Degree functions of graded families of ideals

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.