In this paper, we investigate the construction of two moduli stacks of Kummer varieties. The first one is the stack 𝒦gabs of abstract Kummer varieties and the second one is the stack 𝒦gem of embedded Kummer varieties. We will prove that 𝒦gabs is a Deligne-Mumford stack and its coarse moduli space is isomorphic to Ag, the coarse moduli space of principally polarized abelian varieties of dimension g. On the other hand, we give a modular family 𝒲g → U of embedded Kummer varieties embedded in ℙ2g−1 × ℙ2g−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 K2em of embedded Kummer surfaces and prove that it is obtained from A2 by contracting the locus swept by a particular linear equivalence class of curves. We conjecture that this is a general fact: Kgem could be obtained from Ag via a contraction for all g > 1.

中文翻译：

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抽象和嵌入 Kummer 变体的模空间

在本文中，我们研究了两个 Kummer 变体模堆栈的构造。第一个是堆栈𝒦G腹肌抽象的 Kummer 品种，第二个是堆栈𝒦G时间嵌入式 Kummer 品种。我们将证明𝒦G腹肌是一个 Deligne-Mumford 堆栈，其粗模空间同构于一种G, 维数的主要极化阿贝尔变体的粗模空间G. 另一方面，我们给出了一个模块化的家庭𝒲G → ü嵌入的 Kummer 品种ℙ2G-1 × ℙ2G-1，这意味着该家族的每条几何纤维都是嵌入的 Kummer 变体，并且此类变体的每个同构类至少出现一次作为纤维的类。因此，我们构建了粗模空间ķ2时间嵌入 Kummer 曲面，并证明它是从一种2通过收缩由特定线性等价类曲线扫过的轨迹。我们推测这是一个普遍的事实：ķG时间可以从一种G通过所有人的收缩G > 1.