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

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

1993-11-09

arxiv.org/abs/alg-geom/9311002v1

Mathematics

Algebraic Geometry

24 pages, AMS-LaTeX 1.1

Scientific paper

10.1007/BF01232682

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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