Normal Completions of Toric Varieties Over Rank One Valuation Rings and Completions of Γ-Admissible Fans

Abstract

Abstract We show that any normal toric variety over a rank one valuation ring admits an equivariant open embedding in a normal toric variety which is proper over the valuation ring, after a base-change by a finite extension of valuation rings. If the value group $\Gamma $ is discrete or divisible then no base-change is needed. We give explicit examples that show that existing methods do not produce such normal equivariant completions. Our approach is combinatorial and proceeds by showing that $\Gamma $-admissible fans admit $\Gamma $-admissible completions. In order to show this we prove a combinatorial analog of Noetherian reduction, which we believe will be of independent interest.