平面克雷莫纳群的\(p\)-元非循环子群
$p$-elementary non-cyclic subgroups of the Cremona group of the plane
浏览论文内容
中文总结 AI 辅助
研究平面克雷莫纳群中同构于\((\mathbb{Z}/p\mathbb{Z})^r\)（\(p\)素数，\(r \geq 2\)）的子群，在特定代数闭域上进行共轭分类，给出代表列表和共轭性研究，还给出特征等于\(p\)时的相关结果。
中文摘要 AI 辅助
我们对平面克雷莫纳群中同构于\((\mathbb{Z}/p\mathbb{Z})^r\)（\(p\)为素数且\(r \geq 2\)）在特征不等于\(p\)的代数闭域\(\mathbf{k}\)上的子群进行共轭分类。特别地，证明了若\(p \geq 5\)，则\(r \leq 2\)；若\(p = 3\)，则\(r \leq 3\)；若\(p = 2\)，则\(r \leq 4\)。通过由德容奎尔群子群和德尔佩佐曲面自同构子群组成的20个族给出明确的代表列表，并研究这些族之间通过双有理映射的可能共轭性。最后给出了特征等于\(p\)的代数闭域\(\mathbf{k}\)上平面克雷莫纳群中同构于\((\mathbb{Z}/p\mathbb{Z})^r\)（\(p\)为素数且\(r\)为整数）的子群的一些结果。
英文摘要
We classify, up to conjugacy, the subgroups of the Cremona group of the plane isomorphic to $(\mathbb{Z}/p\mathbb{Z})^r$, where $p$ is prime and $r \geq 2$, over an algebraically closed field $\mathbf{k}$ of characteristic not equal to $p$. In particular, we show that $r \leq 2$ if $p \geq 5$, $r \leq 3$ if $p=3$, and $r \leq 4$ if $p=2$. And hence reprove a well-known statement contained in the article of Arnaud Beauville "$p$-elementary subgroups of the Cremona group", 2007. The main contribution of this article is a concrete description of these groups. In fact, we give an explicit list of representatives via a set of 20 families consisting of subgroups of the de Jonquières group and subgroups of automorphisms of del Pezzo surfaces, and we study the possible conjugacies by birational maps between these families. Finally, we give some results on subgroups of the Cremona group of the plane isomorphic to $(\mathbb{Z}/p\mathbb{Z})^r$, where $p$ is prime and $r$ is an integer, over an algebraically closed field $\mathbf{k}$ of characteristic equal to $p$.
发表机构
- Universität Basel（巴塞尔大学）
机构由 AI 辅助整理，请以论文原文为准。
