Homological and combinatorial aspects of virtually Cohen--Macaulay sheaves

TL;DR

This paper advances the understanding of virtually Cohen--Macaulay sheaves on toric varieties by constructing new classes of such sheaves and developing homological tools to assess their properties, bridging algebraic, geometric, and combinatorial perspectives.

Contribution

It introduces a large class of virtually Cohen--Macaulay Stanley--Reisner rings on products of projective spaces and develops new homological tools for evaluating the virtual Cohen--Macaulay property on general toric varieties.

Findings

Constructed explicit virtual resolutions for new classes of Stanley--Reisner rings.

Developed criteria to identify virtually Cohen--Macaulay modules.

Established relationships among different Cohen--Macaulay properties.

Abstract

When studying a graded module $M$ over the Cox ring of a smooth projective toric variety $X$, there are two standard types of resolutions commonly used to glean information: free resolutions of $M$ and vector bundle resolutions of its sheafification. Each approach comes with its own challenges. There is geometric information that free resolutions fail to encode, while vector bundle resolutions can resist study using algebraic and combinatorial techniques. Recently, Berkesch, Erman, and Smith introduced virtual resolutions, which capture desirable geometric information and are also amenable to algebraic and combinatorial study. The theory of virtual resolutions includes a notion of a virtually Cohen--Macaulay property, though tools for assessing which modules are virtually Cohen--Macaulay have only recently started to be developed. In this paper, we continue this research program in two related ways. The first is that, when $X$ is a product of projective spaces, we produce a large new class of virtually Cohen--Macaulay Stanley--Reisner rings, which we show to be virtually Cohen--Macaulay via explicit constructions of appropriate virtual resolutions reflecting the underlying combinatorial structure. The second is that, for an arbitrary smooth projective toric variety $X$, we develop homological tools for assessing the virtual Cohen--Macaulay property. Some of these tools give exclusionary criteria, and others are constructive methods for producing suitably short virtual resolutions. We also use these tools to establish relationships among the arithmetically, geometrically, and virtually Cohen--Macaulay properties.