Invariants of plane curve singularities and Newton diagrams
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RIS BIB ENDNOTEInvariants of plane curve singularities and Newton diagrams
Data publikacji: 05.06.2012
Universitatis Iagellonicae Acta Mathematica, 2011, Tom 49, s. 9 - 34
https://doi.org/10.4467/20843828AM.12.001.0453Autorzy
Invariants of plane curve singularities and Newton diagrams
The goal of this paper is to present an elementary, intersection-theoretical approach to the local invariants of plane curve singularities. We study in detail three invariants: the Milnor number , the number of double points and the number r of branches of a local plane curve. The technique of Newton diagrams plays an important part in the paper. It is well-known that Newton transformations which arise in a natural way when applying the Newton algorithm provide a useful tool for calculating invariants of singularities. The formulae for the Milnor number in terms of Newton diagrams and Newton transformations presented in the paper grew out of our discussion on Eisenbud{Neumann diagrams. They have counterparts in toric geometry of plane curve singularities and in the case of two dimens
The contents of the article are:
1. Plane local curves
2. The Milnor number: intersection theoretical approach
3. Newton diagrams and power series
4. Newton transformations and factorization of power series
5. Newton transformations, intersection multiplicity and the Milnor number
6. Nondegenerate singularities and equisingularityions imply theorems due to Kouchnirenko, Bernstein and Khovanski.
2000 Mathematics Subject Classification. Primary 32S55; Secondary 14H20.
The first author is partially supported by grants MTM 2010-21740-C02-01 and MTM 2010-21740-C02-02.
1. Bernstein D. N., The number of roots of a system of equations, Funct. Anal. Appl., 9, No. 3 (1975), 183–185.
2. Brieskorn E., Knoerr H., Plane algebraic curves, Birkh¨auser, 1986.
3. Brzostowski Sz., Degenerate singularities and their Milnor numbers, Univ. Iagel. Acta Math., 49 (2011), 37–44.
4. Casas-Alvero E., Singularities of plane curves, London Math. Soc. Lecture Notes, 276, Cambridge University Press, 2000.
5. Cassou-Nogu`es Pi., Veys W., Newton trees for ideals in two variables and applications, arXiv: 1201.0467 v 1 [math.AG] 2 Jan 2012.
6. Duval D., Rational Puiseux expansions, Composito Math., 70 (1989), 119–154.
7. Garc´ıa Barroso E., Lenarcik A., Płoski A., Characterization of non-degenerate plane curve singularities, Univ. Iagel. Acta Math., 45 (2007), 27–36. Erratum, ibidem. 47 (2009), 321–322.
8. Greuel G. M., Lossen Ch., Shustin E., Introduction to singularities and deformations, Springer Verlag, 2006.
9. Greuel G. M., Nguyen Hong Duc, Some remarks on the planar Kouchnirenko’s theorem, arXiv: 1009.4889 v 1 [math. AG] 24 Sep. 2010.
10. Gwo´zdziewicz J., Generalized Noether’s Formulas for plane curve singularities, Univ. Iagel. Acta Math., 48 (2010), 55–62.
11. Gwo´zdziewicz J., Lenarcik A., Płoski A., Polar invariants of plane curve singularities: intersection theoretical approach, Demonstratio Math., 43, N◦ 2 (2010), 303–323.
12. Gwo´zdziewicz J., Płoski A., Formulae for the singularities at infinity of plane algebraic curves, Univ. Iagel. Acta Math., 39 (2001), 109–133.
13. Hironaka H., On the arithmetic genera and the effective genera of algebraic curves, Mem. Coll. Sci. Univ. Kyoto. Ser. A. Math., 30 (1957), 177–195.
14. Jung H. W. E., Einfu¨hrung in die Theorie der algebraischen Funktionen einer Ver¨anderlichen, Walter de Gruyter & Co, Berlin und Leipzig, 1923.
15. de Jong T., Pfister G., Local analytic geometry. Basic theory and applications, Vieweg, 2000.
16. Khovanski A. G., Newton polyhedra and toric varieties, Funkt. Anal. Prloz., 11(4) (1977), 56–67 and 12(1) (1978), 51–61.
