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 $\mathrm{\mid }H\mathrm{\mid }$, 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.