On canonical hypersurfaces of IP4

In this paper we show the existence of algebraic nonsingular 3-folds X of g eneral type having p(g)(X) = 5, h(1)(O-X) = h(2)(O-X) = 0, with K-X(3) = 6, .., 12, such that the canonical morphism phi is birational. In particular, we give an explicit description of families of such 3-folds studying the eq uation of the canonical image, which is a hypersurface of IP4 with ordinary singularities. This is done by looking at the resolution of the canonical ring as a module over the coordinates ring of IP4. In the case K-X(3) = 12 a specialization of the general S-fold of the family is also given, which h as a degree two canonical morphism, producing a new counterexample to the B abbage-Petri type conjecture, in dimension 3. We also introduce how the sam e method could be applied to construct similar examples of hypersurfaces in IP5.