An axiomatic version of Zariski%26apos;s patching theorem

作者：Piltant Olivier^*

来源：Revista de la Real Academia de Ciencias Exactas Fisicas y Naturales Serie A-Matematicas, 2013, 107(1): 91-121.

DOI：10.1007/s13398-012-0090-6

• Valuations
• Local uniformization
• Resolution of singularities
• Zariski
• Hironaka

We state six axioms concerning any regularity property P in a given birational equivalence class of algebraic threefolds. Axiom 5 states the existence of a Local Uniformization in the sense of valuations for P. If Axioms 1 to 5 are satisfied by P, then the function field has a projective model which is everywhere regular with respect to P. Adding Axiom 6 ensures the existence of a P-Resolution of Singularities for any projective model. Applications concern Resolution of Singularities of vector fields and a weak version of Hironaka%26apos;s Strong Factorization Conjecture for birational morphisms of nonsingular projective threefolds, both of them in characteristic zero. The last section contains open problems about axiomatizing regularity properties which have P-Resolution of Singularities.

• 出版日期2013-3