2020/08/24 by İzzet Coşkun, Coskun, Izzet, Jack Huizenga +4 · 1 citation
Mathematics · #14D20 #14F05 (Secondary) #14J26 (Primary) #14J60 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2008.10695
openalex publication_date 2020/08/24 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
Computing the cohomology of the tensor product of two vector bundles is\ncentral in the study of their moduli spaces and in applications to\nrepresentation theory, combinatorics and physics. These computations play a\nfundamental role in the construction of Brill-Noether loci, birational geometry\nand S-duality. Using recent advances in the Minimal Model Program for moduli\nspaces of sheaves on \ℙ2, we compute the cohomology of the tensor\nproduct of general semistable bundles on \ℙ2. This solves a natural\nhigher rank generalization of the polynomial interpolation problem. More\nprecisely, let v and w be two Chern characters of stable bundles on\n\ℙ2 and assume that w is sufficiently divisible depending on v.\nLet V \∈ M(v) and W \∈ M(w) be two general stable bundles. We fully\ncompute the cohomology of V \⊗ W. In particular, we show that if W is\nexceptional, then V \⊗ W has at most one nonzero cohomology group\ndetermined by the slope and the Euler characteristic, generalizing foundational\nresults of Dr 'ezet, G "ottsche and Hirschowitz. We characterize the\ninvariants of effective Brill-Noether divisors on M(v). We also characterize\nwhen V\⊗ W is globally generated. Our computation is canonical given the\nbirational geometry of the moduli space, suggesting a roadmap for tackling\nanalogous problems on other surfaces.\n