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Universal log structures on semi-stable varieties. (English) Zbl 1069.14015
Summary: Given a morphism of schemes which is flat, proper, and ‘fiber-by-fiber semistable’, 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.

MSC:
 14D22 Fine and coarse moduli spaces 14H10 Families, moduli of curves (algebraic) 14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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