Toroidal embedding of Chevalley groups over $\mathbb{Z}$

TL;DR

This paper proves the existence of universal equivariant toroidal embeddings for split reductive group schemes over integers, extending classical embeddings and exploring their geometric properties.

Contribution

It establishes the existence of universal toroidal embeddings for Chevalley group schemes over , generalizing classical results to a base scheme over .

Findings

Existence of universal equivariant toroidal embeddings over 

These embeddings specialize to classical embeddings over fields

Discussion of geometric properties of the embeddings

Abstract

The classification of equivariant toroidal embeddings of a reductive group over an algebraically closed field is combinatorial and does not depend on the characteristic of the base field. This suggests that there should exist ``universal'' toroidal embeddings for a Chevalley group scheme over $\mathbb{Z}$ which specialize to classical toroidal embeddings via base change. In this paper, we establish the existence of ``universal'' equivariant toroidal embeddings for split reductive group schemes over $\mathbb{Z}$. We also discuss several geometric properties of these embeddings.