Construction De Familles Minimales De Courbes Gauches

Let A be a noetherian local ring, and let C in PA be a family of curves, flat over A. We showed in an earlier paper how to associate to C a locally free sheaf N on PA, and we showed that two families of curves C,C′ are in the same biliaison class if and only if the corresponding sheaves N,N ′ are pseudoisomorphic (generalization of the theorem Rao). In this paper we show how to find all the flat families of curves C associated to a given locally free sheaf N and its twists, starting with the minimal family C0. We show also that all other families are obtained from the minimal family by a sequence of elementary biliaisons and a deformation (generalization of the theorem of Lazarsfeld Rao). The calculations are algorithmic in terms of a presentation of N .