Etale Fundamental group of moduli of torsors under Bruhat-Tits group scheme over a curve

Abstract

Let \(X\) be a smooth projective curve over an algebraically closed field \(k\). Let \(\mathcal{G}\) be a Bruhat-Tits group scheme on \(X\) which is generically semi-simple and trivial. We show that the \'etale fundamental group of the moduli stack \(\mathcal{M}_X(\mathcal{G})\) of torsors under \(\mathcal{G}\) is isomorphic to that of the moduli stack \(\mathcal{M}_X(G)\) of principal \(G\)-bundles. For any smooth, noetherian and irreducible stack \(\mathcal{X}\), we show that an inclusion of an open substack \(\mathcal{X}^\circ\), whose complement has codimension at least two, will induce an isomorphism of \'etale fundamental group. Over \(\mathbb{C}\), we show that the open substack of regularly stable torsors in \(\mathcal{M}_X(\mathcal{G})\) has complement of codimension at least two when \(g_X \geq 3\). As an application, we show that the moduli space \(M_X(\mathcal{G})\) of \(\mathcal{G}\)-torsors is simply-connected.