Bounds on Multigraded Regularity

TL;DR

This paper establishes bounds on the multigraded Castelnuovo--Mumford regularity of modules over toric varieties, showing it is contained within certain translated cones and providing asymptotic bounds for powers of ideals.

Contribution

It proves that multigraded regularity of finitely generated modules is contained in a translate of the nef cone, and provides asymptotic bounds for the regularity of ideal powers.

Findings

Regularity is contained in a translate of the nef cone.

Finitely many minimal elements of regularity exist.

Asymptotic bounds for powers of ideals' regularity.

Abstract

Multigraded Castelnuovo--Mumford regularity of a module $M$ over the total coordinate ring $S$ of a smooth projective toric variety $X$ is a region $\operatorname{reg} M \subset \operatorname{Pic} X$ invariant under translation by the nef cone $\operatorname{Nef} X$. We prove that the multigraded regularity of a finitely generated faithful module is contained in a translate of $\operatorname{Nef} X$ determined by the degrees of the generators of $M$, and thus contains only finitely many minimal elements. We show that this condition can fail even for cyclic modules if $M$ has torsion and the rank of the Picard group is at least two. As an application, we exhibit asymptotic bounds for the multigraded regularity of powers of ideals. For $I$ an ideal in $S$, we bound $\operatorname{reg}(I^n)$ by proving that it contains a translate of $\operatorname{reg} S$ and is contained in a translate of $\operatorname{Nef} X$, where each bound translates by a fixed vector as $n$ increases.