G-marked moduli spaces

TL;DR

This paper constructs and analyzes moduli spaces of projective varieties with finite group actions, introducing G-marked models, proving their existence, and exploring their structure and decomposition.

Contribution

It introduces the moduli functor for G-marked models, proves the existence of the coarse moduli scheme, and studies its structure via a canonical decomposition.

Findings

Existence of the coarse moduli scheme $rak{M}_h[G]$ for G-marked models.

Proper and finite morphism from $rak{M}_h[G]$ to $rak{M}_h$ with a closed image.

Canonical decomposition $rak{D}_h[G]$ of the moduli space.

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 $\mathsf{M}^G_h$ 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 $\mathfrak{M}_h[G]$.