Families of smooth curves on surface singularities and wedges

Gérard Gonzalez-Sprinberg; Monique Lejeune-Jalabert

Annales Polonici Mathematici (1997)

  • Volume: 67, Issue: 2, page 179-190
  • ISSN: 0066-2216

Abstract

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Following the study of the arc structure of singularities, initiated by J. Nash, we give criteria for the existence of smooth curves on a surface singularity (S,O) and of smooth branches of its generic hypersurface section. The main applications are the following: the existence of a natural partition of the set of smooth curves on (S,O) into families, a description of each of them by means of chains of infinitely near points and their associated maximal cycle and the existence of smooth curves on any sandwiched surface singularity. A wedge centered at a smooth curve on (S,O) is essentially a one-parameter deformation of the parametrization of the curve. We show that there is no wedge centered at smooth curves of two different families.

How to cite

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Gérard Gonzalez-Sprinberg, and Monique Lejeune-Jalabert. "Families of smooth curves on surface singularities and wedges." Annales Polonici Mathematici 67.2 (1997): 179-190. <http://eudml.org/doc/270573>.

@article{GérardGonzalez1997,
abstract = {Following the study of the arc structure of singularities, initiated by J. Nash, we give criteria for the existence of smooth curves on a surface singularity (S,O) and of smooth branches of its generic hypersurface section. The main applications are the following: the existence of a natural partition of the set of smooth curves on (S,O) into families, a description of each of them by means of chains of infinitely near points and their associated maximal cycle and the existence of smooth curves on any sandwiched surface singularity. A wedge centered at a smooth curve on (S,O) is essentially a one-parameter deformation of the parametrization of the curve. We show that there is no wedge centered at smooth curves of two different families.},
author = {Gérard Gonzalez-Sprinberg, Monique Lejeune-Jalabert},
journal = {Annales Polonici Mathematici},
keywords = {surface singularities; smooth curves; maximal cycle; wedges; arcs; resolution of singularities; surface singularity; desingularization; smooth curves lying on sandwich singularities; wedge morphisms},
language = {eng},
number = {2},
pages = {179-190},
title = {Families of smooth curves on surface singularities and wedges},
url = {http://eudml.org/doc/270573},
volume = {67},
year = {1997},
}

TY - JOUR
AU - Gérard Gonzalez-Sprinberg
AU - Monique Lejeune-Jalabert
TI - Families of smooth curves on surface singularities and wedges
JO - Annales Polonici Mathematici
PY - 1997
VL - 67
IS - 2
SP - 179
EP - 190
AB - Following the study of the arc structure of singularities, initiated by J. Nash, we give criteria for the existence of smooth curves on a surface singularity (S,O) and of smooth branches of its generic hypersurface section. The main applications are the following: the existence of a natural partition of the set of smooth curves on (S,O) into families, a description of each of them by means of chains of infinitely near points and their associated maximal cycle and the existence of smooth curves on any sandwiched surface singularity. A wedge centered at a smooth curve on (S,O) is essentially a one-parameter deformation of the parametrization of the curve. We show that there is no wedge centered at smooth curves of two different families.
LA - eng
KW - surface singularities; smooth curves; maximal cycle; wedges; arcs; resolution of singularities; surface singularity; desingularization; smooth curves lying on sandwich singularities; wedge morphisms
UR - http://eudml.org/doc/270573
ER -

References

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  1. [A] M. Artin, On isolated rational singularities of surfaces, Amer. J. Math. 88 (1966), 129-136 Zbl0142.18602
  2. [E] D. Eisenbud, Open problems in computational algebraic geometry, in: Computational Algebraic Geometry and Commutative Algebra, D. Eisenbud and L. Robbiano (eds.), Cambridge Univ. Press, 1993, 49-70 Zbl0829.14030
  3. [G-S] G. Gonzalez-Sprinberg, Cycle maximal et invariant d'Euler local des singularités isolées de surface, Topology 21 (1982), 401-408 Zbl0538.14026
  4. [G/L1] G. Gonzalez-Sprinberg et M. Lejeune-Jalabert, Courbes lisses sur les singularités de surface, C. R. Acad. Sci. Paris Sér. I Math. 318 (1994), 653-656 Zbl0853.14018
  5. [G/L2] G. Gonzalez-Sprinberg et M. Lejeune-Jalabert, Sur l'espace des courbes tracées sur une singularité, in: Algebraic Geometry and Singularities, A. Campillo and L. Narváez (eds.), Progr. Math. 134, Birkhäuser, 1996, 9-32 
  6. [L] J. Lipman, Desingularization of two-dimensional schemes, Ann. of Math. 107 (1978), 151-207 Zbl0349.14004
  7. [L-J] M. Lejeune-Jalabert, Arcs analytiques et résolution minimale des singularités des surfaces quasi-homogènes, in: Lecture Notes in Math. 777, Springer, 1980, 303-336 
  8. [L/T] M. Lejeune-Jalabert et B. Teissier, Contributions à l'étude des singularités du point de vue du polygone de Newton, Thèse, Université Paris 7, 1973 
  9. [N] J. Nash, Arc structure of singularities, preprint, 1968; Duke Math. J. 81 (1995), 31-38 
  10. [S] M. Spivakovsky, Sandwiched singularities and desingularization of surfaces by normalized Nash transformations, Ann. of Math. 131 (1990), 411-491 Zbl0719.14005
  11. [W] R. Walker, Reduction of the singularities of an algebraic surface, Ann. of Math. 36 (1935), 336-36 Zbl61.0705.02

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