Frobenius Stratification of Moduli Spaces of Rank $3$ Vector Bundles in Characteristic $3$, I

TL;DR

This paper investigates the Frobenius stratification of the moduli space of rank 3 stable vector bundles on a genus 2 curve over a field of characteristic 3, determining the structure and dimensions of the strata.

Contribution

It provides a detailed analysis of Frobenius stratification for rank 3 bundles in characteristic 3, including irreducibility and dimension results for each stratum.

Findings

Irreducibility of Frobenius strata

Dimension formulas for each stratum

Description of Harder-Narasimhan polygons

Abstract

Let $X$ be a smooth projective curve of genus $g\geq 2$ over an algebraically closed field $k$ of characteristic $p>0$, $F_X:X\rightarrow X$ the absolute Frobenius morphism. Let $\M^s_X(r,d)$ be the moduli space of stable vector bundles of rank $r$ and degree $d$ on $X$. We study the Frobenius stratification of $\M^s_X(3,0)$ in terms of Harder-Narasimhan polygons of Frobenius pull backs of stable vector bundles and obtain the irreducibility and dimension of each non-empty Frobenius stratum in the case $(p,g)=(3,2)$.