Nagata's compactification theorem for normal toric varieties over a valuation ring of rank one

TL;DR

This paper extends Nagata's compactification theorem to normal toric varieties over valuation rings of rank one, showing they can be embedded into proper toric varieties, generalizing Sumihiro's result from fields to valuation rings.

Contribution

It proves a Nagata-type compactification theorem for normal toric varieties over valuation rings of rank one, using invariant Zariski-Riemann spaces.

Findings

Normal toric varieties over valuation rings of rank one can be embedded into proper toric varieties.

Extension of Sumihiro's theorem from fields to valuation rings.

Use of invariant Zariski-Riemann spaces in the proof.

Abstract

We prove, using invariant Zariski-Riemann spaces, 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.