Deformation of polar methods

Massey, David B.; Siersma, Dirk

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.

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

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

[Fu] W. Fulton, Intersection Theory, Ergebnisse der Math., Springer-Verlag, 1984. | MR | Zbl

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

[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

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

[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

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

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

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

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

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

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

[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 | Zbl

[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 | Zbl

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

[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

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

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

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

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

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

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Numdam | MR | Zbl

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

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

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

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

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

[Sa] M. Saito, On Steenbrink's Conjecture, Mathematische Annalen, 289, 703-716. | EuDML | MR | Zbl

[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 | Zbl

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

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

[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 | Zbl

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

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

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

[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 | Zbl

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

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