Characterizing Multigraded Regularity and Virtual Resolutions on Products of Projective Spaces

TL;DR

This paper investigates the multigraded regularity and virtual resolutions of modules over products of projective spaces, establishing new bounds and characterizations that enhance understanding of their algebraic and geometric properties.

Contribution

It introduces a new characterization of multigraded regularity via minimal free resolutions of truncations and provides bounds based on Betti numbers, with applications to complete intersections.

Findings

Multigraded regularity region is determined by minimal resolutions of truncations.

A new bound on regularity in terms of Betti numbers is established.

Regularity for complete intersections in products of projective spaces is computed.

Abstract

We explore the relationship between multigraded Castelnuovo--Mumford regularity, truncations, Betti numbers, and virtual resolutions on a product of projective spaces $X$. After proving a uniqueness theorem for certain minimal virtual resolutions, we show that the multigraded regularity region of a module $M$ is determined by the minimal graded free resolutions of the truncations $M_{\geq\mathbf d}$ for $\mathbf d\in\operatorname{Pic} X$. Further, by relating the minimal graded free resolutions of $M$ and $M_{\geq\mathbf d}$ we provide a new bound on multigraded regularity of $M$ in terms of its Betti numbers. Using this characterization of regularity and this bound we also compute the multigraded Castelnuovo--Mumford regularity for a wide class of complete intersections in products of projective spaces.