Frobenius-unstable Vector Bundles and the Generalized Verschiebung

Let C be a smooth curve, and Mr(C) the coarse moduli space of vector bundles of rank r and trivial determinant on C. We discuss the generalized Verschiebung map Vr : Mr(C) 99K Mr(C) induced by pulling back under Frobenius. We begin with a survey of general background results on the Verschiebung, as well as certain more specialized work in fixed genus, characteristic, and rank. We then discuss the case of vector bundles of rank 2 on a curve of genus 2 in some detail, explaining how to conclude the degree of V2 from sufficient knowledge of the locus of semistable bundles which are destabilized by Frobenius. Finally, we sketch how one may obtain such results on the Frobenius-unstable locus by degeneration arguments. 1. Background on the Verschiebung We fix the following notation: Situation 1.1. We suppose C is a smooth, proper curve over an algebraically closed field k of characteristic p, and denote by C the p-twist of C over k, and F : C → C the relative Frobenius morphism. Finally, for any r > 1 we denote by Mr(C) and Mr(C ) the coarse moduli spaces of semistable vector bundles of rank r and trivial determinant on C and C. The basic existence result on the Verschiebung map may be summarized as follows: Proposition 1.2. In the above situation, and given r > 1, the operation of pulling back vector bundles under F induces a generalized Verschiebung rational map Vr : Mr(C ) 99K Mr(C). If we denote by Ur the open subset of Mr(C ) corresponding to bundles E such that F (E) is semi-stable, then we obtain a morphism Vr : Ur → Mr(C). The proof of this proposition is routine, and would be trivial if not for the coarseness of the moduli spaces in question. Definition 1.3. We say that a semistable vector bundle F is Frobenius-unstable if it is in the complement of Ur , which is to say, if F F is unstable. There are several motivations to study this generalized Verschiebung. Of course, the importance of the Verschiebung is well-established in the case of rank 1, so one might naturally want to study its generalization. However, there is also a close relationship between the Verschiebung map and p-adic representations of the fundamental group of C, when the base field k for our curve C is finite (see the introduction to [4]). In particular, de Jong showed that curves in the moduli space of vector bundles which are fixed under some iterate of the Verschiebung will correspond to p-adic representations for which the geometric fundamental group has