Moduli spaces of abstract and embedded Kummer varieties

In this paper, we investigate the construction of two moduli stacks of Kummer varieties. The first one is the stack $K^{abs}_g$ of abstract Kummer varieties and the second one is the stack $K^{em}_g$ of embedded Kummer varieties. We will prove that $K^{abs}_g$ is a Deligne-Mumford stack and its coarse moduli space is isomorphic to $A_g$, the coarse moduli space of principally polarized abelian varieties of dimension $g$. On the other hand we give a modular family $W_g\to U$ of embedded Kummer varieties embedded in $P^{2^g-1}\timesP^{2^g-1}$, 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 $\boldsymbol{K}^{em}_2$ of embedded Kummer surfaces and prove that it is obtained from $A_2$ by contracting the locus swept by a particular linear equivalence class of curves. We conjecture that this is a general fact: $\boldsymbo{K}^{em}_g$ could be obtained from $A_g$ via a contraction for all $g>1$.