On Frobenius-destabilized rank-2 vector bundles over curves

TL;DR

This paper investigates the effects of Frobenius pull-back on rank-2 vector bundles over algebraic curves in positive characteristic, revealing stability properties, base points, and explicit geometric characteristics of the moduli space.

Contribution

It demonstrates stability of Frobenius push-forwards of line bundles, identifies base points of the Frobenius-induced rational map, and computes the base locus and degree in specific cases.

Findings

F_*L is stable for any line bundle L

The rational map V has base points where F^* E is not semistable

The base locus B is a 0-dimensional scheme with explicit length and degree in certain cases

Abstract

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).