On the essential dimension of infinitesimal group schemes

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 greater than or 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.