Associated prime ideals of powers of ideals

Let R be a multivariate polynomial ring over a field and I a (monomial) ideal of R. A prime ideal P of R is called associated to I if it is of the form P = Ann R (x) for a non-zero element x ∈ R/I. For a given ideal I, we aim to understand the sequence (a n ) n∈N of numbers of associated prime ideals of the powers I n of the ideal I. While the sequence is known to stabilize for large n, its initial behaviour is yet not very well understood. This talk serves as an introduction to the topic and presents the latest results of a joint research project with Irena Swanson.