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.


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)
Full Text: DOI arXiv


[1] T. de Fernex, Three-dimensional counter-examples to the Nash problem, 2012. · Zbl 1285.14013
[2] J. Denef and F. Loeser, ”Germs of arcs on singular algebraic varieties and motivic integration,” Invent. Math., vol. 135, iss. 1, pp. 201-232, 1999. · Zbl 0928.14004
[3] J. Fernández de Bobadilla, ”Relative morsification theory,” Topology, vol. 43, iss. 4, pp. 925-982, 2004. · Zbl 1052.32025
[4] J. Fernández de Bobadilla, ”Nash problem for surface singularities is a topological problem,” Adv. Math., vol. 230, iss. 1, pp. 131-176, 2012. · Zbl 1248.14004
[5] J. Fernández de Bobadilla and M. Pe Pereira, Curve Selection Lemma in infinite dimensional algebraic geometry and arc spaces, 2012.
[6] P. D. González Pérez, ”Bijectiveness of the Nash map for quasi-ordinary hypersurface singularities,” Int. Math. Res. Not., vol. 2007, iss. 19, p. 19. · Zbl 1129.14004
[7] H. Grauert and R. Remmert, Coherent Analytic Sheaves, New York: Springer-Verlag, 1984, vol. 265. · Zbl 0537.32001
[8] S. Ishii and J. Kollár, ”The Nash problem on arc families of singularities,” Duke Math. J., vol. 120, iss. 3, pp. 601-620, 2003. · Zbl 1052.14011
[9] S. Ishii, ”The arc space of a toric variety,” J. Algebra, vol. 278, iss. 2, pp. 666-683, 2004. · Zbl 1073.14066
[10] S. Ishii, ”Arcs, valuations and the Nash map,” J. Reine Angew. Math., vol. 588, pp. 71-92, 2005. · Zbl 1082.14007
[11] S. Ishii, ”The local Nash problem on arc families of singularities,” Ann. Inst. Fourier \((\)Grenoble\()\), vol. 56, iss. 4, pp. 1207-1224, 2006. · Zbl 1116.14030
[12] J. Kollár, Arc spaces of \(cA_1\) singularities, 2012.
[13] M. Lejeune-Jalabert, ”Arcs analytiques et résolution minimale des singularités des surfaces quasi-homogenes,” in Séminaire sur les Singularités des Surfaces, New York: Springer-Verlag, 1980, vol. 777, pp. 303-336. · Zbl 0432.14020
[14] M. Lejeune-Jalabert and A. J. Reguera-López, ”Arcs and wedges on sandwiched surface singularities,” Amer. J. Math., vol. 121, iss. 6, pp. 1191-1213, 1999. · Zbl 0960.14015
[15] M. Lejeune-Jalabert and A. J. Reguera-López, Exceptional divisors which are not uniruled belong to the image of the Nash map, 2008. · Zbl 1244.14003
[16] A. Melle-Hernández, ”Euler characteristic of the Milnor fibre of plane singularities,” Proc. Amer. Math. Soc., vol. 127, iss. 9, pp. 2653-2655, 1999. · Zbl 0936.32013
[17] J. Milnor, Singular Points of Complex Hypersurfaces, Princeton, N.J.: Princeton Univ. Press, 1968, vol. 61. · Zbl 0184.48405
[18] M. Morales, ”Some numerical criteria for the Nash problem on arcs for surfaces,” Nagoya Math. J., vol. 191, pp. 1-19, 2008. · Zbl 1178.14004
[19] J. F. Nash Jr., ”Arc structure of singularities. A celebration of John F. Nash, Jr.,” Duke Math. J., vol. 81, iss. 1, pp. 31-38 (1996), 1995. · Zbl 0880.14010
[20] M. Pe Pereira, Nash problem for quotient surface singularities, 2010.
[21] P. Petrov, ”Nash problem for stable toric varieties,” Math. Nachr., vol. 282, iss. 11, pp. 1575-1583, 2009. · Zbl 1182.14008
[22] C. Plénat, ”À propos du problème des arcs de Nash,” Ann. Inst. Fourier \((\)Grenoble\()\), vol. 55, iss. 3, pp. 805-823, 2005. · Zbl 1080.14021
[23] C. Plénat, ”The Nash problem of arcs and the rational double points \(D_n\),” Ann. Inst. Fourier \((\)Grenoble\()\), vol. 58, iss. 7, pp. 2249-2278, 2008. · Zbl 1168.14004
[24] C. Plénat and P. Popescu-Pampu, ”A class of non-rational surface singularities with bijective Nash map,” Bull. Soc. Math. France, vol. 134, iss. 3, pp. 383-394, 2006. · Zbl 1119.14007
[25] C. Plénat and P. Popescu-Pampu, ”Families of higher dimensional germs with bijective Nash map,” Kodai Math. J., vol. 31, iss. 2, pp. 199-218, 2008. · Zbl 1210.14008
[26] C. Plénat and M. Spivakovsky, The Nash problem of arcs and the rational double point \(E_6\), 2010. · Zbl 1271.14006
[27] A. -J. Reguera, ”Families of arcs on rational surface singularities,” Manuscripta Math., vol. 88, iss. 3, pp. 321-333, 1995. · Zbl 0867.14012
[28] A. J. Reguera, ”Image of the Nash map in terms of wedges,” C. R. Math. Acad. Sci. Paris, vol. 338, iss. 5, pp. 385-390, 2004. · Zbl 1044.14032
[29] A. J. Reguera, ”A curve selection lemma in spaces of arcs and the image of the Nash map,” Compos. Math., vol. 142, iss. 1, pp. 119-130, 2006. · Zbl 1118.14004
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