Tori and essential dimension

The present paper deals with algebraic tori and essential dimension in three unrelated contexts. After some preliminaries on essential dimension, versal torsors and tori, we explicitly construct a versal torsor for PGLn, n⩾5 odd, defined over a field of transcendence degree over the base field. This recovers a result of Lorenz, Reichstein, Rowen and Saltman. We also discuss the so-called “tori method” which gives a geometric proof of a result of Ledet on the essential dimension of a cyclic p-group. In the last section we compute the essential dimension of the functor K↦H1(K,GLn(Z)), the latter set being in bijection with the isomorphism classes of n-dimensional K-tori.