论文标题
模量和驯服商奇异性的算术
Fields of moduli and the arithmetic of tame quotient singularities
论文作者
论文摘要
给定一个完美的字段$ k $，带有代数关闭$ \叠加{k} $和$ x $ x $ of $ \ overline {k} $的品种，$ x $的模型领域是$ \ overline {k} $的子字段，该子{ $x_γ$是同构至$ x $。 Moduli的字段包含在所有子延迟中$ k \ subset k'\ subset \ overline {k} $，以便$ x $下降到$ k'$。在本文中，我们扩展了形式主义，并在$ k $并不完美时定义模量领域。 此外，Dèbes和Emsalem确定了一个条件，该条件可确保在其模量范围内定义平滑曲线，并证明具有标记点的平滑曲线在其模量磁场上始终定义。我们的主要定理是这些结果的概括，适用于更高维的品种，以及具有其他结构的品种。 为了应用这一点，我们研究了何时具有商品奇异性的多样性的理性点提升到分辨率。结果，我们证明，$ x $ dimension $ d $带有光滑的点$ p $，使得$ \ permatatorName {aut}（aut}（x，p）$是有限的，étale的，prime to prime to $ d！$在其Moduli领域定义。
Given a perfect field $k$ with algebraic closure $\overline{k}$ and a variety $X$ over $\overline{k}$, the field of moduli of $X$ is the subfield of $\overline{k}$ of elements fixed by field automorphisms $γ\in\operatorname{Gal}(\overline{k}/k)$ such that the twist $X_γ$ is isomorphic to $X$. The field of moduli is contained in all subextensions $k\subset k'\subset\overline{k}$ such that $X$ descends to $k'$. In this paper we extend the formalism, and define the field of moduli when $k$ is not perfect. Furthermore, Dèbes and Emsalem identified a condition that ensures that a smooth curve is defined over its field of moduli, and prove that a smooth curve with a marked point is always defined over its field of moduli. Our main theorem is a generalization of these results that applies to higher dimensional varieties, and to varieties with additional structures. In order to apply this, we study the problem of when a rational point of a variety with quotient singularities lifts to a resolution. As a consequence, we prove that a variety $X$ of dimension $d$ with a smooth marked point $p$ such that $\operatorname{Aut}(X,p)$ is finite, étale and of degree prime to $d!$ is defined over its field of moduli.
