Kato, Kazuya Toric singularities. (English) Zbl 0832.14002 Am. J. Math. 116, No. 5, 1073-1099 (1994). 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. Reviewer: A.G.Aleksandrov (Moskva) Cited in 8 ReviewsCited in 112 Documents 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 Keywords:sheaves of monoids; resolution of singularities; Jungian domain; toric singularity; toroidal embeddings Citations:Zbl 0776.14004; Zbl 0144.031 × Cite Format Result Cite Review PDF Full Text: DOI