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.

Figures (3)

• Figure 1: Shape of a set $\mathcal{B}$ in Proposition \ref{['prop:charact_1singularpt']} if $e\neq D$ and $d = 2$.
• Figure 2: For $\mathcal{A}$ as in Example \ref{['ex:1sing']}, filled circles represent elements in $\mathcal{A}$ and $2 \mathcal{A}$, respectively; while empty squares correspond to elements in $\Delta_{1,2} \setminus \mathcal{A}$ and $\Delta_{2,2} \setminus 2\mathcal{A}$, respectively.
• Figure 3: For $\mathcal{A}$ as in Example \ref{['ex:1sing2_2']}, filled circles represent elements in $\mathcal{A}$ and $2 \mathcal{A}$, respectively; while empty squares correspond to elements in $\Delta_{1,2} \setminus \mathcal{A}$ and $\Delta_{2,2} \setminus 2\mathcal{A}$, respectively.

Theorems & Definitions (52)

• Lemma 1.1: BG2015
• Theorem 1.2: Herzog2003
• Example 1.3
• Proposition 1.4
• Remark 1.5
• proof : Proof of Prop. \ref{['prop:charact_1singularpt']}
• Proposition 2.1
• proof
• Lemma 2.2
• Corollary 2.3