A numerical criterion for simultaneous normalization

Hung-Jen Chiang-Hsieh, Purdue University

Abstract

Let R be a Noetherian local ring. For a reduced R-algebra, one can define an invariant δ R(A) on it. We study the properties of δ and get the following result: Theorem If ( R, A, &phis;) satisfies condition (♠) and kR is algebraically closed, then δ* : Spec (R) → [special characters omitted] is constant if and only if &phis; is a normal map. For a morphism f : X → Y of two reduced complex spaces X and Y , if the fibres are reduced curves, one can define the analogous invariant δ : Y → [special characters omitted] on Y. We apply the above algebraic result to the analytic case and give another proof to the Teissier's Theorem. Theorem If (X, Y, f) satisfies condition (♠′), then the following are equivalent. (i) f admits a weak simultaneous resolution. (ii) X¯y is reduced and δ(Xy) = δ([special characters omitted]) for any y ∈ Y. Moreover, the simultaneous resolution, if it exists, is necessary to be the normalization n : X¯ → X of X.

Degree

Ph.D.

Advisors

Lipman, Purdue University.

Subject Area

Mathematics

COinS