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The cohomology of general tensor products of vector bundles on the projective plane

Banff International Research Station for Mathematical Innovation and Discovery

Using recent advances in the Minimal Model Program for moduli spaces of sheaves on the projective plane, we compute the cohomology of the tensor product of general semistable bundles on the projective plane. More precisely, let V and W be two general stable bundles, and suppose the numerical invariants of W are sufficiently divisible. We fully compute the cohomology of the tensor product of V and W. In particular, we show that if W is exceptional, then the tensor product of V and W has at most one nonzero cohomology group determined by the slope and the Euler characteristic, generalizing foundational results of Drézet, Göttsche and Hirschowitz. We also characterize when the tensor product of V and W is globally generated. Crucially, our computation is canonical given the birational geometry of the moduli space, providing a roadmap for tackling analogous problems on other surfaces. This is joint work with Izzet Coskun and John Kopper.

Mathematics; Algebraic geometry; Category theory; homological algebra

video/mp4

Author affiliation: Pennsylvania State University

2021-05-05

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/78105

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/