Stabilit\'e des fibr\'es $\Lambda^{p}E_{L}$ et condition de Raynaud

Abstract

Let $C$ be a smooth curve of genus $g \geq 2$ 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 $\deg(L) \geq 2g$ or when $L$ is generic. Finally we prove that for each $p$ in $\{2,..., \mathrm{rg}(E_{L})-2\}$, if $\deg(L) \geq 2g+2$ then $\Lambda^{p}E_{L}$ is not verifying $(R)$.