G-marked moduli spaces

Abstract

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 ${\mathsf{M}}_h^G$ of $G$-marked Gorenstein canonical models with Hilbert polynomial $h$, and prove the existence of ${\mathfrak{M}}_h[G]$, the coarse moduli scheme for ${\mathsf{M}}_h^G$. 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 ${\mathfrak{M}}_h[G]$.

25 pages, minor changes, a new section (section 5) containing new results has been added

References in corpus (1)

• Hulls and Husks

Cited by in corpus (2)

• Smooth covers of moduli stacks of Riemann surfaces with symmetry
• A topological construction of families of Galois covers of the line