The aim of this paper is to investigate the closed subschemes of moduli spaces corresponding to projective varieties which admit an effective action by a given finite group $G$. To achieve this, we introduce the moduli functor $\mathsf{ M}^G_h$ of $G$-marked Gorenstein canonical models with Hilbert polynomial $h$, and prove the existence of $\mathfrak{ M}_h[G]$, the coarse moduli scheme for $\mathsf{ M}^G_h$. Then we show that $\mathfrak{ M}_h[G]$ has a proper and finite morphism onto $\mathfrak{ M}_h$ so that its image $\mathfrak{ M}_h(G)$ is a closed subscheme. In the end we obtain the canonical representation type decomposition $\mathcal{ D}_h[G]$ of $\mathfrak{ M}_h[G]$ and use $\mathcal{ D}_h[G]$ to study the structure of $\mathfrak{ M}_h[G]$.
Categories: math.AG