On the second gaussian map for curves on a K3 surface

TL;DR

This paper proves that for high-genus curves on general polarized K3 surfaces, the second gaussian map is surjective, extending understanding of Gaussian maps and their relation to the geometry of K3 surfaces.

Contribution

It establishes the surjectivity of the second gaussian map for general high-genus curves on K3 surfaces, lowering the genus bound from previous results.

Findings

Second gaussian map is surjective for genus > 280

Genus bound for surjectivity is decreased to g > 152

Enhances understanding of Gaussian maps on K3 surface curves

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 (greater than 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.