Let \({\mathcal M}_ g\) be the moduli space of genus \(g\) non-singular curves. Denote by \(\overline{M}_ g\) its natural compactification by means of stable curves. The main result of the paper is that \(\overline{M}_ g\) is the quotient of a smooth variety by a finite group action. The construction of this smooth variety is done as follows: let \(S\) be a topological compact orientable surface of genus \(g\). Denote by \(\widetilde{S} \to S\) the Galois covering corresponding to the subgroup of \(\pi_ 1(S,x)\) generated by the squares. The Galois group \(G\) of this covering is \(H_ 1(S,\mathbb{Z}/2)\). Denote by \(\Gamma_{S\left( \begin{smallmatrix} n\\2\end{smallmatrix}\right)}\) the set of homeomorphisms \(f\) of \(S\) such that a lifting \(\widetilde{f}\) of \(f\) to \(\widetilde{S}\) acts on \(H_ 1(\widetilde{S},\mathbb{Z}/n)\) as an element of \(G\).
A Prym level \(n\) structure on an algebraic curve \(X\) of genus \(g\) is an orientation preserving homeomorphism \(f : S \to X\) modulo the following equivalence relation: \(f\) is equivalent to \(f'\) if and only if \(f^{- 1}f'\) is in \(\Gamma_{S\left(\begin{smallmatrix} n\\2\end{smallmatrix}\right)}\). The compactification of the set of pairs consisting of a curve of genus \(g\) and a Prym level structure gives the required covering of \(\overline{\mathcal M}_ g\).