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 [Formula: see text]. To achieve this, we introduce the moduli functor [Formula: see text] of [Formula: see text]-marked Gorenstein canonical models with Hilbert polynomial [Formula: see text], and prove the existence of [Formula: see text], the coarse moduli scheme for [Formula: see text]. Then we show that [Formula: see text] has a proper and finite morphism onto [Formula: see text] so that its image [Formula: see text] is a closed subscheme. In the end we obtain the canonical representation type decomposition [Formula: see text] of [Formula: see text] and use [Formula: see text] to study the structure of [Formula: see text].