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Published online by Cambridge University Press: 02 November 2018
Chapter III introduces log geometry per se, beginning with the definitions of log structures and log schemes and their charts.It discusses both étale and Zariski log structuresand describes basic examples, including log points and log dashes and the log schemes arising from divisors with normal crossings and semistable reduction.So-called “compactifying” log structures are especially important, and conditions are given for such structures to be coherent or relatively coherent.Also discussed are hollow, solid, and regular log structures. The second part of Chapter II is devoted to morphisms of log schemes.It begins with the construction of fiber products, which is delicate in the category of fine log schemes.Then it discusses special classes of morphisms, including exact, integral, and saturated morphisms, as well as immersions, small morphisms, inseparable morphisms, and Frobenius.It concludes with a definition and brief discussion of log blowups and their functorial properties.
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