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.

We study various measures of irrationality for hypersurfaces of large degree in projective space and other varieties. These include the least degree of a rational covering of projective space, and the minimal gonality of a covering family of curves. The theme is that positivity properties of canonical bundles lead to lower bounds on these invariants. In particular, we prove that if $X\subseteq \mathbf{P}^{n+1}$ is a very general smooth hypersurface of dimension $n$ and degree $d\geqslant 2n+1$, then any dominant rational mapping $f:X{\dashrightarrow}\mathbf{P}^{n}$ must have degree at least $d-1$. We also propose a number of open problems, and we show how our methods lead to simple new proofs of results of Ran and Beheshti–Eisenbud concerning varieties of multi-secant lines.