Open Access
Original Articles
A Numerical Characterization of Equisingularity for Map Germs from N-Space, (N≥3), to the Plane
Terence GaffneyProfessor of Mathematics Northeastern UniversityBostonView full profile → , Ranjit VohraMassachusetts Board of Higher EducationP.O. Box 491Bridgewater, MA, 02324, U.S.A.View full profile →
* Corresponding author · click or hover a name for details
- Received:
- 06 Oct 2003
- Published Online:
- 01 Jan 2004
- Article type:
- Original Articles
- Language:
- EN
- Article no.:
- 1726037X.2004.10698479
- Pages:
- 43–55
Abstract
By a celebrated 1976 result regarding the geometry of families of hypersurfaces with isolated singularities, Lê and Ramanujam showed that the constancy of a single numerical invariant, namely the Milnor number, implies that the family is locally topologically trivial. This classical result has led to generalizations of the Milnor number to various numerical invariants, including ”polar multiplicities”, ”Lê numbers” and ”Segre numbers”, all designed to reflect the geometry of singularities in very general scenarios. In the present study, we consider families of finitely-determined holomorphic map germs from Cundefined to C2 for n ≥ 3, and characterize local topological triviality in terms of the constancy of (n-1) Lê numbers.
Keywords
References
William, Fulton. 1984 . Intersection Theory, Springer Verlag . Terence, Gaffney. 1993 . Polar Multiplicities and Equisingularity of Map Germs . Topology, 32 ( 1 ) : 185 – 223 . Terence, Gaffney. 2000 . “ Plane Sections, W and A, in Real and Complex Singularities ” . In Research Notes in Mathematics, Edited by: Bruce, J.W. and Tari, F.Vol. 412 , Chapman & Hall . edited by Terence, Gaffney. 2000 . L. Math. Proc. Camb. Phil. Soc, 128 : 479 – 496 . in Terence, Gaffney. 1975 . Properties of Finitely Determined Germs, Brandeis University . Ph.D. thesis Terence, Gaffney. 1996 . Multiplicities and Equisingularities of ICIS Germs . Invent. Math, 123 : 200 – 220 . Terence, Gaffney and Robert, Gassler. 1999 . Segre Numbers and Hypersurface Singularities . Algebraic Geometry, 8 : 695 – 736 . Terence, Gaffney and Kleiman, S.L.2000 . undefined, Research Notes in Mathematics Edited by: Bruce, J.W. and Tari, F.Vol. 412 , Chapman & Hall . edited by Terence, Gaffney and David, Massey. 1999 . Trends in Equisingularity Theory, in Singularity Theory, Edited by: Bruce, J.W. and Mond, D.Cambridge University Press . Edited by Martin, Golubitsky and Guillemin, V.1973 . Stable Mappings and Their Singularities, Springer Verlag . Robin, Hartshorne. 1977 . Algebraic Geometry, Springer Verlag . Trang, Lê Dung. 1973 . Une Application d’un Theoreme d’A’Campo a L’equisingularite . jour. Indag. Math, 35 : 403 – 409 . in Trang, Lê Dung and Greuel, Spitzen G.1976 . Doppelpunkte unde verticale tangenten in der diskriminante verseller deformationen von vollstandigen durchschmitten . Math. Ann, 222 : 71 – 99 . Joseph, Lipman. 1982 . “ Equimultiplicity, Reduction, and Blowing Up ” . In Lecture Notes Pure Appl. Math, Vol. 68 , New York: Marcel Dekker . Trang, Lê Dung and Teissier, B.1983 . Cycles evanescents, sections planes et conditions de Whitney . II Proc. Symposia pure math, 40 ( part 2 ) : 65 – 104 . Amer. Math. Soc Trang, Lê Dung and Ramanujam, C.P.1976 . The invariance of Milnor’s number implies the invariance of the topological type . Amer. J. Math, 98 : 67 – 78 . David, Massey. 1995 . “ Lê Cycles and Hypersurface Singularities ” . In Springer Lecture Notes in Mathematics1615 Mather, John N.1969 . Stability of C Mappings, IV: Classification of Stable Germs by R-Algebras . Publ. Math. I.H.B.S, 37 : 223 – 248 . Mather, John N.1970 . Stability of C Mappings, V: Transversality . Advances in Mathematics, 4 Mather, John N.1968 . Stability of C Mappings, III: Finitely Determined Map Germs . Publ. Math. I.H.E.S, 35 : 127 – 156 . Perez, V.H.J.2000 . Polar Multiplicities and Equisingularity of map germs from C, Universidade de Sao Paulo . Ph.D. thesis Bernard, Teissier. 1981 . Varieties Polaires Locales: quelques resultats . Journels Complexes, March Institut Elie Cartan Ranjit, Vohra. 2003 . Equisingularity of Map. Germs from n-Space, (n≥3), to the Plane, Northeastern University . Ph.D. thesis Wall, C.T.C.1981 . Finite Determinacy of Smooth Map-Germa . Bull. London Math. Soc, 13 : 481 – 539 . Wall, C.T.C.1971 . Lectures on C -Stability and Classifications . Springer Lecture Notes in Mathematics, 192 : 178 – 206 .
Views: 7Downloads: 6Citations: 0
