Studying Symbolic Powers with Test Ideals

Abstract

How different is the symbolic power of an ideal from its ordinary power? This is an important question in commutativealgebra. In 2000, Ein–Lazarsfeld–Smith made substantial progress on this question by showing that every regular affine $\mathbb{C}$-algebra has the so-called Uniform Symbolic Topology Property. In particular, if $R$ is such a $\mathbb{C}$-algebra and $\mathfrak{p}$ a prime ideal in $R$, then $\mathfrak{p}^{(dn)} \subseteq \mathfrak{p}^n$ for all $n$, where $d$ is the dimension of $R$. Their proof uses the machinery of multiplier ideals. Their argument extends to the positive-characteristic setting by replacing multiplier ideals with their positive-characteristic analog, test ideals.

In this talk, we will explain one way to extend Ein–Lazarsfeld–Smith’s result to the non-regular setting. In par-ticular, we demonstrate a class of affine semigroup rings in which one can run a version of Ein–Lazarsfeld–Smith’s argument.