Howard Nuer: The Weak Brill–Noether Problem on Abelian surfaces

Abstract: Brill–Noether theory studies the cohomology jumping loci of stable sheave on a given variety inside the moduli space of such sheaves. In the case of line bundles on curves, this forms one of the cornerstones of classical algebraic geometry. We say that a moduli space of stable sheaves satisfies weak Brill–Noether (WBN) if the general sheaf has at most one non-zero cohomology group. In this talk we report on a joint project with Izzet Coskun and Kota Yoshioka addressing the weak Brill-Noether problem for abelian surfaces. In particular, we classify for which polarized abelian surfaces (X,H) all moduli spaces of stable sheaves satisfy WBN and for which polarized abelian surfaces there exists a counterexample to WBN, in which case we show there are infinitely many counterexamples. Time permitting, we will discuss 1) an application of our results to the classification of Chern classes of Ulrich bundles on abelian surfaces; and 2) progress on solving the same problem on other surfaces of Kodaira dimension zero.