Existence of global N\'eron models beyond semi-abelian varieties

TL;DR

This paper proves the existence of global Néron models for certain algebraic groups in perfect residue fields, provides counterexamples in imperfect fields, and classifies specific unipotent groups using duality theory.

Contribution

It confirms conjectures on Néron models in perfect fields, offers counterexamples in imperfect fields, and classifies unipotent groups up to relative perfection.

Findings

Proved conjectures for perfect residue fields

Counterexample in imperfect residue fields

Classified unirational wound unipotent groups

Abstract

We first prove Bosch-L\"utkebohmert-Raynaud's conjectures on existence of global N\'eron models of not necessarily semi-abelian algebraic groups in the perfect residue fields case. We then give a counterexample to the existence in the imperfect residue fields case. Finally, as a complement to the conjectures, we classify unirational wound unipotent groups "up to relative perfection", again in the perfect residue fields case. The key ingredient for all these is the duality for relatively perfect unipotent groups.