The Minkowski Equality for Filtrations

TL;DR

This paper extends the Minkowski equality characterization for multiplicities to divisorial and bounded filtrations in certain local domains, but not for all filtrations.

Contribution

It proves the Minkowski equality characterization holds for divisorial and bounded filtrations, broadening previous results in multiplicity theory.

Findings

Minkowski equality characterization applies to divisorial filtrations

The theorem extends to bounded filtrations

Counterexamples exist for arbitrary filtrations

Abstract

Suppose that R is an analytically irreducible or excellent local domain with maximal ideal m_R. We consider multiplicities and mixed multiplicities of R by filtrations of m_R-primary ideals. We show that the theorem of Teissier, Rees and Sharp, and Katz, characterizing equality in the Minkowski inequality for multiplicities of ideals, is true for divisorial filtrations, and for the larger category of bounded filtrations. This theorem is not true for arbitrary filtrations of m_R-primary ideals.