Triades et familles de courbes gauches

Abstract

Let $A$ be a noetherian ring and $R_A$ be the graded ring $A[X,Y,Z,T]$. In this article we introduce the notion of a triad, which is a generalization to families of curves in ${\mathbf{P}}_A^3$ of the notion of Rao module. A triad is a complex of graded $R_A$-modules $(L_1\to L_0\to L_{-1})$ with certain finiteness hypotheses on its cohomology modules. A pseudo-isomorphism between two triads is a morphism of complexes which induces an isomorphism on the functors $h_0(L\otimes .)$ and a monomorphism on the functors $h_{-1}(L\otimes .)$. One says that two triads are pseudo-isomorphic if they are connected by a chain of pseudo-isomorphisms. We show that to each family of curves is associated a triad, unique up to pseudo-isomorphism, and we show that the map $\{families\; of\; curves\}\to \{triads\}$ has almost all the good properties of the map $\{curves\}\to \{Rao\; modules\}$. In a section of examples, we show how to construct triads and families of curves systematically starting from a graded module and a sub-quotient (that is a submodule of a quotient module), and we apply these results to show the connectedness of $H_{4,0}$.

59 pages, TeX