Gushel–Mukai varieties: Moduli

Abstract

We describe the moduli stack of Gushel–Mukai varieties as a global quotient stack and its coarse moduli space as the corresponding GIT quotient. The construction is based on a comprehensive study of the relation between this stack and the stack of so-called Lagrangian data defined in our previous works; roughly speaking, we show that the former is a generalized root stack of the latter. As an application, we define the period map for Gushel–Mukai varieties and construct some complete nonisotrivial families of smooth Gushel–Mukai varieties. In an appendix, we describe a generalization of the root stack construction used in our approach to the moduli stack.