Differentially fixed ideals in toric varieties

TL;DR

This paper characterizes when monomial ideals in affine semi-group rings are fixed by differential operators, revealing that all such ideals are fixed by infinitely many homogeneous differential operators, thus providing new tools for their study.

Contribution

It offers a complete characterization of monomial ideals fixed by differential operators in affine semi-group rings and demonstrates their applications in multiplier ideals and ideal membership detection.

Findings

Every monomial ideal is fixed by an infinite set of homogeneous differential operators.

Monomial ideals are uniquely determined by the differential operators fixing them.

Applications include detecting ideal membership in various classes of monomial ideals.

Abstract

This article concerns monomial ideals fixed by differential operators of affine semi-group rings over $\mathbb{C}$. We give a complete characterization of when this happens. Perhaps surprisingly, every monomial ideal is fixed by an infinite set of homogeneous differential operators and is in fact determined by them. This opens up a new tool for studying monomial ideals. We explore applications of this to (mixed) multiplier ideals and other variants as well as give examples of detecting ideal membership in integrally closed powers and symbolic powers of squarefree monomial ideals.