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

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.