Multiplicities and Volumes of Filtrations

In this article, we survey some aspects of the theory of multiplicities of m R -primary ideals in a local ring ( R , m R ) and the extension of this theory to multiplicities of graded families of m R -primary ideals. We first discuss the existence of multiplicities as a limit. Then, we focus on a theorem of Rees, characterizing when two m R -primary ideals I ⊂ J have the same multiplicity, and discuss extensions of this theorem to filtrations of m R -primary ideals. In the final sections, we give outlines of the proof of existence of the multiplicity of a graded family of m R -primary ideals as a limit, with mild conditions on R , and the proof of the extension of Rees’ theorem to divisorial filtrations.