Un théorème de Rao pour les familles de courbes gauches

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The aim of this paper is to prove a generalization of a theorem of Rao for families of space curves, which caracterizes the biliaison classes of curves. First we introduce the concept of pseudo-isomorphism of coherent sheaves, which generalizes the concept of stable isomorphism. An N-type resolution for a family of curves C over the local ring A, defined by an ideal J, is an exact sequence 0→ P→ N→ J→ 0 where N is a locally free sheaf on P³ₐ and P is a direct sum of invertible sheaves Oₚ(-nᵢ). We prove the two following results, when the residual field of A is infinite: 1. Let C and C' be two flat families of space curves over the local ring A. Then C and C' are in the same biliaison class if and only if their ideals J and J' are pseudo-isomorphic, up to a shift. 2. Let C and C' be two flat families of space curves over the local ring A, with N-type resolutions, involving sheaves N and N'. Then C and C' are in the same biliaison class if and only if N and N' are pseudo-isomorphic, up to a shift.