  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 $|H|$, is still open. Using Lazarsfeld-Mukai bundle
techniques, we answer this question in the affirmative for polarized K3
surfaces of genus $\leq 19$, which provides a new and unified proof even in the
genera where Mukai models exist.
