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 99K MX. In this paper we show the following results. (1) For any line bundle L over X, the rank-p vector bundle FL is stable. (2) The rational map V has base points, i.e., there exist stable bundles E over X1 such that FE 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 p(p 2 − 1) and the degree of V equals 1 p(p 2 + 2).