On the degree of irrationality of low genus $K3$ surfaces

Given a general polarized $K3$ surface $S\subset {\mathbb{P}}^g$ of genus $g\le 14$, we study projections $S\hookrightarrow {\mathbb{P}}^g⇢{\mathbb{P}}^2$ 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 $S⇢{\mathbb{P}}^2$ 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.