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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

http://www.math.brown.edu/~abrmovic/LOGGEOM/Kato-log.pdf.

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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