Castelnuovo-Mumford regularity of toric varieties with at most one singular point

TL;DR

This paper establishes upper bounds for the Castelnuovo-Mumford regularity of simplicial projective toric varieties with at most one singular point, extending known bounds in the smooth case and for certain singular cases.

Contribution

It provides new bounds for the regularity of toric varieties with singularities, combining combinatorial and homological methods to extend previous results.

Findings

Bounds match Herzog and Hibi in the smooth case

Regularity satisfies Eisenbud-Goto bound when one singular point and dimension ≥ 3

Uses combinatorial and homological techniques to analyze sumsets

Abstract

We establish upper bounds for the Castelnuovo--Mumford regularity of the coordinate ring of a simplicial projective toric variety with at most one singular point. In the smooth case, our results recover the bound of Herzog and Hibi [Proc. Amer. Math. Soc. 131 (2003), 2641--2647], and therefore the Eisenbud--Goto bound. Furthermore, when the variety has exactly one singular point and dimension at least $3$, we prove that its regularity also satisfies the Eisenbud--Goto bound. The proof combines combinatorial and homological methods: we study the asymptotic behavior of the sumsets associated to the toric variety and relate it to Castelnuovo--Mumford regularity via a Hochster-like formula.