The Frobenius action on rank 2 vector bundles over curves in small genus and small characteristic

On Frobenius-destabilized Rank-2 Vector Bundles over Curves

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 ...

Limit Linear Series in Positive Characteristic and Frobenius-Unstable Vector Bundles on Curves

Using limit linear series and a result controlling degeneration from separable maps to inseparable maps, we give a formula for the number of self-maps of P1 with ramification to order ei at general points Pi, in the case that all ei are less than the characteristic. We also develop a new, more functorial construction for the basic theory of limit linear series, which works transparently in posi...

Orthogonal Bundles over Curves in Characteristic Two

Let X be a smooth projective curve of genus g ≥ 2 defined over a field of characteristic two. We give examples of stable orthogonal bundles with unstable underlying vector bundles and use them to give counterexamples to Behrend’s conjecture on the canonical reduction of principal G-bundles for G = SO(n) with n ≥ 7. Let X be a smooth projective curve of genus g ≥ 2 and let G be a connected reduc...

Vector bundles over algebraic curves

Bundles of Rank 2 with Small Clifford Index on Algebraic Curves

In this paper, we construct stable bundles E of rank 2 on suitably chosen curves of any genus g ≥ 12 with maximal Clifford index such that the Clifford index of E takes the minimum possible value for curves with this property.