Bounds on Multigraded Regularity

TL;DR

This work studies multigraded Castelnuovo–Mumford regularity on smooth projective toric varieties, treating reg $M$ as a region in $\operatorname{Pic} X$ invariant under the nef cone $\operatorname{Nef} X$. It proves that for a finitely generated faithful $S$-module $M$, $\operatorname{reg} M$ lies in a translate of $\operatorname{Nef} X$ determined by the degrees of $M$’s generators, yielding finitely many minimal reg degrees; it also shows pathologies when $M$ is not faithful or $\operatorname{rank} \operatorname{Pic} X \ge 2$. The paper then applies these ideas to powers of ideals, establishing asymptotically linear behavior: for any ideal $I$, $\operatorname{reg}(I^n)$ is trapped between a translate of $\operatorname{reg} S$ and a translate of $\operatorname{Nef} X$, with inner bounds governed by the Betti numbers of the Rees ring and outer bounds by the degrees of the generators. Central tools include chamber complexes, truncation arguments, and the Rees ring, which together control multigraded regularity uniformly in $n$ and extend projective-space phenomena to general toric settings.

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.

Figures (6)

• Figure 1: On left, the multigraded regularity (green) of the module in Example \ref{['ex:not-fg']} is an infinite staircase contained in a translate of the effective cone of $\mathcal{H}_2$ (blue). On right, the multigraded regularity (green) of the ideal itself is contained in the nef cone of $\mathcal{H}_2$ (dark blue).
• Figure 2: Left: fan of $\mathcal{H}_2$. Right: the cones $\operatorname{Nef}\mathcal{H}_2$ (dark blue) and $\operatorname{Eff}\mathcal{H}_2$ (blue).
• Figure 3: The regularity of $S$ (dark green) is contained in $\operatorname{Nef}\mathcal{H}_2$ (dark blue).
• Figure 4: A section of a hypothetical chamber complex with $P$ (green, horizontal) and $Q$ (red, vertical) inside $\operatorname{Eff} X$. The chamber $\operatorname{Nef} X$ and its wall $W$ are in blue.
• Figure 5: The multigraded regularity (dark green) of the module $M$ is contained in a translate $(-3,2) + \operatorname{Nef} X$ (light green) of the nef cone of $\mathcal{H}_2$ (dark blue).

Theorems & Definitions (32)

• Example 1.1
• Example 2.1
• Definition 2.2: c.f. maclaganSmith04
• Example 2.3
• Theorem 3.1
• proof
• Lemma 3.2
• Example 3.3
• Lemma 3.4
• proof