Limit trees and generic discriminants of minimal surface singularities
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 17 (2008) no. 1, pp. 37-51.

D’après R. Bondil, le graphe dual de la résolution minimale d’une singularité minimale de surface normale détermine le discriminant générique de cette singularité. Par des arguments combinatoires, nous donnons dans cet article le lien entre les arbres limites et les discriminants génériques des singularités minimales de surfaces normales. Les arbres limites pondérés d’une singularité minimale de surface normale détermine le discriminant générique de cette singularité.

According to R. Bondil the dual graph of the minimal resolution of a minimal normal surface singularity determines the generic discriminant of that singularity. In this article we give with combinatorial arguments the link between the limit trees and the generic discriminants of minimal normal surface singularities. The weighted limit trees of a minimal surface singularity determine the generic discriminant of that singularity.

@article{AFST_2008_6_17_1_37_0,
     author = {Ak\'ek\'e, Eric},
     title = {Limit trees and generic discriminants of minimal surface singularities},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {37--51},
     publisher = {Universit\'e Paul Sabatier, Institut de math\'ematiques},
     address = {Toulouse},
     volume = {Ser. 6, 17},
     number = {1},
     year = {2008},
     doi = {10.5802/afst.1174},
     mrnumber = {2464092},
     zbl = {1159.32017},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/afst.1174/}
}
TY  - JOUR
AU  - Akéké, Eric
TI  - Limit trees and generic discriminants of minimal surface singularities
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2008
DA  - 2008///
SP  - 37
EP  - 51
VL  - Ser. 6, 17
IS  - 1
PB  - Université Paul Sabatier, Institut de mathématiques
PP  - Toulouse
UR  - http://www.numdam.org/articles/10.5802/afst.1174/
UR  - https://www.ams.org/mathscinet-getitem?mr=2464092
UR  - https://zbmath.org/?q=an%3A1159.32017
UR  - https://doi.org/10.5802/afst.1174
DO  - 10.5802/afst.1174
LA  - en
ID  - AFST_2008_6_17_1_37_0
ER  - 
Akéké, Eric. Limit trees and generic discriminants of minimal surface singularities. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 17 (2008) no. 1, pp. 37-51. doi : 10.5802/afst.1174. http://www.numdam.org/articles/10.5802/afst.1174/

[1] Akéké (E.D.).— Classification des singularités minimales de surfaces normales par les discriminants génériques, Université de Provence, Marseille (2005).

[2] Artin (M.).— On isolated rational singularities of surfaces, Amer. J. Math, 88, p.129-136 (1996). | MR 199191 | Zbl 0142.18602

[3] Bondil (R.).— Discriminant of a generic projection of a minimal normal surface singularity, C. R. Acad. Sc. Paris, 337 (2003). | MR 2001134 | Zbl 1053.14040

[4] Bondil (R.).— Fine polar invariants of minimal singularities of surfaces, preprint, Arxiv: AG/0401434, (2004).

[5] Briançon (J.), Galligo (A.) and Granger (M.).— Déformations équisingu-lières des germes de courbes gauches réduites, Mém. Soc. Math. France, 69 (1980/1981). | Numdam | MR 607805 | Zbl 0447.14004

[6] Brieskorn (E.) and Körrer (H.).— Plane algebraic curves, Birkäuser Verlag, Bessel (1986). | MR 886476 | Zbl 0588.14019

[7] De Jong (T.) and Van Straten (D.).— On the deformation theory of rational surface singularities with reduced fundamental cycle, J. Algebraic Geometry, 3, p. 117-172 (1994). | MR 1242008 | Zbl 0822.14004

[8] Kollár (J.).— Toward moduli of singular varieties, Comp. Math., 56, p. 369-398 (1985). | Numdam | MR 814554 | Zbl 0666.14003

[9] Lê (D.T.) and Tosun (M.).— Combinatorics of rational surface singularities, Comment. Math. Helvetici, 79 (2004). | MR 2081727 | Zbl 1060.32016

[10] Spivakovsky (M.).— Sandwiched singularities and desingularisation of surfaces by normalized Nash transformations, Ann. Math., 131, p. 411-491 (1990). | MR 1053487 | Zbl 0719.14005

[11] Teissier (N.).— Variétés polaires II, Multiplicités polaires, Sections planes et conditions de Whitney, Algebraic Geometry, 961, Lecture notes in math., p. 314-491, La Rabida 1981, Springer-Verlag (1982). | MR 708342 | Zbl 0585.14008

[12] Zariski (O.).— Studies in equisingularity I, equivalent singularities of plane algebroid curves, Amer. J. Math., 87, p. 507-536 (1965). | MR 177985 | Zbl 0132.41601

[13] Zariski (O.).— General theory of saturation and of satured local rings, Amer. J. Math., 93 (1971). | Zbl 0226.13013

Cité par Sources :