Geometric nullstellensatz and symbolic powers on arbitrary varieties

TL;DR

This paper extends classical algebraic geometry results, including the geometric nullstellensatz and symbolic power comparisons, from nonsingular to arbitrary projective varieties using Mather multiplier ideals.

Contribution

It introduces new generalizations of the nullstellensatz and symbolic power results applicable to all varieties, not just nonsingular ones.

Findings

Extended the geometric nullstellensatz to arbitrary projective varieties.

Proved comparison results between symbolic and ordinary powers on any varieties.

Utilized Mather multiplier ideals to generalize classical theorems.

Abstract

In recent years, a multiplier ideal defined on arbitrary varieties, so called Mather multiplier ideal, has been developed independently by Ein-Ishii-Mustata, and de Fernex-Docampo. With this new tool, we have a chance of extending some classical results proved in nonsingular case to arbitrary varieties to establish their general forms. In this paper, we first extend a result of geometric nullstellensatz due to Ein-Lazarsfeld in nonsingular case to any projective varieties. Then we prove a result on comparison of symbolic powers with ordinary powers on any varieties, which extends results of Ein-Lazarsfeld-Smith and Hochster-Huneke.