Classification des formes de Seifert rationnelles des germes de courbe plane
Annales de l'Institut Fourier, Volume 46 (1996) no. 2, p. 371-410
We give an explicit description of the rational Seifert form associated with a plane curve germ, up to isomorphism or up to Witt-equivalence, in terms of a complete set of invariants determined by the topological type of the germ. The invariants are related to the classification of hermitian forms on cyclotomic extensions of and of quadratic forms on .As an application, we find cobordant and nonisotopic algebraic knots, the monodromy of which is of finite order.
Nous donnons une description explicite de la forme de Seifert rationnelle associée à un germe de courbe plane, à isomorphisme près ou à Witt-équivalences près, en termes d’un ensemble complet d’invariants déterminé à partir du type topologique du germe. Ces invariants sont liés à la classification des formes hermitiennes sur les extensions cyclotomiques de et à celle des formes quadratiques sur .En application, nous trouvons des nœuds algébriques cobordants et non isotopes dont la monodromie est d’ordre fini.
@article{AIF_1996__46_2_371_0,
     author = {Bois, Philippe Du and Hunault, Ollivier},
     title = {Classification des formes de Seifert rationnelles des germes de courbe plane},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {46},
     number = {2},
     year = {1996},
     pages = {371-410},
     doi = {10.5802/aif.1518},
     zbl = {0854.32021},
     mrnumber = {97g:32048},
     language = {fr},
     url = {http://www.numdam.org/item/AIF_1996__46_2_371_0}
}
Bois, Philippe Du; Hunault, Ollivier. Classification des formes de Seifert rationnelles des germes de courbe plane. Annales de l'Institut Fourier, Volume 46 (1996) no. 2, pp. 371-410. doi : 10.5802/aif.1518. http://www.numdam.org/item/AIF_1996__46_2_371_0/

[C] A. Chenciner, Courbes algébriques planes, Publications Mathématiques de l'Université Paris VII, 1978. | MR 84k:14023 | Zbl 0557.14016

[DM1] P. Du Bois, F. Michel, Filtration par le poids et monodromie entière, Bull. Soc. Math. de France, 120 (1992), 129-167. | Numdam | MR 93h:32049 | Zbl 0771.14005

[DM2] P. Du Bois, F. Michel, Cobordism of algebraic knots via Seifert forms, Invent. Math., 111 (1993), 151-169. | MR 94d:57051 | Zbl 0789.57015

[DM3] P. Du Bois, F. Michel, The integral Seifert form does not determine the topology of plane curves, J. Alg. Geometry, 3 (1994), 1-38. | MR 94k:32062 | Zbl 0810.32005

[DS] M. Van Doorn, J. Steenbrink, A supplement to monodromy theorem, Abh. Math. Sem. Hamburg Univ., 59 (1989), 225-233. | MR 91e:32036 | Zbl 0712.32022

[Du] A. Durfee, Fibred knots and algebraic singularities, Topology, 13 (1974), 47-59. | MR 49 #1523 | Zbl 0275.57007

[EGAIII]A. Grothendieck, Éléments de Géométrie Algébrique III, Publ. Math. IHES, 11 (1961). | Numdam

[J] N. Jacobson, A Note on Hermitian Forms, Bull. Amer. Math. Soc., 46 (1940), 264-268. | JFM 66.0048.03 | MR 1,325d | Zbl 0024.24503

[Ka] R. Kaenders, The Seifert Form of a Plane Curve Singularity determines its Intersection Multiplicities, à paraître dans Indag. Math.. | Zbl 0873.14032

[Ke] W. Landherr, Äquivalenz Hermitescher Formen über einem beliebigen algebraischen Zahlkörper, Abh. Math. Sem. Hamburg Univ., 11 (1935), 245-248. | JFM 62.0170.01 | Zbl 0013.38901

[Le] J. Levine, Knot cobordism groups in codimension two, Comment. Math. Helv., 44 (1969), 229-244. | MR 39 #7618 | Zbl 0176.22101

[Mi1] J. Milnor, Singular points of complex hypersurfaces, Annals Math. Studies 61, Princeton Univ. Press, 1968. | MR 39 #969 | Zbl 0184.48405

[Mi2] J. Milnor, On isometries of inner product spaces, Invent. Math., 8 (1969), 83-97. | MR 40 #2764 | Zbl 0177.05204

[MiH] J. Milnor, D. Husemoller, Symmetric bilinear forms, Springer-Verlag, 1973. | MR 58 #22129 | Zbl 0292.10016

[Ne] W.D. Neumann, Invariants of plane curves singularities, Noeuds, tresses et sing., Monog. de l'Enseignement. Math., Univ. de Genève, 1983. | MR 85c:14019 | Zbl 0586.14023

[Sa] K. Sakamoto, The Seifert matrices of Milnor fiberings defined by holomorphic functions, J. Math. Soc. Japan, 26 (1974), 4. | MR 50 #14771 | Zbl 0286.32010

[SSS] R. Schrauwen, J. Steenbrink, J. Stevens, Spectral pairs and the topology of curve singularities, Proc. Sympos. Pure Math., 53 (1991), 305-328. | MR 93f:32042 | Zbl 0749.14003

[Se] J.-P. Serre, Cours d'Arithmétique, Presses Universitaires de France, 1970. | MR 41 #138 | Zbl 0225.12002

[St] J. Steenbrink, Mixed Hodge structure on the vanishing cohomology, Nordic Summer School NAVF, Symposium in Math. Oslo, 1976. | Zbl 0373.14007