Weak Brill–Noether on abelian surfaces

• Izzet Coskun ^[1] ; Howard Nuer ^[2] ; Kōta Yoshioka ^[3]
  1. [1] University of Illinois at Chicago

     University of Illinois at Chicago

     City of Chicago, Estados Unidos

     
  2. [2] Technion – Israel Institute of Technology

     Technion – Israel Institute of Technology

     Israel

     
  3. [3] Kobe University

     Kobe University

     Chuo-ku, Japón

     

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• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 31, Nº. 3, 2025
• Idioma: inglés
• DOI: 10.1007/s00029-025-01044-7
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• Resumen

  • We study the cohomology of a general stable sheaf on an abelian surface. We say that a moduli space satisfies weak Brill–Noether if the general sheaf has at most one non-zero cohomology group. Let (X, H) be a polarized abelian surface and let v = (r,ξ, a) be a Mukai vector on X with v2 0, r > 0 and ξ · H > 0. We show that if ρ(X) = 1 or ρ(X) = 2 and X contains an elliptic curve, then all the moduli spaces MX,H (v)satisfy weak Brill–Noether. Conversely, if ρ(X) > 2 or ρ(X) = 2 and X does not contain an elliptic curve, we show that there are infinitely many moduli spaces MX,H (v) that fail weak Brill–Noether. As a consequence, we classify Chern classes of Ulrich bundles on abelian surfaces.

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