A short resolution of the diagonal for smooth projective toric varieties of Picard rank 2
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866912082530664448 |
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| author | Brown, Michael K. Sayrafi, Mahrud |
| author_facet | Brown, Michael K. Sayrafi, Mahrud |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_00562 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A short resolution of the diagonal for smooth projective toric varieties of Picard rank 2 Brown, Michael K. Sayrafi, Mahrud Algebraic Geometry Commutative Algebra 13D02, 14F06, 14F08 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. |
| title | A short resolution of the diagonal for smooth projective toric varieties of Picard rank 2 |
| topic | Algebraic Geometry Commutative Algebra 13D02, 14F06, 14F08 |
| url | https://arxiv.org/abs/2208.00562 |