An axiomatic version of Zariski's patching theorem

Abstract

30 pagesWe 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 4 are satisfied by P, then the function field has a projective model which is everywhere regular w.r.t. P. Axiom 6 ensures the existence of P-resolution of singularities for any projective model. Applications concern resolution of singularities of vector fields and a weak version of Hironaka's Strong Factorization Conjecture for birational morphisms of nonsingular projective threefolds, both of them in characteristic zero