17. Khovanski A. G., The index of polynomial vector field, (Russian), Funkt. Anal. Prloz., 13(1) (1979), 49–58.
18. Kouchnirenko A. G., Poly`edres de Newton et nombres de Milnor, Invent. Math., 32 (1976), 1–31.
19. Kuo T. C., Generalized Newton-Puiseux theory and Hensel’s lemma in C[[x, y]], Canad. J. Math., 41, No. 6 (1989), 1101–1116.
20. Kucharz W., Newton polygons and topological determinancy of analytic germs, Periodica Math. Hungarica, 22(2) (1991), 129–132.
21. Lenarcik A., On the Jacobian Newton polygon of plane curve singularities, Manuscripta Math., 125 (2008), 309–324.
22. Lipovski A., Newton polyhedra and irreducibility, Math. Z., 199 (1988), 119–127.
23. Maurer J., Puiseux expansion for space curves, Manuscripta Math., 32 (1980), 91–100.
24. Masternak M., Invariants of singularities of polynomials in two complex variables and the Newton diagrams, Univ. Iagel. Acta Math., 39 (2001), 179–188.
25. Merle M., Teissier B., Conditions d’adjonction d’apr``es du Val, S´´eminaire sur les Singularit´´es des Surfaces, Palaiseau, France 1976–77, Lecture Notes in Math. Springer, 777, 230–245.
26. Milnor J., Singular Points of Complex Hypersurfaces, Princeton University Press, 1968.
27. Oka M., Geometry of plane curves via toroidal resolution, Algebraic Geometry and Singularities (eds. A. Campillo L´´opez and Norrvaa´ez Macarro), Progr. Math., 134, Birkh¨¨auser, Basel (1996), 95–121.
28. Pfister G., Invariants of singularities and Newton polyhedron, Proceedings of the Week of Algebraic Geometry, Bucharest 1980, Teuber-Texte zur Mathematik, Band 40, 123–138.
29. Pham E., Courbes discriminantes des singularit´´es planes d’ordre 3, Ast´´erisque, 7–8 (1973), 363–391.
30. Risler J. J., Sur l’id´´eal jacobian d’une courbe planes, Bull. Soc. Math. Fr., 99(4) (1971), 305–311.
31. Teissier B., Th´´ese, 2◦ partie, Paris 1973.
32. Teissier B., Cycles ´´evanescents, section planes et conditions de Whitney, Ast´´erisque, No 7–8 (1973), 285–362.
33. Teissier B., The hunting of invariants in the geometry of discriminants, Nordic Summer School/NAVF Symposium in Mathematics, Oslo, August 5–25, 1976.
34. Teissier B., Introduction to Curve Singularities, Singularity Theory, Eitors D. T. Lˆˆe, K. Saito, B. Teissier, World Scientific 1991.
35. Varchenko A. N., Zeta function of monodromy and Newton’s diagram, Invent. Math., 37 (1976), 253–262.
36. Wall C. T. C., Newton polytopes and nondegeneracy, J. reine angew. Math., 509 (1999).
37. Wall C. T. C., Singular Points of Plane Curves, London Mathematical Society, Student Text 63, Cambridge University Press, 2004.
38. van der Waerden B. L., Einfuu¨hrung in die algebraische Geometrie, Springer, Berlin, 1939.
39. Zariski O., Le probl``eme des modules pour les branches planes. Centre de Math´´ematiques de l’EE´ cole Polytechnique, Paris, 1973.
Informacje: Universitatis Iagellonicae Acta Mathematica, 2011, Tom 49, s. 9 - 34
Typ artykułu: Oryginalny artykuł naukowy
Tytuły:
Invariants of plane curve singularities and Newton diagrams
Invariants of plane curve singularities and Newton diagrams
Université Michel De Montaigne Bordeaux; 146 rue Léo Saignat CS 61292 33 076 Bordeaux CEDEX
Katedra Matematyki, Wydział Zarządzania i Modelowania Komputerowego, Politechnika Świętokrzyska, Kielce, Polska
Publikacja: 05.06.2012
Status artykułu: Otwarte
Licencja: Żadna
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