Mixed multiplicities of graded families of ideals

TL;DR

This paper establishes the existence and definition of mixed multiplicities for arbitrary graded families of ideals, including m-primary families, and presents a comprehensive volume-multiplicity formula linking these concepts.

Contribution

It introduces a general framework for mixed multiplicities of graded families of ideals and proves a volume equals multiplicity formula applicable in broad settings.

Findings

Existence and definition of mixed multiplicities for graded families.

A general volume = multiplicity formula for mixed multiplicities.

Applicability to arbitrary m-primary graded families.

Abstract

We show the existence (and define) the mixed multiplicities of arbitrary graded families of ideals under mild assumptions. In particular, our methods and results are valid for the case of arbitrary $\mathfrak{m}$-primary graded families. Furthermore, we provide a far-reaching "Volume = Multiplicity formula" for the mixed multiplicities of graded families of ideals.