On automorphisms of moduli spaces of parabolic vector bundles

Abstract

Fix $n\ge 5$ 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 $0\le a_i\le 1$. Consider the moduli space ${\mathcal{M}}_{\mathcal{A}}$ parametrizing rank two parabolic vector bundles with trivial determinant on $({\mathbb{P}}^1,p_1,\dots ,p_n)$ 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.

13 pages