On Nash problem of arc

Camille Plénat[1]

  • [1] Université du Maine, département de mathématiques, avenue Messiaen, 72085 Le Mans cedex 9 (France)

Annales de l’institut Fourier (2005)

  • Volume: 55, Issue: 3, page 805-823
  • ISSN: 0373-0956

Abstract

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Let ℋ = ⋃ N i the canonical decomposition of the space of arcs passing through a normal surface singularity. In this paper, we give two new conditions to ensure that N i is not included in N j . These conditions allow us to give two different direct demonstration of Nah problem for minimal sandwiched singularities.

How to cite

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Plénat, Camille. "A propos du problème des arcs de Nash." Annales de l’institut Fourier 55.3 (2005): 805-823. <http://eudml.org/doc/116209>.

@article{Plénat2005,
abstract = {Soit $\{\mathcal \{H\}\}=\bigcup N_i$ la décomposition canonique de l’espace des arcs $\{\mathcal \{H\}\}$ passant par une singularité normale de surface. Dans cet article, on propose deux nouvelles conditions qui si elles sont vérifiées permettent de montrer que $N_i$ n’est pas inclus dans $N_j$. On applique ces conditions pour donner deux nouvelles preuves du problème de Nash pour les singularités sandwich minimales.},
affiliation = {Université du Maine, département de mathématiques, avenue Messiaen, 72085 Le Mans cedex 9 (France)},
author = {Plénat, Camille},
journal = {Annales de l’institut Fourier},
keywords = {geometry algebraic; space of arcs; minimal sandwiched singularities; problem of Nash},
language = {fre},
number = {3},
pages = {805-823},
publisher = {Association des Annales de l'Institut Fourier},
title = {A propos du problème des arcs de Nash},
url = {http://eudml.org/doc/116209},
volume = {55},
year = {2005},
}

TY - JOUR
AU - Plénat, Camille
TI - A propos du problème des arcs de Nash
JO - Annales de l’institut Fourier
PY - 2005
PB - Association des Annales de l'Institut Fourier
VL - 55
IS - 3
SP - 805
EP - 823
AB - Soit ${\mathcal {H}}=\bigcup N_i$ la décomposition canonique de l’espace des arcs ${\mathcal {H}}$ passant par une singularité normale de surface. Dans cet article, on propose deux nouvelles conditions qui si elles sont vérifiées permettent de montrer que $N_i$ n’est pas inclus dans $N_j$. On applique ces conditions pour donner deux nouvelles preuves du problème de Nash pour les singularités sandwich minimales.
LA - fre
KW - geometry algebraic; space of arcs; minimal sandwiched singularities; problem of Nash
UR - http://eudml.org/doc/116209
ER -

References

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  1. M. Artin, On isolated rational singularities of surfaces, Amer. J. Math. 88 (1966), 129-136 Zbl0142.18602MR199191
  2. J. Fernandez-Sanchez, Equivalence of the Nash conjecture for primitive and sandwiched singularities, Proc. Amer. Math. Soc. 133 (2005), 677-679 Zbl1056.14004MR2113914
  3. S. Ishii, J. Kollár, The Nash problem on arc families of singularities, Duke Math. J. 120 (2003), 601-620 Zbl1052.14011MR2030097
  4. H.B. Laufer, Taut two-dimensional singularities, Math. Ann. 205 (1973), 131-164 Zbl0281.32010MR333238
  5. M. Lejeune-Jalabert, Courbes tracées sur un germe d'hypersurface, Amer. J. Math. 112 (1990), 525-568 Zbl0743.14002MR1064990
  6. J.F. Nash Jr., Arc structure of singularities, A celebration of John F. Nash, Jr., Duke Math. J. 81 (1995), 31-38 Zbl0880.14010MR1381967
  7. A. Reguera, Families of arcs on rational surface singularities, Manuscripta Math. 88 (1995), 321-333 Zbl0867.14012MR1359701
  8. Mark Spivakovsky, Sandwiched singularities and desingularization of surfaces by normalized Nash transformations, Annals of Math. 131 (1990), 411-491 Zbl0719.14005MR1053487
  9. Mark Spivakovsky, Valuation in function fields of surfaces, Amer. J. Math. 112 (1990), 107-156 Zbl0716.13003MR1037606

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