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arXiv · 2208.00562

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

Michael K. Brown·Mahrud Sayrafi
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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.

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Michael K. Brown, Mahrud Sayrafi. 2022-08-01. A short resolution of the diagonal for smooth projective toric varieties of Picard rank 2. https://doi.org/10.2140/ant.2024.18.1923

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