On Nash Blowing-Up

  • Heisuke Hironaka
Part of the Progress in Mathematics book series (PM, volume 36)


Let X be an algebraic variety, reduced and equidimensional, over the base field k of characteristic zero. Let us consider a sequence of transformations
$$ {X_0} = X\xleftarrow{{{\sigma _1}}}{X_1}\xleftarrow{{{\sigma _2}}}{X_2} \leftarrow \cdots $$
where \( {\sigma _i}:{X_i} \to {X_{i - 1}} \) for each \( i \geqslant 1 \) is
  1. (1)

    birational, i.e., proper and almost everywhere isomorphic, while X i is reduced and equidimensional, and

  2. (2)

    \( \sigma _i^*\left( {{\Omega _{{X_{i - 1}}}}} \right) \) /(its torsion) is locally free as \( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{O} _{Xi}} \) -module. Here Ω denotes the sheaf of Kähler differentials on the variety and the torsion means the subsheaf consisting of those local sections whose supports are nowhere dense.



Nash NASH 


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  1. [1]
    Conzalez-Sprinberg, C., "Resolution de Nash des points doubles rationnels," Mimeographed Note, Centre de Math., Ecole Polytech., France, ( October, 1980 ).Google Scholar
  2. [2]
    Nobile, A., "Some properties of the Nash blowing-up," Pacific J. Math. 60, pp. 297 - 305 (1975).MathSciNetMATHGoogle Scholar
  3. [3]
    Zariski, O., "Some open questions in the theory of singularities," Bull. Am. Math. Soc. 77 pp. 481 - 491 (1971).MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1983

Authors and Affiliations

  • Heisuke Hironaka
    • 1
  1. 1.Department of MathematicsHarvard UniversityCambridgeUSA

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