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

Cutkosky, Steven Dale

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.

Reviewer: Ivan V. Arzhantsev (Moskva)

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

Citations:

Zbl 1003.14004