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Local monomialization and factorization of morphisms. (English) Zbl 0941.14001
Astérisque. 260. Paris: Société Mathématique de France, 149 p. (1999).
The book provides a contribution to the local study of birational maps in characteristic $$0$$. For any birational map $$\phi: X\to Y$$ of complete non-singular varieties one looks for a description of $$\phi$$ by means of blow-ups along non-singular subvarieties. There are two main questions:
(1) Show the existence of a chain of birational maps $$X=X_0 \to X_1\to \dots \to X_n=Y$$ which factorizes $$\phi$$, such that each map $$X_i\to X_{i+1}$$ is either a blow-up or its inverse. This is the ”weak factorization problem”, whose solution has been recently announced by Abramovich-Karu-Matsuki-Włodarczyk.
(2) Show the existence of a factorization of $$\phi$$ by means of two chains of birational maps $X=X_0\leftarrow X_1 \leftarrow \dots \leftarrow Z\to \dots\to Y_1\to Y_0=Y$ where all the maps $$X_{i+1}\to X_i$$ and $$Y_{i+1}\to Y_i$$ are blow-ups. This is the “strong factorization problem”, still open by now.
The author takes a local point of view. He considers pairs of regular local rings $$R\subset S$$ essentially of finite type over a field $$k$$, having the same quotient field $$K$$ (char$$(K)=0$$). Then he shows a monomialization theorem for the inclusion $$R\subset S$$: For any valuation ring $$V$$ of $$K$$ which dominates $$R$$ and $$S$$ there are sequences of monoidal transforms $$R\to R'$$ and $$S\to S'$$ such that $$V$$ dominates $$S'$$, $$S'$$ dominates $$R'$$ and a set of regular parameters of $$R'$$ is given by monomials in a set of regular parameters for $$S'$$.
Using monomial reduction and going back to schemes, the author proves a local version of the strong factorization problem:
For any birational morphism of proper non-singular varieties $$\phi:X\to Y$$ there exists a non-singular complete variety $$Z$$ and birational morphisms $$X\leftarrow Z\to Y$$ which are locally a product of monoidal transforms and factor $$\phi$$.

##### MSC:
 14E05 Rational and birational maps 13B10 Morphisms of commutative rings 14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry 13-02 Research exposition (monographs, survey articles) pertaining to commutative algebra 13A18 Valuations and their generalizations for commutative rings
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