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Chiang-Hsieh, Hung-Jen; Lipman, Joseph

A numerical criterion for simultaneous normalization. (English) Zbl 1101.14004

Duke Math. J. 133, No. 2, 347-390 (2006).
Let \(f:X\to Y\) be a scheme map, flat with all fibres geometrically reduced. A simultaneous normalization of \(f\) is a finite map \(v:Z\to X\) such that \(\bar{f}:=f\circ v\) is flat with all fibres geometrically normal and such that for each \(y\in f(X)\) the induced map of fibres \(v_y:\bar{f}^{-1}(y)\to f^{-1} (y)\) is a normalization map.
The main result is that a flat family \(f:X\to Y\) of reduced equidimensional projective \(\mathbb C\)-varieties with normal parameter space admits a simultaneous normalization if and only if the Hilbert polynomial of the integral closure \(\overline{\mathcal O}_{f^{-1}(y)}\) is locally independent of \(y\). In case the fibres are curves projectivity is not needed and the statement reduces to the \(\delta\)-constant criterion of Teissier.
Reviewer: Gerhard Pfister (Kaiserslautern)

Cited in 10 Documents

MSC:

14B05 Singularities in algebraic geometry
32S15 Equisingularity (topological and analytic)

Keywords:

simultaneous normalization; families of schemes
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\textit{H.-J. Chiang-Hsieh} and \textit{J. Lipman}, Duke Math. J. 133, No. 2, 347--390 (2006; Zbl 1101.14004)
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