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.



Maximal Ideal Algebraic Variety Free Base Valuation Ring Residue Field 
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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).MathSciNetzbMATHGoogle Scholar
  3. [3]
    Zariski, O., "Some open questions in the theory of singularities," Bull. Am. Math. Soc. 77 pp. 481 - 491 (1971).MathSciNetzbMATHCrossRefGoogle 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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