The Uniform Bound Problem for Local Birational Nonsingular Morphisms

Abstract

It is known that any factorization of a local birational morphism $f:\operatorname {Spec}\;S \to \operatorname {Spec}\;R$ of nonsingular (affine) schemes of arbitrary dimension via other nonsingular schemes must be finite in length. This fact generalizes the classical Local Factorization Theorem of Zariski and Abhyankar, which states that there is a unique such factorization, that given by quadratic transformations, in the surface case. A much stronger generalization is given here, namely, that there exists a uniform bound on the lengths of all such factorizations, provided that $R$ is excellent. This bound is explicitly calculated for some concrete extensions and examples are given to show that this is the strongest generalization possible in some sense.