Un théorème de Rao pour les familles de courbes gauches

Details Un théorème de Rao pour les familles de courbes gauches Un théorème de Rao pour les familles de courbes gauches

1997-10-15

arxiv.org/abs/alg-geom/9710017v1

Mathematics

Algebraic Geometry

22 pages, TeX

Scientific paper

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 $0\to P\to N\to J\to 0$ where $N$ is a locally free sheaf on $P^3_A$ and $P$ is a direct sum of invertible sheaves $O_P(-n_i)$. 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.

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