On Automorphisms of Moduli Spaces of Parabolic Vector Bundles

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 $\big ({\mathbb{P}}^1, p_1,\dots , p_n\big )$ that 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.