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By a theorem of Wahl, for canonically embedded curves which are hyperplanesections of K3 surfaces, the first gaussian map is not surjective. In thispaper we prove that if C is a general hyperplane section of high genus (greaterthan 280) of a general polarized K3 surface, then the second gaussian map of Cis surjective. The resulting bound for the genus g of a general curve withsurjective second gaussian map is decreased to g >152.

On the second gaussian map for curves on a K3 surface