Brill-Noether-Petri curves on K3 surfaces

In '86 Lazarsfeld proved that a general hyperplane section of a rank one K3 surface is Brill-Noether-Petri (BNP) general, i.e. the map $ H^0( L)\otimes H^0( \omega_CL^{-1})\to H^0( \omega_C) $ is injective $\forall$ line bundles $L$ on $C$. In '97, Wahl showed that the non-surjectivity of the Gaussian map $\nu_1:\wedge ^2H^0( \omega_C)\to H^0( \omega^3_C)$ is a necessary condition for a curve $C$ to be a hyperplane section of a K3 surface (or a limit of such). He conjectured that, for a BNP general curve of genus $g\geq 12$, this condition should also be sufficient. We prove this conjecture (jointly with A. Bruno and E. Sernesi). In a separate work (with A. Bruno, G. Farkas, and G. Saccà) we produce concrete examples of BNP general curves, of any given genus $g$, as degree-$3g$ plane curves with eight $g$-tuple points and one $(g-1)$-tuple point and no other singularities (the Du Val curves).