2005/11/27 by Maxim Leyenson, Leyenson, Maxim
Mathematics · #14J10 #14J28 #14J60 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.math/0511659
openalex publication_date 2005/11/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let (S,H) be a polarized K3 surface. We define Brill-Noether filtration on moduli spaces of vector bundles on S. Assume that (c1(E),H) > 0 for a sheaf E in the moduli space. We give a formula for the expected dimension of the Brill-Noether subschemes. Following the classical theory for curves, we give a notion of Brill-Noether generic K3 surfaces. Studying correspondences between moduli spaces of sheaves of different ranks on S, we prove our main theorem: polarized K3 surface which is generic in sense of moduli is also generic in sense of Brill-Noether theory (here H is the positive generator of the Picard group of S). In case of algebraic curves such a theorem, proved by Griffiths and Harris and, independently, by Lazarsfeld, is sometimes called ``the strong theorem of the Brill-Noether theory''. We finish by considering a number of projective examples. In particular, we construct explicitly Brill-Noether special K3 surfaces of genus 5 and 6 and show the relation with the theory of Brill-Noether special curves.