Local factorization and monomialization of morphisms

Abstract

Suppose that X to Y is a generically finite map of nonsingular varieties over a field of characteristic zero, and v is a valuation of the function field of X. We prove that it is possible to perform a sequence of monoidal transforms X' to X and Y' to Y so that X' to Y' is a monomial mapping at the center of v. We deduce from this that a birational morphism of nonsingular varieties can be factored along a valuation by a sequence of blowups and blowdowns with nonsingular centers.

145 pages In the revision a new chapter (chapter 6) has been added which proves a stronger local factorization Theorem. The introduction has also been modified. There are no other changes from the original March '98 submission

Cited by in corpus (13)

• Valuation Semigroups of Two Dimensional Local Rings
• The role of defect and splitting in finite generation of extensions of associated graded rings along a valuation
• Semigroups of valuations dominating local domains
• Torification and Factorization of Birational Maps
• Toroidal varieties and the weak Factorization Theorem
• Finite Generation of Extensions of Associated Graded Rings Along a Valuation
• Ramification of local rings along valuations
• A generalization of the Abhyankar Jung theorem to associated graded rings of valuations
• Toroidalization of generating sequences in dimension two function fields
• Generically Finite Morphisms
• Ramification of valuations and local rings
• Value groups, residue fields and bad places of rational function fields
• Simultaneous Resolution of Singularities