Seshadri Regions and the Asymptotic Shape of Multigraded Regularity

TL;DR

This paper introduces Seshadri regions as a convex measure of positivity for subvarieties, and uses them to analyze the asymptotic behavior of Castelnuovo-Mumford regularity for ideal and symmetric powers on smooth projective toric varieties.

Contribution

It develops the theory of Seshadri regions and applies it to determine the asymptotic shape of multigraded regularity in algebraic geometry.

Findings

Seshadri regions unify classical constants across line bundles.

Asymptotic regularity can be characterized using Seshadri regions.

Results apply specifically to smooth projective toric varieties.

Abstract

We introduce the Seshadri region of a subvariety, a convex region packaging the classical Seshadri constants with respect to every line bundle simultaneously. We develop the theory of Seshadri regions as a measure of positivity along subvarieties and apply it to determine asymptotic Castelnuovo-Mumford regularity for ideal powers and symmetric powers on smooth projective toric varieties.