no. 4
Deformation of polar methods
Annales de l'Institut Fourier, Tome 42 (1992) no. 4, pp. 737-778.

Nous étudions les déformations des hypersurfaces avec lieu singulier de dimension un par deux méthodes différentes. La première méthode utilise les nombres de Lê d’une singularité d’hypersurface, ceci est un exemple de méthode “polaire”. La seconde consiste à étudier le nombre de certains types spéciaux de singularités des déformations génériques de l’hypersurface donnée. Nous comparons ces deux méthodes et donnons beaucoup d’exemples.

We study deformations of hypersurfaces with one-dimensional singular loci by two different methods. The first method is by using the Le numbers of a hypersurfaces singularity — this falls under the general heading of a “polar” method. The second method is by studying the number of certain special types of singularities which occur in generic deformations of the original hypersurface. We compare and contrast these two methods, and provide a large number of examples.

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     title = {Deformation of polar methods},
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Massey, David B.; Siersma, Dirk. Deformation of polar methods. Annales de l'Institut Fourier, Tome 42 (1992) no. 4, pp. 737-778. doi : 10.5802/aif.1308. http://www.numdam.org/articles/10.5802/aif.1308/

[Di1] A. Dimca, Betti numbers of hypersurfaces and defects of linear systems, Duke Math. Journ., 60 (1) (1990), 285-298. | MR 91f:14041 | Zbl 0729.14017

[Di2] A. Dimca, On the Milnor fibration of weighted homogeneous polynomials, Compositio Mathematica, 76 (1990), 19-47. | Numdam | MR 91m:32038 | Zbl 0726.14002

[Fu] W. Fulton, Intersection Theory, Ergebnisse der Math., Springer-Verlag, 1984. | MR 85k:14004 | Zbl 0541.14005

[Ga1] T. Gaffney, Polar Multiplicities and Equisingularity of Map Germs, Topology (to appear). | Zbl 0790.57020

[Ga2] T. Gaffney, handwritten note.

[Ga-Mo] T. Gaffney and D. Q. M. Mond, Cusps and double folds of germs of analytic maps ℂ2 → ℂ2, Journ. London Math. Soc. (to appear). | Zbl 0737.58007

[Gi1] M. Giusti, Sur les singularités isolées d'intersections complète quasi-homogènes, Ann. Inst. Fourier, Grenoble, 27-3 (1977), 163-192. | Numdam | MR 57 #702 | Zbl 0353.14003

[Gi2] M. Giusti, Intersections complète quasi-homogènes : calcule d'invariants (These: Université Paris VII, 1981) Prépublication du Centre Mathématiques de l'Ecole Polytechnique (1979), 69-99.

[GoMac] M. Goresky and R. Macpherson, Ergebnisse der Math. 14 (1988), Stratified Morse Theory, Springer-Verlag. | Zbl 0639.14012

[Io1] I.N. Iomdin, Local topological properties of complex algebraic sets, Sibirsk. Mat. Z., 15 (4) (1974), 784-805. | MR 447620 | Zbl 0305.14001

[Io2] I.N. Iomdin, Variétés complexes avec singularités de dimension un, Sibirsk. Mat. Z., 15 (1974), 1061-1082. | MR 447621 | Zbl 0325.32003

[Jo1] Th. De Jong, Some classes of line singularities, Math. Zeitschrift, 198 (1988), 493-517. | EuDML 173940 | MR 950580 | Zbl 0628.32028

[Jo2] Th. De Jong, The virtual number of D∞ points I, Topology, 29 (1990), 175-184. | MR 1056268 | Zbl 0752.32010

[JoSt1] Th. De Jong, D. Van Straten, A deformation Theory for Non-Isolated Singularities, Abh. Math. Sem. Univ. Hamburg, 60, 177-208. | MR 1087127 | Zbl 0721.32014

[JoSt2] Th. De Jong, D. Van Straten, Deformations of the Normalizations of Hypersurfaces, Math. Annalen, 228, 527-547. | EuDML 164749 | MR 1079877 | Zbl 0715.32013

[JoSt3] Th. De Jong, D. Van Straten, Disentanglements, Singularity Theory and its Applications, Warwick 1989, Part I ; Lecture Notes in Mathematics, 1462, 199-211. | MR 1129033 | Zbl 0734.32020

[KM] M. Kato and Y. Matsumoto, On the connectivity of the Milnor fibre of a holomorphic function at a critical point, Proc. 1973 Tokyo Manifolds Conf., 131-136. | MR 372880 | Zbl 0309.32008

[Lê1] D.T. Lê, Calcul du nombre de cycles évanouissants d'une hypersurface complexe, Ann. Inst. Fourier, Grenoble, 23-4 (1973), 261-270. | EuDML 74151 | Numdam | MR 330501 | Zbl 0293.32013

[Lê2] D.T. Lê, Ensembles analytiques complexes avec lieu singulier de dimension un (d'après I.N. Jomdin), Séminaire sur les Singularités (Paris, 1976-1977), Publ. Math. Univ. Paris VII (1980), 87-95. | MR 683620

[Lê3] D.T. Lê, Une application d'un théorème d'A'Campo a l'équisingularité, Indag. Math., 35 (1973), 403-409. | MR 326001 | Zbl 0271.14001

