## Four-Dimensional Algebraic Tori. (arXiv:1511.00315v3 [math.AG] UPDATED)

The study of the birational properties of algebraic $k$-tori began in the
sixties and seventies with work of Voskresenkii, Endo, Miyata,
Colliot-Th\'el\`ene and Sansuc. There was particular interest in determining
the rationality of a given algebraic $k$-tori. As rationality problems for
algebraic varieties are in general difficult, it is natural to consider relaxed
notions such as stable rationality, or even retract rationality. Work of the
above authors and later Saltman in the eighties determined necessary and
sufficient conditions to determine when an algebraic torus is stably rational,
respectively retract rational in terms of the integral representations of its
associated character lattice. An interesting question is to ask whether a
stably rational algebraic $k$-torus is always rational. In the general case,
there exist examples of non-rational stably rational $k$-varieties. Algebraic
$k$-tori of dimension $r$ are classified up to isomorphism by conjugacy classes
of finite subgr查看全文