An equivariant compactification for adjoint reductive group schemes

TL;DR

This paper introduces a new intrinsic equivariant compactification for adjoint reductive group schemes over arbitrary bases, generalizing classical wonderful compactifications and enabling new applications in torsor studies.

Contribution

It constructs an equivariant compactification for adjoint reductive groups over arbitrary base schemes using a variant of the Artin-Weil method, providing a new intrinsic approach.

Findings

Parameterizes classical wonderful compactifications as fibers

Computes the Picard group scheme of the compactification

Discusses applications to torsors under reductive group schemes

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.