Extendability of automorphisms of K3 surfaces

IF 0.6 3区 数学 Q3 MATHEMATICS
Yuya Matsumoto
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引用次数: 0

Abstract

A K3 surface $X$ over a $p$-adic field $K$ is said to have good reduction if it admits a proper smooth model over the ring of integers of $K$. Assuming this, we say that a subgroup $G$ of $\operatorname{Aut}(X)$ is extendable if $X$ admits a proper smooth model equipped with $G$-action (compatible with the action on $X$). We show that $G$ is extendable if it is of finite order prime to $p$ and acts symplectically (that is, preserves the global $2$-form on $X$). The proof relies on birational geometry of models of K3 surfaces, and equivariant simultaneous resolutions of certain singularities. We also give some examples of non-extendable actions.
K3 曲面的可扩展性
如果在 $K$ 的整数环上有一个适当的光滑模型,那么在 $p$-adic field $K$ 上的 K3 曲面 $X$ 就被称为具有良好的还原性。假定如此,如果 $X$ 允许一个适当的光滑模型,并配有 $G$ 作用(与 $X$ 上的作用相容),我们就说 $operatorname{Aut}(X)$ 的子群 $G$ 是可扩展的。我们证明,如果 $G$ 是 $p$ 的有限阶素数,并且交义作用(即保留 $X$ 上的全局 $2$-形式),那么 $G$ 是可扩展的。证明依赖于 K3 曲面模型的双向几何以及某些奇点的等变同步解析。我们还给出了一些不可扩展作用的例子。
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来源期刊
CiteScore
1.40
自引率
0.00%
发文量
9
审稿时长
6.0 months
期刊介绍: Dedicated to publication of complete and important papers of original research in all areas of mathematics. Expository papers and research announcements of exceptional interest are also occasionally published. High standards are applied in evaluating submissions; the entire editorial board must approve the acceptance of any paper.
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