The Stacks project

112.5.12 Group actions on stacks

Actions of groups on algebraic stacks naturally appear. For instance, symmetric group $S_ n$ acts on $\overline{\mathcal{M}}_{g, n}$ and for an action of a group $G$ on a scheme $X$, the normalizer of $G$ in $\text{Aut}(X)$ acts on $[X/G]$. Furthermore, torus actions on stacks often appear in Gromov-Witten theory.

• Romagny: Group actions on stacks and applications [romagny_actions]

  This paper makes precise what it means for a group to act on an algebraic stack and proves existence of fixed points as well as existence of quotients for actions of group schemes on algebraic stacks. See also Romagny's earlier note [romagny_notes].