Stabilité des fibrés ${\Lambda }^p{E}_L$ et condition de Raynaud

Abstract

Let $C$ be a smooth curve of genus $g\ge 2$ on $\C$. Let $L$ be a line bundle on $C$ generated by its global sections and let $E_L$ be the dual of the kernel of the evaluation map $e_L$. We are studying here the relation between the stability the fact that the bundle is verifying a condition $(R)$ introduced by Raynaud : we prove that $E_L$ is semi stable when $C$ is general. We also prove that $E_L$ is verifying $(R)$ when $°(L)\ge 2g$ or when $L$ is generic. Finally we prove that for each $p$ in $\{2,...,\mathrm{rg}(E_L)-2\}$, if $°(L)\ge 2g+2$ then ${\Lambda }^p{E}_L$ is not verifying $(R)$.

11 pages