On the Corank of Gaussian Maps for General Embedded K3 Surfaces

Details On the Corank of Gaussian Maps for General Embedded K3 Surfaces On the Corank of Gaussian Maps for General Embedded K3 Surfaces

1994-11-15

arxiv.org/abs/alg-geom/9411008v1

Mathematics

Algebraic Geometry

21 pages, fairly plain TeX

Scientific paper

Let $S_{g}$ be a general prime K3 surface in $P^g$ of genus $g \geq 3$ 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 $\Phi:\bigwedge^2 H^0(S,O_S(1)) \to H^0(S,\Omega_S^1(2))$ for $i \geq 2, g \geq 2$ and $i=1, g \geq 17$. The main idea is to reduce the surjectivity of $\Phi$ 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 $i \geq 3$ or on some K3 surfaces, constructed using the surjectivity of the period mapping, when $i = 1, g \geq 17$ or $i=2, g \geq 7$.

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