Normal completions of toric varieties over rank one valuation rings and completions of $\Gamma$admissible fans
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 basechange by a finite extension of valuation rings. If the value group $\Gamma$ is discrete or divisible then no basechange 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.
 Publication:

arXiv eprints
 Pub Date:
 July 2019
 arXiv:
 arXiv:1908.00064
 Bibcode:
 2019arXiv190800064F
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Combinatorics
 EPrint:
 23 pages, 3 figures. Comments welcome!