The action of the Frobenius map on rank 2 vector bundles in characteristic 2

Abstract

We compute the rational morphism of “inverse image by Frobenius” acting on the coarse moduli space of semi-stable rank 2 2 bundles of trivial determinant over an ordinary genus curve in characteristic 2 2 .

Mochizuki’s Crys-Stable Bundles: A Lexicon and Applications

“…This in turn has immediate implications to a subject which has recently been studied by a number of different people (see, for instance, [15], [14], [8], [20]): Frobeniusunstable vector bundles, and by extension the generalized Verschiebung rational map induced on moduli spaces of vector bundles by pulling back under Frobenius. Furthermore, together with the results of [22], one can use Mochizuki's work to describe rational functions with prescribed ramification in positive characteristic.…”

A smooth counterexample to Nori’s conjecture on the fundamental group scheme

“…We briefly recall some results from [LP1] and [LP2]. Let X be a smooth projective ordinary curve of genus 2 defined over an algebraically closed field k of characteristic 2.…”

“…We identify M X with the projective space P 3 (see [LP1] Proposition 5.1). We denote by V : P , P 10 (x) = x 00 x 10 + x 01 x 11 , P 01 (x) = x 00 x 01 + x 10 x 11 , P 11 (x) = x 00 x 11 + x 10 x 01 .…”

“…We recall (see, e.g., [LP1], Lemma 2.11 (i)) that the coordinates of A 0 ∈ P 3 in the coordinate system (x 00 : x 01 : x 10 : x 11 ) are (1 : 0 : 0 : 0). It follows from Proposition 3.1 and equations (3.2) that V −1 (A 0 ) consists of the 4 points which correspond to the 2-torsion points of the Jacobian of X. Abusing notation we denote by A 1 both the 2-torsion line bundle on X and the point (0 : 1 : 0 : 0) ∈ P 3 .…”

The generalized Verschiebung map for curves of genus 2

“…Aside from the importance of the Verschiebung in the case of Jacobians, and the consequent desire to understand its generalization to higher rank, motivation for understanding the geometry of the Verschiebung map was provided by the close relationship between the Verschiebung map and characteristic-p representations of the fundamental group of C, when the base field k for our curve C is finite (see the introduction to [16]). In particular, A. J. de Jong showed that curves in the moduli space of vector bundles which are fixed under some iterate of the Verschiebung will correspond to characteristic-p representations for which the geometric fundamental group has infinite image, which he conjectures in [4] cannot happen for characteristic-ℓ representations.…”