Christian Pauly

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 bundle FL is stable. (2) The rational map V has base points, i.e., there exist stable bundles E over X1 such that FE is not semistable. (3) Let B ⊂ MX1 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 p(p 2 − 1) and the degree of V equals 1 p(p 2 + 2).

On the Hitchin morphism in positive characteristic

Let X be a smooth projective curve over a field of characteristic p > 0. We show that the Hitchin morphism, which associates to a Higgs bundle its characteristic polynomial, has a nontrivial deformation over the affine line. This deformation is constructed by considering the moduli stack of t-connections on vector bundles on X and an analogue of the p-curvature, and by observing that the associated characteristic polynomial is, in a suitable sense, a p th -power.

HEISENBERG INVARIANT QUARTICS AND LUC(2) FOR A CURVE OF GENUS FOUR

The projective moduli variety SUC(2) of semistable rank 2 vector bundles with trivial determinant on a smooth projective curve C comes with a natural morphism φ to the linear series |2Θ| where Θ is the theta divisor on the Jacobian of C. Well-known results of Narasimhan and Ramanan say that φ is an isomorphism to P if C has genus 2 [16], and when C is nonhyperelliptic of genus 3 it is an isomorphism to a special Heisenberg-invariant quartic QC ⊂ P 7 [18]. The present paper is an attempt to extend these results to higher genus. In the nonhyperelliptic genus 3 case the so-called Coble quartic QC ⊂ |2Θ| = P is characterised by either of two properties: (i) QC is the unique Heisenberg-invariant quartic containing the Kummer variety, i.e. the image of Kum : JC → |2Θ|, x 7→ Θx + Θ−x, in its singular locus; and (ii) QC is precisely the set of 2Θ-divisors containing some translate of the curve W1 ⊂ J C .

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

Let

The Frobenius map, rank 2 vector bundles and Kummer's quartic surface in characteristic 2 and 3

X

PRYM VARIETIES OF SPECTRAL COVERS

be an ordinary smooth curve defined over an algebraically closed field of characteristic 2. The absolute Frobenius induces a rational map

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

F

Polarizations of Prym varieties for Weyl groups via abelianization

on the moduli space

A Duality for Spin Verlinde Spaces and Prym Theta Functions

M_X

FIBRES PARABOLIQUES DE RANG 2 ET FONCTIONS THETA GENERALISEES

of rank 2 vector bundles with fixed trivial determinant. If the genus of