The Nash problem for surfaces.(English)Zbl 1264.14049

Let consider a surface $$S$$ with an isolated singular point $$O$$ over a closed field of characteristic 0. J. F. Nash, jun. proved in the sixties (see [Duke Math. J. 81, No.1, 31–38 (1995; Zbl 0880.14010)]) that the space of arcs passing trough $$O$$ has a finite number of irreducible components, and he raised the question to know if this number coincides with the number of exceptional components of the minimal resolution of singularities of $$S$$. This problem was considered to be difficult but recently there has been a lot of progress, and many partial answers were given by several authors (including the reviewer).
In the paper under review, the authors solve completely this problem. Their proof uses many tools developed by people trying to prove the Nash problem and some new ideas developed recently by the authors. One main point is the translation in terms of wedges by Monique Lejeune-Jalabert, and recent work done by the two authors [J. Fernández de Bobadilla, Adv. Math. 230, No. 1, 131–176 (2012; Zbl 1248.14004); M. Pe Pereira, J. Lond. Math. Soc., II. Ser. 87, No. 1, 177–203 (2013; Zbl 1272.32027)]. The proof is by contradiction, if the Nash’s problem is not true for $$S$$, there exists a convergent wedge $$\alpha$$ with precise properties, $$\alpha$$ can be viewed as a family of mappings $$\alpha_s :{\mathcal U}_s\rightarrow (S,O)$$, so is natural to consider the lifts $$\tilde\alpha_s :{\mathcal U}_s\rightarrow \tilde S$$, where $$\tilde S$$ is the minimal resolution of singularities of $$S$$. Their images $$Y_s\subset \tilde S$$ converge to a limit divisor $$Y_0$$, then by some computations on the Euler characteristics of $$Y_0$$ they prove that this leads to a contradiction which finishes the proof.

MSC:

 14J17 Singularities of surfaces or higher-dimensional varieties 32S45 Modifications; resolution of singularities (complex-analytic aspects) 14B05 Singularities in algebraic geometry 14E15 Global theory and resolution of singularities (algebro-geometric aspects)
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