On the essential dimension of infinitesimal group schemes

Abstract

We discuss essential dimension of group schemes, with particular attention to infinitesimal group schemes. We prove that the essential dimension of a group scheme of finite type over a field k is at least equal to the difference between the dimension of its Lie algebra and its dimension. Furthermore, we show that the essential dimension of a trigonalizable group scheme of length p^{n} over a field of characteristic p>0 is at most n. We give several examples.

11 pages; proof of Theorem 1.2 slightly changed; improved the exposition in section 4 and added Proposition 4.3. Accepted for publication in The American Journal of Mathematics