Projective Degenerations of K Surfaces Gaussian Maps and Fano Threefolds

In this article we exhibit certain projective degenerations of smoothK surfaces of degree g in P 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 K surfaces has a corank one Gaussian map if g or g We also prove that the general such hyperplane section lies on a unique K surface up to projectivities Finally we present a new approach to the classi cation of prime Fano threefolds of index one which does not rely on the existence of a line