Compositio Mathematica

Research Article

Higher Nash blowups

Takehiko Yasudaa1

a1 Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan (email:


For each non-negative integer n we define the nth Nash blowup of an algebraic variety, and call them all higher Nash blowups. When n=1, it coincides with the classical Nash blowup. We study higher Nash blowups of curves in detail and prove that any curve in characteristic zero can be desingularized by its nth Nash blowup with n large enough. Moreover, we completely determine for which n the nth Nash blowup of an analytically irreducible curve singularity in characteristic zero is normal, in terms of the associated numerical monoid.

(Received October 07 2006)

(Accepted November 24 2006)

(Online publication October 01 2007)


  • Nash blowup;
  • resolution of singularities;
  • Hilbert scheme of points

2000 Mathematics subject classification

  • 14E15;
  • 14B12


Financial support has been provided by the Japan Society for the Promotion of Science.