Moduli spaces of abstract and embedded Kummer varieties

TL;DR

This paper constructs and analyzes moduli stacks of Kummer varieties, proving their properties and relationships with abelian varieties, and introduces a modular family of embedded Kummer varieties with implications for their moduli spaces.

Contribution

It establishes the Deligne-Mumford property of the stack of abstract Kummer varieties and constructs a modular family of embedded Kummer varieties, linking their moduli spaces.

Findings

The stack of abstract Kummer varieties is a Deligne-Mumford stack.

The coarse moduli space of abstract Kummer varieties is isomorphic to the space of principally polarized abelian varieties.

The coarse moduli space of embedded Kummer surfaces is obtained from the abelian varieties space by contracting a curve.

Abstract

In this paper, we investigate the construction of two moduli stacks of Kummer varieties. The first one is the stack $\mathcal K^{\text{abs}}_g$ of abstract Kummer varieties and the second one is the stack $\mathcal K^{\text{em}}_g$ of embedded Kummer varieties. We will prove that $\mathcal K^{\text{abs}}_g$ is a Deligne-Mumford stack and its coarse moduli space is isomorphic to $\boldsymbol A_g$, the coarse moduli space of principally polarized abelian varieties of dimension $g$. On the other hand we give a modular family $\mathcal W_g\to U$ of embedded Kummer varieties embedded in $\mathbb P^{2^g-1}\times\mathbb P^{2^g-1}$, meaning that every geometric fiber of this family is an embedded Kummer variety and every isomorphic class of such varieties appears at least once as the class of a fiber. As a consequence, we construct the coarse moduli space $\boldsymbol{\mathsf K}^{\text{em}}_2$ of embedded Kummer surfaces and prove that it is obtained from $\boldsymbol A_2$ by contracting a particular curve inside this space. We conjecture that this is a general fact: $\boldsymbol{\mathsf K}^{\text{em}}_g$ could be obtained from $\boldsymbol A_g$ via a contraction for all $g>1$.