On automorphisms of moduli spaces of parabolic vector bundles (1902.04136v1)

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.