Stable equivalence of algebraic tori

We show that the Picard modules $\operatorname{Pic}V_L(T)$ of projective models of algebraic tori $T$, computed over a field of characteristic zero, also determine birational invariants over a field of positive characteristic. We study the semigroup of classes of stably equivalent tori, and in particular prove that, for tori that are split over a cyclic extension, this is a group.