## Monomialization of morphisms from 3-folds to surfaces.(English)Zbl 1057.14009

Lecture Notes in Mathematics 1786. Berlin: Springer (ISBN 3-540-43780-0/pbk). v, 235 p. (2002).
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Let us say that a dominant morphism $$\Phi: X\to Y$$ of nonsingular $$k$$-varieties (where $$k$$ is a field of characteristic zero) is monomial if for any point $$p\in X$$ there exists an étale neighborhood $$U$$ in $$X$$, uniformizing parameters $$(x_1,\dots,x_n)$$ on $$U$$, and regular parameters $$(y_1,\dots,y_m)$$ in the local ring of $$\Phi(p)$$, such that $$\Phi^*(y_j)$$ are monomials in $$x_i$$. Clearly, monomial morphisms are easier to analyze then morphisms in general. On the other hand, most morphisms $$\Phi:X\to Y$$ are not monomial.
These reasons stimulate the authors to seek to monomialize a morphism. By definition, a monomialization of a dominant morphism $$\Phi:X\to Y$$ of $$k$$-varieties is a commutative diagram $\begin{tikzcd} X_1 \ar[r,"\Psi"]\ar[d] & Y_1\ar[d] \\ X \ar[r,"\Phi" ']& Y\rlap{\,,} \end{tikzcd}$ where $$X_1, Y_1$$ are nonsingular, $$\Psi$$ is a monomial morphism and vertical arrows are sequences of blow-ups of non-singular subvarieties.
The existence of a monomialization of a morphisms between two surfaces was proved by S. D. Cutkosky and O. Piltant [Commun. Algebra 28, 5935–5959 (2000; Zbl 1003.14004)].
The main aim of the present lecture notes is to give a systematic development of the theory of monomialization of morphisms between algebraic varieties and to prove the existence of a monomialization for a dominant morphism from a 3-fold to a surface.

### MSC:

 14B25 Local structure of morphisms in algebraic geometry: étale, flat, etc. 14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry 14D06 Fibrations, degenerations in algebraic geometry 12E10 Special polynomials in general fields 14E15 Global theory and resolution of singularities (algebro-geometric aspects) 14J30 $$3$$-folds

### Keywords:

monomial morphisms; étale neighborhoods; proper morphisms

Zbl 1003.14004
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