By a theorem of Wahl, for canonically embedded curves which are hyperplane sections of K3 surfaces, the first Gaussian map is not surjective. In this paper we prove that if is a general hyperplane section of high genus (> 280) of a general polarized K3 surface, then the second Gaussian map of is surjective. The resulting bound for the genus of a general curve with surjective second Gaussian map is decreased to .
"On the second Gaussian map for curves on a K3 surface." Nagoya Math. J. 199 123 - 136, September 2010. https://doi.org/10.1215/00277630-2010-006