On Frobenius-destabilized rank-2 vector bundles over curves

Abstract

Let X be a smooth projective curve of genus g ≥ 2 over an algebraically closed field k of characteristic p > 0. Let MX be the moduli space of semistable rank-2 vector bundles over X with trivial determinant. The relative Frobenius map F : X → X1 induces by pull-back a rational map V : MX1 - - → MX. In this paper we show the following results.

1. For any line bundle L over X, the rank-p vector bundle F∗ L is stable.
2. The rational map V has base points, i.e., there exist stable bundles E over X1 such that F∗E is not semistable.
3. Let ℬ ⊂ MX1 denote the scheme-theoretical base locus of V. If g = 2, p > 2 and X ordinary, then ℬ is a 0-dimensional local complete intersection of length 2/3 · p(p2 − 2) and the degree of V equals 1/3 · p(p2 + 2).