Automorphism group of a moduli space of framed bundles over a curve

TL;DR

This paper determines the automorphism group of the moduli space of framed vector bundles over a smooth curve, revealing it is generated by curve automorphisms fixing a point, line bundle tensorizations, and PGL actions.

Contribution

It explicitly computes the automorphism group of the moduli space of framed bundles, identifying its generators and structure.

Findings

Automorphism group generated by curve automorphisms fixing the point

Includes tensorization with line bundles

Contains PGL(r,C) action on the framing

Abstract

Let $X$ be a smooth complex projective curve, and let $x\in X$ be a point. We compute the automorphism group of the moduli space of framed vector bundles on $X$ of rank $r \geq 2$ with a framing over $x$. It is shown that this automorphism group is generated by the following three: (1) pullbacks using automorphisms of the curve $X$ that fix the marked point $x$, (2) tensorization with certain line bundles over $X$ and (3) the action of $\operatorname{PGL}_r(\mathbb{C})$ through composition with the framing.