Abstract

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 curves. First we introduce the concept of pseudo-isomorphism: let A be a noetherian ring, a morphism f:N→N′, where N and N′ are coherent sheaves on PA3, flat over A, is a pseudo-isomorphism if the induced morphism of functors H0(N(n)⊗A⋅)→H0(N′(n)⊗A⋅) (resp. H1(N(n)⊗A⋅)→H1(N′(n)⊗A⋅), resp. H2(N(n)⊗A⋅)→H2(N′(n)⊗A⋅)) is an isomorphism for all n⪡0 (resp. an isomorphism for all n, resp. a monomorphism for all n). Two sheaves are pseudo-isomorphic, if there exists a chain of pseudo-isomorphisms between them. An N-type resolution for a family of curves C defined by an ideal JC is an exact sequence 0→P→N→JC→0 where N is a locally free sheaf on PA3 and P is (in the case when A is a local ring) a direct sum of invertible sheaves OPA3(-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 JC and JC′ 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, 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.

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