On the degree of irrationality of low genus K3 surfaces

Given a general polarized K3 surface S⊂Pg of genus g≤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.