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arXiv · alg-geom/9311002

Projective Degenerations of K3 Surfaces, Gaussian Maps, and Fano Threefolds

Ciro Ciliberto·Angelo Lopez·Rick Miranda
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Abstract

In this article we exhibit certain projective degenerations of smooth $K3$ surfaces of degree $2g-2$ in $\Bbb P^g$ (whose Picard group is generated by the hyperplane class), to a union of two rational normal scrolls, and also to a union of planes. As a consequence we prove that the general hyperplane section of such $K3$ surfaces has a corank one Gaussian map, if $g=11$ or $g\geq 13$. We also prove that the general such hyperplane section lies on a unique $K3$ surface, up to projectivities. Finally we present a new approach to the classification of prime Fano threefolds of index one, which does not rely on the existence of a line.

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Ciro Ciliberto, Angelo Lopez, Rick Miranda. 1993-11-09. Projective Degenerations of K3 Surfaces, Gaussian Maps, and Fano Threefolds. https://doi.org/10.1007/bf01232682

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