Tohoku Mathematical Journal
2003
September
SECOND SERIES VOL. 55, NO. 3

Tohoku Math. J.
55 (2003), 397-438

Title UNIVERSAL LOG STRUCTURES ON SEMI-STABLE VARIETIES

Author Martin C. Olsson

(Received August 10, 2001, revised September 18, 2002)
Abstract. Given a morphism of schemes which is flat, proper, and "fiber-by-fiber semi-stable", we study the problem of extending the morphism to a morphism of fine log schemes, which is log smooth, integral, and vertical. The problem is rephrased in terms of a functor on the category of fine log schemes over the base, and the main result of the paper is that this functor is representable by a fine log scheme whose underlying scheme maps naturally to the base by a monomorphism of finite type. In the course of the proof, we also generalize results of Kato on the existence of log structures of embedding and semi-stable type.

2000 Mathematics Subject Classification. Primary 14M25; Secondary 14D22, 14H10.
Key words and phrases. Logarithmic structures, semi-stable schemes, moduli spaces.

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