Demazure Models of Algebraic Tori in Dimension 4

Source abstract

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.