On the existence of Enriques-Fano threefolds of index greater than one

Let X be an irreducible threefold in P^N having a hyperplane section Y that is a smooth Enriques surface and such that X is not a cone over Y . In 1938 Fano claimed a classification of such threefolds; however, due to gaps in his proof, the problem still remains open. In this article we solve the case when Y is the r-th Veronese embedding, for r > 1, of another Enriques surface, by proving that there are no such X. The latter is achieved, among other things, by a careful study of trisecant lines to Enriques surfaces. As another consequence we get precise information on the ideal of an Enriques surface. In a previous paper we had proved that any smooth linearly normal Enriques surface has homogeneous ideal generated by quadrics and cubics. Here we are able to specify when the quadrics are enough, at least scheme-theoretically.