Essential dimension of group schemes over a local scheme

TL;DR

This paper develops the theory of essential dimension for group schemes over local bases, explores invariance over fields, discusses cases over discrete valuation rings, and proposes a generalized Ledet conjecture for unipotent group schemes.

Contribution

It introduces a new framework for essential dimension over local schemes and generalizes the Ledet conjecture to finite commutative unipotent group schemes.

Findings

Invariance of essential dimension over fields established.

Analysis of group schemes over discrete valuation rings.

Proposed and discussed a generalized Ledet conjecture.

Abstract

In this paper we develop the theory of essential dimension of group schemes over an integral base. Shortly we concentrate over a local base. As a consequence of our theory we give a result of invariance of the essential dimension over a field. The case of group schemes over a discrete valuation ring is discussed. Moreover we propose a generalization of Ledet conjecture, which predicts the essential dimension of cyclic $p$-groups in positive characteristic, for finite commutative unipotent group schemes. And we show some results and some consequences of this new conjecture.