Explicit constructions of short virtual resolutions of truncations

TL;DR

This paper introduces a new truncation concept for smooth projective toric varieties, constructing explicit cellular resolutions for nef truncations and connecting them with existing symplectic geometry-inspired resolutions.

Contribution

It provides explicit cellular resolutions for nef truncations of total coordinate rings and links these to prior symplectic geometry motivated resolutions.

Findings

Resolutions match those of Hanlon, Hicks, and Lazarev

Nontrivial homology demonstrated in algebraic analogue

Truncation concept applicable to arbitrary smooth projective toric varieties

Abstract

We propose a concept of truncation for arbitrary smooth projective toric varieties and construct explicit cellular resolutions for nef truncations of their total coordinate rings. We show that these resolutions agree with the short resolutions of Hanlon, Hicks, and Lazarev, which were motivated by symplectic geometry, and we use our definition to exhibit nontrivial homology in the commutative algebraic analogue of their construction.