Back to Algebraic Geometry Seminar: Fall 2022
Algebraic Geometry Seminar
Date: Tuesday, Oct 25, 2022
Time: 3:30PM - 4:30PM
Location: LCB 323
Mahrud Sayrafi
University of Minnesota
Title |
Short resolutions of the diagonal and a Horrocks-type splitting criterion in Picard rank 2 |
Abstract |
In 1964, Horrocks proved that a vector bundle on a projective space splits as a sum of line bundles if and only if it has no intermediate cohomology. Motivated by the study of indecomposable vector bundles, in 1978 Beilinson constructed a resolution of the diagonal on P^n which has been used to great effect in algebraic geometry. We obtain a Horrocks-type splitting criterion (under an additional hypothesis) for vector bundles over a smooth projective toric variety X of Picard rank 2 using a linear resolution of the diagonal consisting of finite direct sums of line bundles. Since this resolution has length dim(X), we also prove a new case of a conjecture of Berkesch–Erman–Smith that predicts a version of Hilbert’s Syzygy Theorem for virtual resolutions. This is joint work with Michael Brown. |
Algebraic Geometry