${\mathbb{A}}^1$-connectedness of moduli of vector bundles on a stacky curve

Abstract

Let $\mathfrak{C}$ be a projective stacky curve over an infinite field $k$ with the coarse space $C$, a geometrically connected smooth projective curve, together with a generating sheaf $\mathcal{E}$ on $\mathfrak{C}$. We consider the moduli stack of $\mathcal{E}$-semistable vector bundles of fixed determinant and multiplicities on $\mathfrak{C}$, and show that it is ${\mathbb{A}}^1$-connected if and only if it admits a $k$-rational point. We show that this holds whenever genus of $C$ is at least $2$ and $C(k)\ne \varnothing$. As a consequence, we conclude that, for any system of weights $\mathbf{\alpha }$, the moduli stack of parabolic $\mathbf{\alpha }$-semistable vector bundles of fixed determinant and quasi-parabolic data on a geometrically connected smooth projective curve is ${\mathbb{A}}^1$-connected.

arXiv comment15 pages. Assumptions in the main results are clarified and also the exposition in sections 2 and 3 is simplified. Comments are welcome