The variety of module structures
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Introduction. In this paper we study how is "structured" the set of curves of the projective space having assigned dimensions for the cohomology groups Hi (P3, Jx (t)). In particular, we show that almost always this locally closed subset of the Hilbert scheme, let us call it HF, is reducible, whilst it is known that if we fix also a multiplicative structure on the cohomology in dimension one (that is to say, we fix a liaison class) we get an irreducible subset. First, we perform a study of the variety V which parametrizes all possible structures of graded k [x0, x1, ..., xn]-module which are compatible with a "graded" k-vector space structure. Then we construct two families of maps: the ones from (a suitable subset of) HF to V, the other ones from (a suitable fibration over) V to HF. Then we show that, given V, there exist infinitely many cohomologies F such that the irreducible components of HF are in 1-1 correspondence with those of V. This approach shows that Buchsbaum curves are very particular and play a special role in HF. We sketch some applications, for instance to the so-called Zeuthen problem. We also concentrate on cohomologies of curves of maximal rank, showing that every liaison class whose Hartshorne-Rao module has length at most 2 contains maximal rank curves, and giving conditions for modules of greater length. We thank A. Hirschowitz for having suggested us this problem, and the referee for his suggestions to the first version of this paper.