Toric Models of Algebraic Tori

For a field k, algebraic k-tori are classified up to isomorphism by integral representations of the absolute Galois group over k. Work of Demazure, Voskresenskii and Klyachko, among others, established, in principle, the existence of a smooth projective model of an algebraic k-torus, as the quotient of a smooth projective toric variety by the action of the splitting group of the algebraic torus.  Kunyavskii’s birational classification of algebraic k-tori in dimension 3 involved constructions of the smooth projective toric models of the algebraic tori corresponding to maximal finite subgroups of GL(3,Z). We discuss some explicit constructions of toric models of low-dimensional algebraic tori, making connections with the defining integral representations of their splitting groups.