# On automorphisms of moduli spaces of parabolic vector bundles

## TL;DR

This paper characterizes the automorphism groups of moduli spaces of rank two parabolic vector bundles on the projective line, revealing their structure as elementary transformations and identifying the case with the largest automorphism group.

## Contribution

It provides a detailed description and modular interpretation of automorphism groups of these moduli spaces, especially for the central weight case, including their structure and geometric properties.

## Key findings

- Automorphism group is isomorphic to a product of cyclic groups of order 2.
- Largest automorphism group occurs at the central weight with all weights 1/2.
- The moduli space at the central weight is a Fano variety with specific smoothness and singularity properties.

## Abstract

Fix $n\geq 5$ general points $p_1, \dots, p_n\in\mathbb{P}^1$, and a weight vector $\mathcal{A} = (a_{1}, \dots, a_{n})$ of real numbers $0 \leq a_{i} \leq 1$. Consider the moduli space $\mathcal{M}_{\mathcal{A}}$ parametrizing rank two parabolic vector bundles with trivial determinant on $\big(\mathbb{P}^1, p_1,\dots , p_n\big)$ which are semistable with respect to $\mathcal{A}$. Under some conditions on the weights, we determine and give a modular interpretation for the automorphism group of the moduli space $\mathcal{M}_{\mathcal{A}}$. It is isomorphic to $\left(\frac{\mathbb{Z}}{2\mathbb{Z}}\right)^{k}$ for some $k\in \{0,\dots, n-1\}$, and is generated by admissible elementary transformations of parabolic vector bundles. The largest of these automorphism groups, with $k=n-1$, occurs for the central weight $\mathcal{A}_{F}= \left(\frac{1}{2},\dots,\frac{1}{2}\right)$. The corresponding moduli space ${\mathcal M}_{\mathcal{A}_F}$ is a Fano variety of dimension $n-3$, which is smooth if $n$ is odd, and has isolated singularities if $n$ is even.