On the Corank of Gaussian Maps for General Embedded K3 Surfaces

Abstract

Let $S_g$ be a general prime K3 surface in $P^g$ of genus $g\ge 3$ or a general double cover of $P^2$ ramified along a sextic curve for $g=2$ and $S=S_{i,g}$ its {\it i}-th Veronese embedding. In this article we compute the corank of the Gaussian map $\hat{I}¦:{\bigwedge }^2{H}^0(S,O_S(1))\to H^0(S,\hat{I}{\text{c◯}}_S^1(2))$ for $i\ge 2,g\ge 2$ and $i=1,g\ge 17$. The main idea is to reduce the surjectivity of $\hat{I}¦$ to an application of the Kawamata-Viehweg vanishing theorem on the blow-up of $S\times S$ along,the diagonal. This is seen to apply once the hyperplane divisor of the K3 surface $S$ can be decomposed as a sum of three suitable birationally ample divisors. We show that such a decomposition exists when $i\ge 3$ or on some K3 surfaces, constructed using the surjectivity of the period mapping, when $i=1,g\ge 17$ or $i=2,g\ge 7$.

21 pages, fairly plain TeX