The cohomology of the general stable sheaf on a K3 surface

Abstract: Let X be a K3 surface of Picard rank one and degree 2n with ample generator H. Let M_H(v) be the moduli space of Gieseker semistable sheaves on X with Mukai vector v. In this talk, we consider the weak Brill-Noether property for v, namely that the general sheaf in M_H(v) has at most one nonzero cohomology group. We show that given any positive rank r, there are only finitely many Mukai vectors of rank r failing weak Brill-Noether over all K3 surfaces of Picard rank one. We discuss our algorithm for finding the potential counterexamples and demonstrate the utility of our approach by discussing how we were able to classify all such counterexamples up to rank 20 and calculate the cohomology of the general sheaf in each case. Moreover, for fixed rank r, we give sharp bounds on n, d, and a that guarantee that a Mukai vector v=(r,dH,a) satisfies weak Brill-Noether. As a corollary, we provide another proof of the classification of Ulrich bundles on K3 surfaces of Picard rank one. In addition, we discuss the question of when the general sheaf in M_H(v) is globally generated. This joint work with Izzet Coskun and Kota Yoshioka makes crucial use of Bridgeland stability conditions and wall-crossing.

algebraic geometry

Audience: researchers in the topic

ZAG (Zoom Algebraic Geometry) seminar

Series comments: Description: ZAG seminar

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