Irreducible families of curves with fixed cohomology
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One of the classical ways for studying the problem of the classification of curves in the projective space was the investigation of the irreducible families of curves, and this approach led to the notion of Hilbert scheme. In this paper we try to give an answer to the following question: do curves with fixed cohomology (see below for a precise definition) form an irreducible "family" (that is to say, an irreducible subscheme of a suitable Hilbert scheme)? Or, at least, if two curves have same postulation, deficiency and speciality, do they belong to the same irreducible component of the Hilbert scheme? In fact, all these informations are contained, from the numerical point of view, in the dimensions of the cohomology groups H1 (P3, Jx (t)), i = 0, 1, 2, t € 7Z. We will see that this question has a positive answer if we consider also the multiplicative structure induced on the direct sum + H1 (P3, Jx (t)) by the linear forms on P3. Several steps of t the proof follow arguments contained in , and in fact all this paper is inspired by the work of Lazarsfeld and Rao. We thank the referee of this paper for his suggestions and remarks to the first version of this paper. From now on, a curve will be a one-dimensional, locally Cohen-Macaulay and equidimensional subscheme of P3, where k is an algebraically closed field.