On the essential dimension of infinitesimal group schemes

TL;DR

This paper investigates the essential dimension of infinitesimal group schemes, establishing bounds related to their Lie algebra dimensions and providing specific results for trigonalizable schemes in characteristic p.

Contribution

It introduces bounds on the essential dimension of infinitesimal group schemes and explores specific cases like trigonalizable schemes in characteristic p.

Findings

Essential dimension is at least the difference between Lie algebra dimension and scheme dimension.

For trigonalizable schemes of length p^n, the essential dimension is at most n.

Several examples illustrate the bounds and properties of these 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.