Compactification of Reductive Group Schemes

TL;DR

This paper constructs a smooth projective scheme compactifying a reductive group scheme over a base, extending group actions and confirming a conjecture, with special cases and counterexamples included.

Contribution

It provides a new construction of compactifications for reductive group schemes over arbitrary bases, confirming a conjecture and extending known cases like the wonderful compactification.

Findings

Constructed a smooth projective scheme containing the group as a dense open subscheme.

Verified a conjecture of Cesnavicius regarding such compactifications.

Presented a non-isotrivial torus without an equivariant compactification.

Abstract

Let $\mathrm G$ be an isotrivial reductive group over a scheme $S$. We construct a smooth projective $S$-scheme containing $\mathrm G$ as a fiberwise-dense open subscheme equipped with left and right actions of $\mathrm G$ which extend the translation actions of $\mathrm G$ on itself. This verifies a conjecture of \v{C}esnavi\v{c}ius (arXiv:2201.06424). When $\mathrm G$ is adjoint, we recover fiberwise the wonderful compactification. Finally, we give an example of a non-isotrivial torus admitting no equivariant compactification.