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 \(\phi\).

(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 \(\phi\).

Reviewer: L.Chiantini (Siena)

##### 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 |