Weak Brill-Noether on Abelian Surfaces

2024/08/12 by İzzet Coşkun, Coskun, Izzet, Howard Nuer +3 · 2 citations
Mathematics · #14F08 #14K99. Secondary: 14F06 #Algebraic Geometry (math.AG) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Primary 14D20

paper · pdf · doi:10.48550/arxiv.2408.06095

openalex publication_date 2024/08/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 M_X,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 M_X,H(v) that fail weak Brill-Noether. As a consequence, we classify Chern classes of Ulrich bundles on abelian surfaces.

Cited by

• Preservation of stability under the Fourier-Mukai transform whose kernel is the Poincare line bundle
• Vector bundles on bielliptic surfaces: Ulrich bundles and degree of irrationality

Related

• Weak Brill-Noether for rational surfaces
• The Brill-Noether Theory of the Moduli Spaces of Sheaves on Surfaces
• On the Brill-Noether theory for K3 surfaces, II
• Brill-Noether and existence of semistable sheaves on del Pezzo surfaces
• The cohomology of the general stable sheaf on a K3 surface