Frobenius stratification of moduli spaces of rank $3$ vector bundles in positive characteristic $3$, I

Abstract:

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