Degree functions of graded families of ideals

TL;DR

This paper investigates the behavior of degree functions and multiplicities of graded families of primary ideals in excellent normal local rings, providing limit formulas and examples illustrating the structure of Rees valuations.

Contribution

It introduces limit expressions for multiplicities and degree functions of graded ideal families and explores Rees valuation sets in dimension two, including novel examples.

Findings

Expressed multiplicities as limits of intersection products.

Provided refined results for divisorial filtrations in dimension 2.

Constructed examples of divisorial filtrations with finite and infinite Rees valuation sets.

Abstract

We express multiplicities and degree functions of graded families of $\mathfrak{m}_R$-primary ideals in an excellent normal local ring $(R,\mathfrak{m}_R)$ as limits of intersection products. Moreover, in dimension 2, we show more refined results for divisorial filtrations. Finally, also in dimension 2, we give an example of a non-Noetherian divisorial filtration $\{I_n\}_{n\geqslant 0}$ of $\mathfrak{m}_R$-primary ideals such that the union of all the sets of Rees valuations of all the $I_n$ is a finite set, and another example of a (necessarily non-Noetherian) divisorial filtration of $\mathfrak{m}_R$-primary ideals such that the set of all Rees valuations is infinite.