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A boundedness theorem for morphisms between threefolds

Ekatarina Amerik, Marat Rovinsky, Antonius Van de Ven (1999)

Annales de l'institut Fourier

The main result of this paper is as follows: let X , Y be smooth projective threefolds (over a field of characteristic zero) such that b 2 ( X ) = b 2 ( Y ) = 1 . If Y is not a projective space, then the degree of a morphism f : X Y is bounded in terms of discrete invariants of X and Y . Moreover, suppose that X and Y are smooth projective n -dimensional with cyclic Néron-Severi groups. If c 1 ( Y ) = 0 , then the degree of f is bounded iff Y is not a flat variety. In particular, to prove our main theorem we show the non-existence of a flat 3-dimensional...

A simpler proof of toroidalization of morphisms from 3-folds to surfaces

Steven Dale Cutkosky (2013)

Annales de l’institut Fourier

We give a simpler and more conceptual proof of toroidalization of morphisms of 3-folds to surfaces, over an algebraically closed field of characteristic zero. A toroidalization is obtained by performing sequences of blow ups of nonsingular subvarieties above the domain and range, to make a morphism toroidal. The original proof of toroidalization of morphisms of 3-folds to surfaces is much more complicated.

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