On The Global Quotient Structure of The Space of Twisted Stable Maps to a Quotient Stack

Abstract

Let $\mathcal{X}$ be a tame proper Deligne-Mumford stack of the form $[M/G]$ where $M$ is a scheme and $G$ is an algebraic group. We prove that the stack $\mathcal{K}_{g,n}(\mathcal{X},d)$ of twisted stable maps is a quotient stack and can be embedded into a smooth Deligne-Mumford stack. When $G$ is finite, we give a more precise construction of $\mathcal{K}_{g,n}(\mathcal{X},d)$ using Hilbert schemes and admissible $G$-covers.