TY - JOUR

T1 - The cohomology of general tensor products of vector bundles on P2

AU - Coskun, Izzet

AU - Huizenga, Jack

AU - Kopper, John

N1 - Funding Information:
We would like to thank Arend Bayer, Aaron Bertram, Lawrence Ein, Joe Harris, Emanuele Macr?, Sam Shideler, and Jonathan Wolf for valuable conversations related to the subject matter of the paper. We would also like to thank the referee who provided several helpful comments on the paper.
Funding Information:
During the preparation of this article the first author was partially supported by the NSF FRG Grant DMS 1664296 and the second author was partially supported by NSF FRG Grant DMS 1664303.
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

PY - 2021/11

Y1 - 2021/11

N2 - Computing the cohomology of the tensor product of two vector bundles is central in the study of their moduli spaces and in applications to representation theory, combinatorics and physics. These computations play a fundamental role in the construction of Brill–Noether loci, birational geometry and S-duality. Using recent advances in the Minimal Model Program for moduli spaces of sheaves on P2, we compute the cohomology of the tensor product of general semistable bundles on P2. This solves a natural higher rank generalization of the polynomial interpolation problem. More precisely, let v and w be two Chern characters of stable bundles on P2 and assume that w is sufficiently divisible depending on v. Let V∈ M(v) and W∈ M(w) be two general stable bundles. We fully compute the cohomology of V⊗ W. In particular, we show that if W is exceptional, then V⊗ 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 characterize the invariants of effective Brill–Noether divisors on M(v). We also characterize when V⊗ W is globally generated. Our computation is canonical given the birational geometry of the moduli space, suggesting a roadmap for tackling analogous problems on other surfaces.

AB - Computing the cohomology of the tensor product of two vector bundles is central in the study of their moduli spaces and in applications to representation theory, combinatorics and physics. These computations play a fundamental role in the construction of Brill–Noether loci, birational geometry and S-duality. Using recent advances in the Minimal Model Program for moduli spaces of sheaves on P2, we compute the cohomology of the tensor product of general semistable bundles on P2. This solves a natural higher rank generalization of the polynomial interpolation problem. More precisely, let v and w be two Chern characters of stable bundles on P2 and assume that w is sufficiently divisible depending on v. Let V∈ M(v) and W∈ M(w) be two general stable bundles. We fully compute the cohomology of V⊗ W. In particular, we show that if W is exceptional, then V⊗ 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 characterize the invariants of effective Brill–Noether divisors on M(v). We also characterize when V⊗ W is globally generated. Our computation is canonical given the birational geometry of the moduli space, suggesting a roadmap for tackling analogous problems on other surfaces.

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U2 - 10.1007/s00029-021-00707-5

DO - 10.1007/s00029-021-00707-5

M3 - Article

AN - SCOPUS:85116007878

VL - 27

JO - Selecta Mathematica, New Series

JF - Selecta Mathematica, New Series

SN - 1022-1824

IS - 5

M1 - 94

ER -