[Ma1] D. Massey, The Lê - Ramanujam Problem for Hypersurfaces with One - Dimensional Singular Sets, Math. Ann., 282 (1988), 33-49. | EuDML 164448 | MR 960832 | Zbl 0657.32005

[Ma2] D. Massey, The Lê Varieties, I, Invent. Math., 99 (1990), 357-376. | EuDML 143762 | MR 1031905 | Zbl 0712.32020

[Ma3] D. Massey, The Lê Varieties, II, Invent. Math., 104 (1991), 113-148. | EuDML 143876 | MR 1094048 | Zbl 0727.32015

[Ma4] D. Massey, The Thom Condition along a Line, Duke Math. Journ., 60 (1990), 631-642. | MR 1054528 | Zbl 0709.32026

[Mi] J. Milnor, Singular Points of Complex Hypersurfaces, Princeton University Press, Ann. Math. Studies, 1968. | MR 239612 | Zbl 0184.48405

[MiOr] J. Milnor and P. Orlik, Isolated Singularities Defined by Weighted Homogeneous Polynomials, Topology, 9 (1969), 385-393. | MR 293680 | Zbl 0204.56503

[Mo1] D.M.Q. Mond, Some remarks on the geometry and classification of germs of maps from surfaces to 3-space, Topology, 26, 3 (1987), 361-383. | MR 899055 | Zbl 0654.32008

[Mo2] D.M.Q. Mond, Vanishing cycles for analytic maps, Singularity Theory and its Applications, Warwick 1989, Part I ; Lecture Notes in Mathematics, 1462, 221-234. | MR 1129035 | Zbl 0745.32020

[Ne1] A. Nemethi, On the fundamental group of the complement of certain singular plane curves, Math. Proc. Camb. Phil. Soc., 102 (1987), 453-457. | MR 88m:14020 | Zbl 0679.14004

[Ne2] A. Nemethi, The Milnor fibre and the zeta function of the singularities of the type f = P(h,g), Compositio Mathematica, 79, 63-97. | EuDML 90099 | Numdam | MR 93a:32058 | Zbl 0724.32020

[Ok] M. Oka, On the fundamental group of the complement of certain plane curves, J. Math. Soc. Japan, (4) 30 (1978), 579-597. | MR 80d:14017 | Zbl 0387.14004

[Pe1] G. R. Pellikaan, Hypersurface singularities and resolutions of Jacobi modules, Thesis, Rijksuniversiteit Utrecht (1985). | MR 87j:32024 | Zbl 0589.32017

[Pe2] G.R. Pellikaan, Deformations of hypersurfaces with a one-dimensional critical locus, Journal of Pure and Applied Algebra, 67, 49-71. | Zbl 0733.32026

[Pe3] G.R. Pellikaan, Series of isolated singularities, Contemp. Math., 90 (1989), 241-259. | MR 90k:32044 | Zbl 0675.32009

[Ra] R. Randell, On the topology of non-isolated singularities, Proc. 1977 Georgia Topology Conference, (1979), 445-473. | MR 80f:14002 | Zbl 0473.57007

[Sa] M. Saito, On Steenbrink's Conjecture, Mathematische Annalen, 289, 703-716. | EuDML 164806 | MR 92e:32021 | Zbl 0712.14002

[Sc] R. Schrauwen, Deformations and the Milnor number of non-isolated plane curve singularities, Singularity Theory and its Applications, Warwick 1989, Part I ; Lecture Notes in Mathematics, 1462, 276-291. | MR 92j:32128 | Zbl 0734.32021

[Si1] D. Siersma, Isolated Line Singularities, Proc. Symp. Pure Math., 40 (part 2) (1983), 485-496. | MR 85d:32017 | Zbl 0514.32007

[Si2] D. Siersma, Hypersurfaces with singular locus a plane curve and transversal type A1, Singularities, 40 (1988), 397-410, Banach Center Publ., Warsaw. | EuDML 267775 | MR 92a:32044 | Zbl 0662.32011

[Si3] D. Siersma, Singularities with critical locus a 1-dimensional complete intersection and transversal type A1, Topology and its Applications, 27 (1987), 51-73. | MR 88m:32015 | Zbl 0635.32006

[Si4] D. Siersma, Quasi-homogeneous singularities with transversal type A1, Contemporary Math., 90 (1989), 261-294. | MR 91a:32050 | Zbl 0682.32012

[Si5] D. Siersma, The monodromy of a series of hypersurface singularities, Comment. Math. Helvetici, 65, 181-195. | EuDML 140190 | MR 91h:32034 | Zbl 0723.32015

[Si6] D. Siersma, Variation Mappings on singularities with a 1-dimensional critical locus, Topology, 30, 445-469. | MR 92e:32022 | Zbl 0746.32014

[Si7] D. Siersma, Vanishing cycles and special fibres, Singularity Theory and its Applications, Warwick 1989, Part I ; Lecture Notes in Mathematics, 1462, 292-301. | MR 92j:32129 | Zbl 0746.32015

[SiTr] Y. T. Siu and G. Trautmann, Gap-Sheaves and Extension of Coherent Analytic Subsheaves, S.L.N. 172, Springer-Verlag, 1971. | MR 44 #4240 | Zbl 0208.10403

[St] J. H. M. Steenbrink, The spectrum of hypersurface singularities, Theory de Hodge ; Luminy, juin 1987 ; Astérisque, 179-180, 163-184. | Zbl 0725.14031

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