On Frobenius-destabilized rank-2 vector bundles over curves

Details On Frobenius-destabilized rank-2 vector bundles over curves On Frobenius-destabilized rank-2 vector bundles over curves

2003-09-29

arxiv.org/abs/math/0309456v2

Mathematics

Algebraic Geometry

22 pages

Scientific paper

Let X be a smooth projective curve of genus g \geq 2 over an algebraically closed field k of characteristic p > 0. Let 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: M_{X_1} \to M_{X}. 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 X_1 such that F^* E is not semistable. 3) Let B \subset M_{X_1} 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(p^2 -1) and the degree of V equals {1/3}p(p^2 +2).

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