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The theory of regular log scheme, roughly specking is that scheme theory with singularities like those of toric varieties.

Dose anyone could explain this idea in detail or give a reference about the relations between log schemes and toric varieties?

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The source text

Kato, Kazuya Logarithmic structures of Fontaine-Illusie. Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), 191–224, Johns Hopkins Univ. Press, Baltimore, MD, 1989.

available here

Example (3.7) (1) the bottom line is

"... essentially, the theory of toroidal embeddings is nothing but the theory of schemes with smooth fine log. str.'s over a field (with respect to the trivial log. str. on the base field)."

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