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A simpler proof of toroidalization of morphisms from 3-folds to surfaces. (English. French summary) Zbl 1282.14029

Let \(k\) be an algebraically closed field of characteristic zero. Let \(\Phi :X\to Y\) be a dominant morphism of non-singular algebraic varieties over \(k\). Roughly speaking \(\Phi\) is called toroidal if and only if for each closed point \(x\in X\) the formal neighborhood of \(X\) at \(x\) and the formal neighborhood of \(Y\) at \(\Phi(x)\) are isomorphic to formal neighborhoods of toric varieties and the morphism induced by \(\Phi\) between these toric varieties is a toric morphism (i.e. this morphism is given by monomials in toric coordinates).
In this paper the author gives a new proof of the following result (proved previously in [S. D. Cutkosky, Monomialization of morphisms from 3-folds to surfaces. Berlin: Springer (2002; Zbl 1057.14009)]):
Let \(\Phi : X\to Y\) be a dominant morphism from a threefold to a surface. Then there exist \(\pi : X_1\to X\) and \(\rho : Y_1\to Y\), compositions of blow-ups with non-singular centers, and a toroidal morphism \(f :X_1\to Y_1\) such that \(\Phi\circ\pi=\rho\circ f\).
The proof of Cutkosky given in [loc. cit.] consists of two steps: the first one consists to replace \(\Phi\) by a “strongly prepared” morphism, the second one consists to prove that strongly prepared morphism can be toroidalized. The aim of the paper under review is to give a shorter proof of this first step (it takes 170 pages in [loc. cit.]) with the hope that this one may be generalized to higher dimension (the second step is generalized to higher dimension in [S. D. Cutkosky and O. Kashcheyeva, J. Algebra 275, No. 1, 275–320 (2004; Zbl 1057.14008)]).

MSC:

14E99 Birational geometry
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
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References:

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