For every fan supported in the negative Weyl chamber, universal equivariant toroidal embeddings of split reductive group schemes over Z exist and specialize to the classical embeddings over every algebraically closed field.
An equivariant compactification for adjoint reductive group schemes
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abstract
Wonderful compactifications of adjoint reductive groups over an algebraically closed field play an important role in algebraic geometry and representation theory. In this paper, we construct an equivariant compactification for adjoint reductive groups over arbitrary base schemes. Our compactifications parameterize classical wonderful compactifications of De Concini and Procesi as geometric fibers. Our construction is based on a variant of the Artin-Weil method of birational group laws. In particular, our construction gives a new intrinsic construction of wonderful compactifications. The Picard group scheme of our compactifications is computed. We also discuss several applications of our compactification in the study of torsors under reductive group schemes.
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Toroidal embedding of Chevalley groups over $\mathbb{Z}$
For every fan supported in the negative Weyl chamber, universal equivariant toroidal embeddings of split reductive group schemes over Z exist and specialize to the classical embeddings over every algebraically closed field.