Restriction of stable bundles in characteristic $p$

Abstract:

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $X$ be a nonsingular projective variety defined over $k$ and $H$ an ample line bundle on $X$. We shall prove that there exists an explicit number $m_{0}$ such that if $E$ is a $\mu$-stable vector bundle of rank at most three, then the restriction $E_{\vert D}$ is $\mu$-stable for all $m\geq m_{0}$ and all smooth irreducible divisors $D\in \vert mH\vert$. This result has implications to the geometry of the moduli space of $\mu$-stable bundles on a surface or a projective space.