Lazarsfeld-Mukai Bundles of Rank 2 on a Polarized K3 Surface of Low Genus

Abstract

Let g and c be any integers satisfying g ≥ 3 and 0 ≤ c ≤ ⌊ g−1 2 ⌋. It is known that there exists a polarized K3 surface (X, H) such that X is a K3 surface of Picard number 2, and H is a very ample line bundle on X of sectional genus g and Clifford index c, by Johnsen and Knutsen([J-K] and [Kn]). In this paper, we give a necessary and sufficient condition for a Lazarsfeld-Mukai bundles of rank 2 associated with a smooth curve C belonging to the linear system |H| and a base point free pencil on C not to be H-slope stable.