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An axiomatic version of Zariski’s patching theorem

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Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas Aims and scope Submit manuscript

Abstract

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’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.

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Authors and Affiliations

  1. CNRS and LMV-UMR 8100, University of Versailles, LMV-Bât. Fermat, 45 av. des Etats-Unis, 78035, Versailles Cedex, France

    Olivier Piltant

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  1. Olivier Piltant
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Correspondence to Olivier Piltant.

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Dedicated to Heisuke Hironaka on the occasion of his 80th birthday

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Piltant, O. An axiomatic version of Zariski’s patching theorem. RACSAM 107, 91–121 (2013). https://doi.org/10.1007/s13398-012-0090-6

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  • DOI: https://doi.org/10.1007/s13398-012-0090-6

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