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arXiv · 1908.00064

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

Netanel Friedenberg
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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.

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Netanel Friedenberg. 2019-07-31. Normal completions of toric varieties over rank one valuation rings and completions of $\Gamma$-admissible fans. https://arxiv.org/abs/1908.00064

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