On Frobenius-destabilized rank-2 vector bundles over curves

Abstract

Let $X$ be a smooth projective curve of genus $g\ge 2$ over an algebraically closed field $k$ of characteristic $p>0$. Let ${\mathcal{M}}_X$ be the moduli space of semistable rank-2 vector bundles over $X$ with trivial determinant. The relative Frobenius map $F:X\to X_1$ induces by pull-back a rational map $V:{\mathcal{M}}_{X_1}⇢{\mathcal{M}}_X$. In this paper we show the following results.

1. For any line bundle $L$ over $X$, the rank-$p$ vector bundle $F_{\ast }L$ is stable.
2. The rational map $V$ has base points, i.e., there exist stable bundles $E$ over $X_1$ such that $F^{\ast }E$ is not semistable.
3. Let $\mathcal{B}\subset {\mathcal{M}}_{X_1}$ denote the scheme-theoretical base locus of $V$. If $g=2$, $p>2$ and $X$ ordinary, then $\mathcal{B}$ is a 0-dimensional local complete intersection of length $\frac{2}{3}p(p^2-1)$ and the degree of $V$ equals $\frac{1}{3}p(p^2+2)$.