# Normal completions of toric varieties over rank one valuation rings and completions of $\Gamma$-admissible fans

## TL;DR

This paper proves that normal toric varieties over rank one valuation rings can be embedded into proper ones after finite base change, using combinatorial methods involving $ ext{Gamma}$-admissible fans and their completions.

## Contribution

It introduces a combinatorial approach to construct $ ext{Gamma}$-admissible completions of fans, enabling equivariant compactifications of toric varieties over valuation rings.

## Key findings

- Normal toric varieties admit equivariant open embeddings into proper ones after finite extension.
- Explicit examples show existing methods are insufficient for such completions.
- A combinatorial analog of noetherian reduction is developed, potentially of independent interest.

## 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 which 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.