On automorphisms of moduli spaces of parabolic vector bundles

Fix n≥5 general points p1,…,pn∈P1 and a weight vector A=(a1,…,an) of real numbers 0≤ai≤1⁠. Consider the moduli space MA parametrizing rank two parabolic vector bundles with trivial determinant on (P1,p1,…,pn) that are semistable with respect to A⁠. Under some conditions on the weights, we determine and give a modular interpretation for the automorphism group of the moduli space MA⁠. It is isomorphic to (Z2Z)k for some k∈{0,…,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 AF=(12,…,12)⁠. The corresponding moduli space MAF is a Fano variety of dimension n−3⁠, which is smooth if n is odd, and has isolated singularities if n is even.