# Gushel-Mukai varieties: moduli

## TL;DR

This paper constructs the moduli space of Gushel-Mukai varieties as a quotient stack, studies its relation to Lagrangian data, and defines the period map, providing new tools for understanding these varieties.

## Contribution

It offers a novel description of the moduli stack as a root stack and constructs the period map for Gushel-Mukai varieties.

## Key findings

- Moduli stack described as a global quotient stack.
- Constructed the period map for these varieties.
- Provided complete families of smooth Gushel-Mukai varieties.

## 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 Lagrangian data; 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 space.