# Mixed multiplicities of Divisorial Filtrations

## TL;DR

This paper extends classical multiplicity theory to divisorial filtrations in excellent local domains, proving that key equalities and inequalities hold in this special case, unlike for arbitrary filtrations.

## Contribution

It demonstrates that important theorems on multiplicities and mixed multiplicities, valid for ideals, also hold for divisorial filtrations, highlighting their special properties.

## Key findings

- Rees's theorem on ideal multiplicities extends to divisorial filtrations.
- Teissier-Rees-Sharp-Katz inequality holds for divisorial filtrations in dimension two.
- Mixed multiplicities correspond to anti-positive intersection products on a normal scheme.

## Abstract

Suppose that $R$ is an excellent local domain with maximal ideal $m_R$. The theory of multiplicities and mixed multiplicities of $m_R$-primary ideals extends to (possibly non Noetherian) filtrations of $R$ by $m_R$-primary ideals, and many of the classical theorems for $m_R$-primary ideals continue to hold for filtrations. The celebrated theorems involving inequalities continue to hold for filtrations, but the good conclusions that hold in the case of equality for $m_R$-primary ideals do not hold for filtrations. In this article, we consider multiplicities and mixed multiplicities of $R$ by $m_R$-primary divisorial filtrations. We show that some important theorems on equalities of multiplicities and mixed multiplicities of $m_R$-primary ideals, which are not true in general for filtrations, are true for divisorial filtrations. We prove that a theorem of Rees showing that if there is an inclusion of $m_R$-primary ideals $I\subset I'$ with the same multiplicity then $I$ and $I'$ have the same integral closure also holds for divisorial filtrations. This theorem does not hold for arbitrary filtrations. We show that the Teissier Rees Sharp Katz theorem on equality in the Minkowski inequality holds for divisorial filtrations in an excellent domain of dimension two. We also show that the mixed multiplicities of divisorial filtrations are anti-positive intersection products on a suitable normal scheme $X$ birationally dominating $R$, when $R$ is an algebraic local domain.