PROJECTIVE DEGENERATIONS OF K3 SURFACES, GAUSSIAN MAPS, AND FANO THREEFOLDS

In this article we exhibit certain projective degenerations of smooth K3 surfaces of degree 2g - 2 in P(g) (whose Picard group is generated by the hyperplane class), to a union of two rational normal scrolls, a nd also to a union of planes. As a consequence we prove that the gener al hyperplane section of such K3 surfaces has a corank one Gaussian ma p, if g = 11 or g greater-than-or-equal-to 13. We also prove that the general such hyperplane section lies on a unique K3 surface, up to pro jectivities. Finally we present a new approach to the classification o f prime Fano threefolds of index one, which does not rely on the exist ence of a line.