Compactification of Reductive Group Schemes

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 Česnavičius (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.