A Rao theorm for families of space curves

The aim of this paper is to prove a generalization of a theorem of Rao for families of space curves, which characterizes the biliaison classes of curv es. First we introduce the concept of pseudo-isomorphism: let A be a noethe rian ring, a morphism f: N --> N', where N and N' are coherent sheaves on P -A(3), flat over A, is a pseudo-isomorphism if the induced morphism of func tors H-0(N(n) x(A).) --> H-0(N'(n) x(A).) (resp. H-1(N(n) x(A).)--> H-1(N'( n)x(A).), resp. H-2(N(n) x(A) .) --> H-2(N'(n)x(A) .)) is an isomorphism fo r all n much less than0 (resp. an isomorphism for all n, resp. a monomorphi sm for all n). Two sheaves are pseudo-isomorphic, if there exists a chain o f pseudo-isomorphisms between them. An N-type resolution for a family of cu rves L; defined by an ideal J(e) is an exact sequence 0 --> P --> N -->J(e) --> 0 where N is a locally free sheaf on P-A(3), and P is (in the case whe n A is a local ring) a direct sum of invertible sheaves O-P3A(-n(i)). We pr ove the two following results, when the residual field of A is infinite: 1. Let L and L' be two flat families of space curves over the local ring A. Then L and L' are in the same biliaison class if and only if J(e) and J(e) ' are pseudo-isomorphic, up to a shift. 2. Let L and L' be two flat families of space curves over the local ring A, with N-type resolutions, involving sheaves N, N' Then L and L' are in the same biliaison class if and only if N and N' are pseudo-isomorphic, up to a shift. (C) 2001 Elsevier Science B.V. All rights reserved. MSC. 14H50; 14H 60.