Curves on Brill–Noether special K3 surfaces

Mukai showed that projective models of Brill–Noether general polarized K3 surfaces of genus 6–10 and 12 are obtained as linear sections of projective homogeneous varieties, and that their hyperplane sections are Brill–Noether general curves. In general, the question, raised by Knutsen, and attributed to Mukai, of whether the Brill–Noether generality of any polarized K3 surface $(S,H)$ is equivalent to the Brill–Noether generality of smooth curves $C$ in the linear system $\left|H\right|$, is still open. Using Lazarsfeld–Mukai bundle techniques, we answer this question in the affirmative for polarized K3 surfaces of genus $\le 19$, which provides a new and unified proof even in the genera where Mukai models exist.