On the global quotient structure of the space of twisted stable maps to a quotient stack

Abstract

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