Classification of Normal Toric Varieties over a Valuation Ring of Rank One

Normal toric varieties over a field or a discrete valuation ring are classified by rational polyhedral fans. We generalize this classification to normal toric varieties over an arbitrary valuation ring of rank one. The proof is based on a generalization of Sumihiro's theorem to this non-noetherian setting. These toric varieties play an important role for tropicalizations.

2010 Mathematics Subject Classification: 14M25, 14L30, 13F30

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