Construction de familles minimales de courbes gauches

Abstract

Let $A$ be a local noetherian ring and $N$ be a locally sheaf on the projective space $P^3_A$ : one proves easily that there exists a family $C$ of (smooth connected) curves contained in $P^3_A$, flat over $A$, and an integer $h$ such that the ideal sheaf $J$ of $C$ has a resolution $0\to P\to N\to J\to 0$ where $P$ is a direct sum of invertible sheaves $O_P(-n_i)$. In this paper we determine, for a given sheaf $N$, all the families of curves with such a resolution, especially the minimal ones (corresponding to the minimum value of $h$). It gives a description of the biliaison class related to $N$, and a tool for constructing families of space curves.