Nagata’s Compactification Theorem for Normal Toric Varieties over a Valuation Ring of Rank One

Abstract

Using invariant Zariski–Riemann spaces, we prove that every normal toric variety over a valuation ring of rank one can be embedded as an open dense subset into a proper toric variety equivariantly. This extends a well-known theorem of Sumihiro for toric varieties over a field to this more general setting.