certain classes of ideals.

Contribution

It develops a general framework linking mixed multiplicities to projective degrees of rational maps and derives explicit formulas for perfect and Gorenstein ideals.

Findings

Explicit formulas for projective degrees of rational maps from perfect ideals of height two.

Formulas for Gorenstein ideals of height three.

A unified method to compute projective degrees via multiplicity of the saturated special fiber ring.

Abstract

We consider the notion of mixed multiplicities for multigraded modules by using Hilbert series, and this is later applied to study the projective degrees of rational maps. We use a general framework to determine the projective degrees of a rational map via a computation of the multiplicity of the saturated special fiber ring. As specific applications, we provide explicit formulas for all the projective degrees of rational maps determined by perfect ideals of height two or by Gorenstein ideals of height three.