Claudia Polini

ABSTRACT

A classical theorem of Rees says that, for zero-dimensional ideals in a suitable local ring, integral dependence is detected by Hilbert-Samuel multiplicity. For arbitrary ideals, one must replace this single invariant with generalized multiplicities such as mixed multiplicities and the multiplicity sequence.

In this talk I will discuss how these invariants detect and organize integral dependence. I will emphasize recent work with Ngo Viet Trung and Bernd Ulrich on the behavior of mixed multiplicities and the multiplicity sequence under generic hyperplane sections, leading to lifting criteria for integral dependence. I will also explain new graded criteria showing that, for homogeneous ideals, integral dependence can often be detected by one global number, such as the multiplicity of a Rees algebra or the j-multiplicity. The talk will end by contrasting these positive results with examples from multidegree theory showing that closely related polar-type invariants do not always detect integral dependence.