# 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

## Access Full Article

top## Abstract

top## How to cite

topGérard 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

top- [A] M. Artin, On isolated rational singularities of surfaces, Amer. J. Math. 88 (1966), 129-136 Zbl0142.18602
- [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
- [G-S] G. Gonzalez-Sprinberg, Cycle maximal et invariant d'Euler local des singularités isolées de surface, Topology 21 (1982), 401-408 Zbl0538.14026
- [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
- [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
- [L] J. Lipman, Desingularization of two-dimensional schemes, Ann. of Math. 107 (1978), 151-207 Zbl0349.14004
- [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
- [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
- [N] J. Nash, Arc structure of singularities, preprint, 1968; Duke Math. J. 81 (1995), 31-38
- [S] M. Spivakovsky, Sandwiched singularities and desingularization of surfaces by normalized Nash transformations, Ann. of Math. 131 (1990), 411-491 Zbl0719.14005
- [W] R. Walker, Reduction of the singularities of an algebraic surface, Ann. of Math. 36 (1935), 336-36 Zbl61.0705.02

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.