A numerical criterion for simultaneous normalization

Hung-Jen Chiang-Hsieh; Joseph Lipman

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A numerical criterion for simultaneous normalization

Hung-Jen Chiang-Hsieh·Joseph Lipman

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Abstract

We investigate conditions for "simultaneous normalizability" of a family of reduced schemes, i.e., the normalization of the total space normalizes, fiber by fiber, each member of the family. The main result (under more general conditions) is that a flat family of reduced equidimensional projective complex varieties X_y with parameter y ranging over a normal space--algebraic or analytic--admits a simultaneous normalization if and only if the Hilbert polynomial of the integral closure of the structure sheaf O_{X_y} is locally independent of y. When the X_y are curves projectivity is not needed, and the statement reduces to the well known δ-constant criterion of Teissier. Proofs are basically algebraic, analytic results being related via standard techniques (Stein compacta, etc.) to more abstract algebraic ones.

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Hung-Jen Chiang-Hsieh, Joseph Lipman. 2005-10-20. A numerical criterion for simultaneous normalization. https://arxiv.org/abs/math/0408394

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