This paper deals with the Nash problem, which consists in comparing the number of families of arcs on a singular germ of surface with the number of essential components of the exceptional divisor in the minimal resolution of this singularity. We prove their equality in the case of the rational double points ().
Dans cet article, on étudie le problème des arcs de Nash, qui consiste à comparer le nombre de composantes irréductibles de l’espace des arcs passant par une singularité isolée de surface normale avec les courbes exceptionnelles apparaissant dans la résolution minimale de cette singularité. On montre que les deux nombres sont égaux dans le cas des points doubles rationnels .
Keywords: Space of arcs, Nash map, Nash problem, rational double points
Mot clés : espace des arcs, problème de Nash, points doubles rationnels
@article{AIF_2008__58_7_2249_0, author = {Pl\'enat, Camille}, title = {The {Nash} problem of arcs and the rational double points $D_n$}, journal = {Annales de l'Institut Fourier}, pages = {2249--2278}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {58}, number = {7}, year = {2008}, doi = {10.5802/aif.2413}, zbl = {1168.14004}, mrnumber = {2498350}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.2413/} }
TY - JOUR AU - Plénat, Camille TI - The Nash problem of arcs and the rational double points $D_n$ JO - Annales de l'Institut Fourier PY - 2008 SP - 2249 EP - 2278 VL - 58 IS - 7 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.2413/ DO - 10.5802/aif.2413 LA - en ID - AIF_2008__58_7_2249_0 ER -
%0 Journal Article %A Plénat, Camille %T The Nash problem of arcs and the rational double points $D_n$ %J Annales de l'Institut Fourier %D 2008 %P 2249-2278 %V 58 %N 7 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.2413/ %R 10.5802/aif.2413 %G en %F AIF_2008__58_7_2249_0
Plénat, Camille. The Nash problem of arcs and the rational double points $D_n$. Annales de l'Institut Fourier, Volume 58 (2008) no. 7, pp. 2249-2278. doi : 10.5802/aif.2413. http://archive.numdam.org/articles/10.5802/aif.2413/
[1] On isolated rational singularities of surfaces, Amer. J. Math., Volume 88 (1966), pp. 129-136 | DOI | MR | Zbl
[2] Diviseurs essentiels, composantes essentielles des variétés toriques singulières, Duke Math. J., Volume 91 (1998), pp. 609-620 | DOI | MR | Zbl
[3] Système générateur minimal, diviseurs essentiels et G-désingularisations de varitétés toriques, Tohoku Math. J., Volume 47 (1995), pp. 125-149 | DOI | MR | Zbl
[4] Commutative Algebra with a view toward Algebraic Geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, New York, 1995 | MR | Zbl
[5] Equivalence of the Nash conjecture for primitive and sandwiched singularities, Proc. Amer. Math. Soc., Volume 133 (2005), pp. 677-679 | DOI | MR | Zbl
[6] Arcs, valuations and the Nash map, arXiv: math.AG/0410526 | Zbl
[7] The local Nash problem on arc families of singularities, arXiv: math.AG/0507530 | Numdam | Zbl
[8] The Nash problem on arc families of singularities, Duke Math. J., Volume 120, 3 (2003), pp. 601-620 | MR | Zbl
[9] Arcs analytiques et résolution minimale des singularités des surfaces quasi-homogènes, Séminaire sur les Singularités des Surfaces, Lecture Notes in Math., Volume 777, Springer-Verlag (1980), pp. 303-336 | Numdam | Zbl
[10] Désingularisation explicite des surfaces quasi-homogènes dans , Nova Acta Leopoldina, Volume NF 52, Nr 240 (1981), pp. 139-160 | MR | Zbl
[11] Courbes tracées sur un germe d’hypersurface, Amer. J. Math., Volume 112 (1990), pp. 525-568 | DOI | Zbl
[12] Arcs and wedges on sandwiched surface singularities, Amer. J. Math., Volume 121 (1999), pp. 1191-1213 | DOI | MR | Zbl
[13] Commutative ring theory. Translated from the Japanese by M. Reid, Cambridge Studies in Advanced Mathematics, 8, Cambridge University Press, Cambridge, 1986 | MR | Zbl
[14] Arc structure of singularities, A celebration of John F. Nash, Jr. Duke Math. J., Volume 81, 1 (1995), pp. 31-38 | DOI | MR | Zbl
[15] A propos du problème des arcs de Nash, Annales de l’Institut Fourier, Volume 55 (2005) no. 3, pp. 805-823 | DOI | Numdam | Zbl
[16] Résolution du problème des arcs de Nash pour les points doubles rationnels ., Note C.R.A.S, Série I , Volume 340 (2005), pp. 747-750 | MR | Zbl
[17] A class of non-rational surface singularities for which the Nash map is bijective, Bulletin de la SMF, Volume 134 (2006) no. 3, pp. 383-394 | Numdam | MR | Zbl
[18] Families of arcs on rational surface singularities, Manuscripta Math, Volume 88, 3 (1995), pp. 321-333 | DOI | MR | Zbl
[19] Image of the Nash map in terms of wedges, C. R. Acad. Sci. Paris, Ser. I , Volume 338 (2004), pp. 385-390 | MR | Zbl
Cited by Sources: