Complexes inattendus de droites de saut (Unexpected complex of jumping lines)

Abstract

We prove here the following results: \begin{th} Let $E$ a rank 2 vector bundle over ${\bf P}_3$, if $C$ is a reduced irreducible curve of ${\bf P}_3^{\vee}$ such that $E_H$ is unstable for all $H\in C$ then $C$ is a line. \end{th} We define now the set $W(E)$ as the set of planes $H$ such that the restricted bundle $E_H$ is unstable (that means non semi-stable). \begin{th} Let $E$ a rank 2 vector bundle over ${\bf P}_3$, with first chern class $c_1=c_1(E)$, $L$ a line and an integer $n\ge 0$. The following conditions are equivalent: \begin{description} \item[(i)] $L^{\vee}\subset W(E)$ and $H^0(E_H(-n+[-c_1/2]))\neq 0$ for a general point $H\in L^{\vee}$. \item[(ii)] There exist $m>0$ and a section $t\in H^0(E(m+[-c_1/2]))$ such that the zero variety of $t$ contains the infinitesimal neighbourhood of order $(m+n-1)$ of $L$. \end{description}