Resumen de Weak Brill–Noether on abelian surfaces

Izzet Coskun, Howard Nuer, Kota Yoshioka

• 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.