On the second Gaussian map for curves on a K3 surface

Abstract

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 $C$ is a general hyperplane section of high genus (> 280) of a general polarized K3 surface, then the second Gaussian map of $C$ is surjective. The resulting bound for the genus $g$ of a general curve with surjective second Gaussian map is decreased to $g>152$.