Sandwiched singularities and desingularization of surfaces by normalized Nash transformations.

*(English)*Zbl 0719.14005For complex algebraic surfaces, this paper provides the final step in the proof of the conjecture (due to John Nash, about 30 years ago) that the singularities of an algebraic variety can be resolved by a finite succession of proper birational maps \(\mu\) : \(\bar S\to S\), \(\mu^*\Omega_{S/k}/\)torsion locally free of rank dim(S) and \(\mu\) universal with respect to that property (so called “Nash transformations”). It is known by A. Nobile [Pac. J. Math. 60, 297-305 (1975; Zbl 0324.32012)] that (in characteristic 0) \(\mu\) is an isomorphism iff S is nonsingular, and in particular, for plane curve singularities the conjecture is true. G. GonzĂˇlez-Sprinberg [Ann. Inst. Fourier 32, No.2, 111-178 (1982; Zbl 0469.14019)] proved for complex surfaces, that normalized Nash transformations resolve rational double points and cyclic quotient singularities. By a theorem of H. Hironaka, any surface singularity is transformed by a finite succession of Nash transformations into “sandwiched singularities”, i.e. singularities of a surface which birationally dominates a nonsingular surface.

The main theorem of the article under review states that sandwiched singularities are resolved by normalized Nash transformations, thus completing the proof of the above mentioned conjecture for \(\dim(S)=2\), \(k={\mathbb{C}}.\)

Chapter II of the paper gives a classification of sandwiched singularities, a problem which is shown to be equivalent to the classification of plane curve singularities, complete ideals in 2- dimensional regular local rings or valuations with center in a regular 2- dimensional local ring [partially, this is an overview of results of the same author, cf. Am. J. Math. 112, No.1, 107-156 (1990; Zbl 0716.13003)].

In chapter III, the main theorem is proved in two steps: First of all, minimal singularities are considered, i.e. here: rational surface singularities with reduced fundamental cycle. If \((S,\xi)\) is such a singularity, \(\Gamma\) its dual graph and \(\mu:S'\to S\) the normalized Nash transformation, \(\Gamma '_ i\) the dual graphs of the singularities of \(S'\), then \(\#\{\)vertices of \(\Gamma'_ i\}\leq \#\{\)vertices of \(\Gamma\},\) i.e. such procedure terminates after finitely many steps. The proof is completed by showing that the problem can be reduced to minimal singularities, i.e. any sandwiched singularity is transformed into a minimal one after finitely many normalized Nash transformations.

The main theorem of the article under review states that sandwiched singularities are resolved by normalized Nash transformations, thus completing the proof of the above mentioned conjecture for \(\dim(S)=2\), \(k={\mathbb{C}}.\)

Chapter II of the paper gives a classification of sandwiched singularities, a problem which is shown to be equivalent to the classification of plane curve singularities, complete ideals in 2- dimensional regular local rings or valuations with center in a regular 2- dimensional local ring [partially, this is an overview of results of the same author, cf. Am. J. Math. 112, No.1, 107-156 (1990; Zbl 0716.13003)].

In chapter III, the main theorem is proved in two steps: First of all, minimal singularities are considered, i.e. here: rational surface singularities with reduced fundamental cycle. If \((S,\xi)\) is such a singularity, \(\Gamma\) its dual graph and \(\mu:S'\to S\) the normalized Nash transformation, \(\Gamma '_ i\) the dual graphs of the singularities of \(S'\), then \(\#\{\)vertices of \(\Gamma'_ i\}\leq \#\{\)vertices of \(\Gamma\},\) i.e. such procedure terminates after finitely many steps. The proof is completed by showing that the problem can be reduced to minimal singularities, i.e. any sandwiched singularity is transformed into a minimal one after finitely many normalized Nash transformations.

Reviewer: M.Roczen (Berlin)

##### MSC:

14E15 | Global theory and resolution of singularities (algebro-geometric aspects) |

32S45 | Modifications; resolution of singularities (complex-analytic aspects) |

14B05 | Singularities in algebraic geometry |

14J17 | Singularities of surfaces or higher-dimensional varieties |

14H20 | Singularities of curves, local rings |