A short resolution of the diagonal for smooth projective toric varieties of Picard rank 2

TL;DR

This paper constructs a linear resolution of the diagonal sheaf for smooth projective toric varieties of Picard rank 2, leading to new insights in syzygies, vector bundle splitting, and derived category dimensions.

Contribution

It provides the first explicit linear resolution of the diagonal for these varieties, enabling progress on conjectures related to syzygies, vector bundles, and derived categories.

Findings

Resolved the diagonal sheaf with a linear complex of length equal to the dimension of the variety.

Proved a new case of a conjecture on virtual resolutions and Hilbert's Syzygy Theorem.

Established a Horrocks-type splitting criterion and confirmed a conjecture on Rouquier dimension.

Abstract

Given a smooth projective toric variety $X$ of Picard rank 2, we resolve the diagonal sheaf on $X \times X$ by a linear complex of length $\dim{X}$ consisting of finite direct sums of line bundles. As applications, we prove a new case of a conjecture of Berkesch-Erman-Smith that predicts a version of Hilbert's Syzygy Theorem for virtual resolutions, and we obtain a Horrocks-type splitting criterion for vector bundles over smooth projective toric varieties of Picard rank 2, extending a result of Eisenbud-Erman-Schreyer. We also apply our results to give a new proof, in the case of smooth projective toric varieties of Picard rank 2, of a conjecture of Orlov concerning the Rouquier dimension of derived categories.