Frobenius splitting of moduli spaces of parabolic bundles

TL;DR

This paper proves that the moduli space of semistable parabolic bundles on a generic curve over a field of characteristic p>3r is Frobenius split, which has implications for its geometric and cohomological properties.

Contribution

It establishes Frobenius splitting for moduli spaces of parabolic bundles in characteristic p>3r, extending understanding of their geometric structure.

Findings

Moduli space is F-split for generic curves and parabolic structures.

F-splitting holds when characteristic p exceeds thrice the rank r.

Results apply to generic curves and parabolic data.

Abstract

Let $C$ be a nonsingular projective curve over an algebraically closed field of characteristic $p>0$ and $I\subset C$ be a finite set. If $\mathcal{U}_{C,\,\omega}$ denotes the moduli space of semistable parabolic bundles of rank $r$ and degree $d$ on $C$ with parabolic structures determined by $\omega=(k,\{\vec n(x),\vec a(x)\}_{x\in I})$, we prove that $\mathcal{U}_{C,\,\omega}$ is \textit{$F$-split} for generic $C$ and generic choice of $I$ when $p>3r$.