AI 中文总结
本文通过显式构造与分裂群积分表示相关的对称多胞体，给出了GL(4,Z)极大有限子群对应代数环面的光滑射影Demazure模型。
AI 中文摘要
一个代数$k$-环面$T$的Demazure模型，其中$T$被有限伽罗瓦域扩张$L/k$分裂，分裂群为$G=\text{Gal}(L/k)$，是一个光滑射影$T_L$-环面簇$X_{\Sigma}$，使得$G$在$T_L$上的自然作用扩展到$X_{\Sigma}$。代数$k$-环面$T$的光滑射影模型因此可以取为Demazure模型除以其分裂群的商。代数$k$-环面的Demazure模型的存在性由Brylinski确定，并由Colliot-Thélène、Harari和Skorobogatov进一步细化。Voskresenskii和Kunyavskii对维度2和3中代数$k$-环面的双有理分类涉及构造对应于$\text{GL}(r,\mathbb{Z})$（$r=2,3$）的极大有限子群的代数$k$-环面的Demazure模型。我们讨论代数$k$-环面的Demazure模型的一些显式构造，并将其与其分裂群的定义积分表示联系起来。这些构造确定了对应于$\text{GL}(4,\mathbb{Z})$的极大有限子群的代数环面的光滑射影Demazure模型。由于射影环面簇由格多胞体决定，我们的构造聚焦于一些高度对称的多胞体族，例如根多胞体和中心运输多胞体。
英文摘要
A Demazure model of an algebraic $k$-torus $T$ split by a finite Galois extension of fields $L/k$ and splitting group $G=\text{Gal}(L/k)$ is a smooth projective $T_L$-toric variety $X_Σ$ such that the natural action of $G$ on $T_L$ extends to $X_Σ$. A smooth projective model of the algebraic $k$-torus $T$ can then be taken to be the quotient of a Demazure model by its splitting group. The existence of Demazure models of algebraic $k$-tori was determined by Brylinski and further refined by Colliot-Thélène, Harari and Skorobogatov. Voskresenskii and Kunyavskii's birational classifications of algebraic $k$-tori in dimensions 2 and 3 involved constructions of Demazure models of the algebraic $k$-tori corresponding to maximal finite subgroups of $\text{GL}(r,\mathbb{Z})$ for $r=2,3$. We discuss some explicit constructions of Demazure models of algebraic $k$-tori, making connections with the defining integral representations of their splitting groups. The constructions determine smooth projective Demazure models of the algebraic tori corresponding to maximal finite subgroups of $\text{GL}(4,\mathbb{Z})$. Since projective toric varieties are determined by lattice polytopes, our constructions focus on some highly symmetric families of polytopes, such as root polytopes and central transportation polytopes.
