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

Let 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 Kg,n(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 Kg,n(X , d) using Hilbert schemes and admissible G-covers.

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