Hecke curves in Frobenius strata of moduli space of rank 2 vector bundles

TL;DR

This paper investigates the structure of the moduli space of rank 2 stable vector bundles over a genus g curve in characteristic 2, demonstrating that certain Frobenius strata are covered by Hecke curves, revealing geometric properties of these strata.

Contribution

It establishes the existence of Frobenius strata in the moduli space that are covered by Hecke curves, linking Frobenius instability with geometric curve structures.

Findings

Existence of Frobenius strata covered by Hecke curves.

Connection between Frobenius instability and geometric properties.

Insights into the structure of moduli space in characteristic 2.

Abstract

Let $k$ be an algebraically closed field with characteristic $2$, and let $X$ be a smooth projective algebraic curve of genus $g \geqslant 2$ over $k$. Let $\mathcal{M}^s_X(2,\mathcal{L})$ be the moduli space of rank $2$ stable vector bundles with determinant $\mathcal{L}$ on $X$. The Frobenius stratification measures the instability of bundles in $\mathcal{M}^s_X(r,\mathcal{L})$ under pullback by the Frobenius map. We show that there exists a Frobenius stratum in $\mathcal{M}^s_X(2,\mathcal{L})$ which is covered by Hecke curves.