The action of the Frobenius map on rank 2 vector bundles over a supersingular genus 2 curve in characteristic 2

TL;DR

This paper studies how the Frobenius map acts on rank 2 vector bundles over a supersingular genus 2 curve in characteristic 2, providing explicit equations and analyzing the map's properties.

Contribution

It extends previous work by deriving the Frobenius action equations for supersingular curves using deformation techniques, revealing new geometric features.

Findings

Frobenius map has a single base point on the moduli space.

The Zariski open set where the map is stable is identified.

Explicit equations for the Frobenius map on supersingular curves are obtained.

Abstract

Let $X$ be a smooth proper genus 2 curve over an algebraically closed field of characteristic 2. The absolute Frobenius induces a rational map $F$ on the the moduli space $M\_X$ of semi-stable rank 2 vector bundles over $X$, which is isomorphic to a 3-dimensional projective space. Y. Laszlo and C. Pauly recently gave the equations of $F$ for an ordinary $X$. Using deformation, we give these equations for a supersingular $X$ and draw some consequences such as the base locus of $F$ (one point), or the stability of the complementary Zariski open set.