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Toric singularities. (English) Zbl 0832.14002
The aim of the paper is to give a definition of toric singularity in the most general context without referrings neither to the ambient variety of toroidal embeddings nor to a base field or scheme. At first the author recalls the notion of logarithmic structure on a scheme due to Fontaine and Illusie [see the author in: Algebraic Analysis, Geometry, and Number Theory, Proc. JAMI Inaugur. Conf., Baltimore 1988, 191-224; Zbl 0776.14004)]. He then defines logarithmically regular points (or equivalently, toric singularities) on a scheme with logarithmic structure. For example, the “Jungian domain” from S. S. Abhyankar [Wiss. Abh. Arbeitsgemeinschaft Nordrhein-Westfalen 33, Festschr. Gedächtnisfeier K. Weierstraß, 243-317 (1966; Zbl 0144.031)] is a toric singularity in the author’s sense. The rest of the paper is devoted to the extension of basic properties of classical regularity and standard results in the modern theory of toroidal embeddings to the case of logarithmically regular schemes.

MSC:
14B05 Singularities in algebraic geometry
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
14J20 Arithmetic ground fields for surfaces or higher-dimensional varieties
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