Abstract. In this article we exhibit certain projective degenerations of smooth K3 surfaces of degree 2g;2 i n 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 p r o ve that the general hyperplane section of such K3 surfaces has a corank one Gaussian map, if g = 1 1 o r g 13. We also prove that the general such h yperplane section lies on a unique K3 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. Contents

We study the existence of linear series on curves lying on an Enriques surface and general in their complete linear system. Using a method that works also below the Bogomolov-Reider range, we compute, in all cases, the gonality of such curves. We also give a new result about the positive cone of line bundles on an Enriques surface and we show how this relates to the gonality.

We introduce a new technique, based on Gaussian maps, to study the possibility, for a given surface, to lie on a threefold as a very ample divisor with given normal bundle. We give several applications, among which one to surfaces of general type and another one to Enriques surfaces. For the latter we prove that any threefold (with no assumption on its singularities) having as hyperplane section a smooth Enriques surface (by definition an Enriques-Fano threefold) has genus g ≤ 17 (where g is the genus of its smooth curve sections). Moreover we find a new Enriques-Fano threefold of genus 9 whose normalization has canonical but not terminal singularities and does not admit Q-smoothings.Besides the mentioned applications, in the present article we will concentrate most of our attention on the case of Enriques-Fano threefolds: in analogy with Fano threefolds where an anticanonical divisor is a K3 surface, we define Definition 1.3. An Enriques-Fano threefold is an irreducible three-dimensional variety X ⊂ P N having a hyperplane section S that is a smooth Enriques surface, and such that X is not a cone over S. We will say that X has genus g if g is the genus of its general curve section.Fano himself, in a 1938 article [Fa], claimed a classification of such threefolds, but his proof contains several gaps. Conte and Murre [CM] were the first to remark that an Enriques-Fano threefold must have some isolated singularities, typical examples of which are quadruple points with tangent cone the cone over the Veronese surface. Filling out some of the gaps in [Fa] and making some special assumptions on the singularities, Conte and Murre recovered some of the results of Fano, but not enough to give a classification, nor to bound the numerical invariants. On the opposite extreme, with the strong assumption that the Enriques-Fano threefold is a quotient of a smooth Fano threefold by an involution (this corresponds to having only cyclic quotient terminal singularities), a list was given by Bayle [Ba, Thm.A] and Sano [Sa, Thm.1.1], by using the classification of smooth Fano threefolds and studying which of them have such involutions.Moreover, by the results of Minagawa [Mi, MainThm2], any Enriques-Fano threefold with at most terminal singularities admits a Q-smoothing, that is [Mi, R1], it appears as central fiber of a small deformation over the 1-parameter unit disk, such that a general fiber has only cyclic quotient terminal singularities. This, together with the results of Bayle and Sano, gives then the bound g ≤ 13 for Enriques-Fano threefolds with at most terminal singularities.Bayle and Sano recovered all of the examples of Enriques-Fano threefolds given by Fano and Conte-Murre. As these were the only known examples, it has been conjectured for some time now that this list is complete or, at least, that the genus is bounded, in analogy with the celebrated genus bound for smooth Fano threefolds [I1, I2, Sh2].In Section 16, we will show that the list of known Enriques-Fano threefolds of Fano, Conte-Murre, Bayle and Sano is not...

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