ON THE DEGREE OF IRRATIONALITY OF LOW GENUS $K3$ SURFACES

Abstract Given a general polarized $K3$ surface $S\subset \mathbb P^g$ of genus $g\le 14$ , we study projections of minimal degree and their variational structure. In particular, we prove that the degree of irrationality of all such surfaces is at most $4$ , and that for $g=7,8,9,11$ there are no rational maps of degree $3$ induced by the primitive linear system. Our methods combine vector bundle techniques à la Lazarsfeld with derived category tools and also make use of the rich theory of singular curves on $K3$ surfaces.