# On Nash Blowing-Up

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

## Abstract

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.

## Keywords

Maximal Ideal Algebraic Variety Free Base Valuation Ring Residue Field
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

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).
3. [3]
Zariski, O., "Some open questions in the theory of singularities," Bull. Am. Math. Soc. 77 pp. 481 - 491 (1971).