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

Brown, Michael K.; Sayrafi, Mahrud

doi:10.2140/ant.2024.18.1923

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

Michael K. Brown and Mahrud Sayrafi

Vol. 18 (2024), No. 10, 1923–1943

DOI: 10.2140/ant.2024.18.1923

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 and 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 and 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.

Keywords

Horrocks splitting criterion, resolution of the diagonal, toric variety, virtual resolution

Mathematical Subject Classification

Primary: 13D02

Secondary: 14F06, 14F08

Milestones

Received: 11 November 2022

Revised: 30 August 2023

Accepted: 12 October 2023

Published: 7 October 2024

Authors

Michael K. Brown
Department of Mathematics and Statistics Auburn University Auburn, AL United States

Mahrud Sayrafi
School of Mathematics University of Minnesota Minneapolis, MN United States

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Vol. 18, No. 10, 2024