Equisingular deformations of Puiseux expansions

Nobile, Augusto

doi:10.1090/S0002-9947-1975-0414560-8

Equisingular deformations of Puiseux expansions

Author: Augusto Nobile
Journal: Trans. Amer. Math. Soc. 214 (1975), 113-135
MSC: Primary 14H20; Secondary 14D15
DOI: https://doi.org/10.1090/S0002-9947-1975-0414560-8
MathSciNet review: 0414560
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Parametrizations of a deformation (over a complete local ring) of an irreducible algebroid curve are studied. With these parametrizations another definition of equisingularity (equivalent to the known ones) is given, by using their characteristic numbers. These methods are also used in the complex analytic case. The following applications of these techniques are given: a proof of the existence of a versal equisingular deformation in the complex analytic case; a proof that an equisingular formal family of branches is determined by its $\nu$ -truncation ( $\nu$ large enough, depending only on the characteristic of the special fiber).

[1] Shreeram Abhyankar, On the ramification of algebraic functions, Amer. J. Math. 77 (1955), 575–592. MR 0071851, https://doi.org/10.2307/2372643
[2] Shreeram Shankar Abhyankar, Inversion and invariance of characteristic pairs, Amer. J. Math. 89 (1967), 363–372. MR 0220732, https://doi.org/10.2307/2373126
[3] Sherwood Ebey, The classification of singular points of algebraic curves, Trans. Amer. Math. Soc. 118 (1965), 454–471. MR 0176983, https://doi.org/10.1090/S0002-9947-1965-0176983-8
[4] Heisuke Hironaka, On the equivalence of singularities. I, Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963) Harper & Row, New York, 1965, pp. 153–200. MR 0201433
[5] C. Hozuel, Séminaire Cartan 13ième année: 1960/61, Familles d'espaces complexes et fondements de la géométrie analytique, Exposés 18, 21, École Normale Supérieure, Secrétariat Mathématique, Paris, 1962. MR 26 #3562.
[6] Arnold Kas and Michael Schlessinger, On the versal deformation of a complex space with an isolated singularity, Math. Ann. 196 (1972), 23–29. MR 0294701, https://doi.org/10.1007/BF01419428
[7] Wolfgang Krull, Jacobsonsche Ringe, Hilbertscher Nullstellensatz, Dimensionstheorie, Math. Z. 54 (1951), 354–387 (German). MR 0047622, https://doi.org/10.1007/BF01238035
[8] Hideyuki Matsumura, Commutative algebra, W. A. Benjamin, Inc., New York, 1970. MR 0266911
[9] Michel Raynaud, Anneaux locaux henséliens, Lecture Notes in Mathematics, Vol. 169, Springer-Verlag, Berlin-New York, 1970 (French). MR 0277519
[10] Michael Schlessinger, Functors of Artin rings, Trans. Amer. Math. Soc. 130 (1968), 208–222. MR 0217093, https://doi.org/10.1090/S0002-9947-1968-0217093-3
[11] A. Seidenberg, Analytic products, Amer. J. Math. 91 (1969), 577–590. MR 0254048, https://doi.org/10.2307/2373339
[12] G. N. Tjurina, Locally semi-universal flat deformations of isolated singularities of complex spaces, Izv. Akad. Nauk SSSR Ser. Mat. 33 (1969), 1026–1058 (Russian). MR 0252685
[13] Jonathan M. Wahl, Equisingular deformations of plane algebroid curves, Trans. Amer. Math. Soc. 193 (1974), 143–170. MR 0419439, https://doi.org/10.1090/S0002-9947-1974-0419439-2
[14] Robert J. Walker, Algebraic Curves, Princeton Mathematical Series, vol. 13, Princeton University Press, Princeton, N. J., 1950. MR 0033083
[15] O. Zariski, Studies in equisingularity. I, II, III, Amer. J. Math. 87 (1965), 507-536, 972-1006; ibid. 90 (1968), 961-1023. MR 31 #2243; 33 #125; 38 #5775.
[16] O. Zariski, Contributions to the problem of equisingularity, Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969) Edizioni Cremonese, Rome, 1970, pp. 261–343. MR 0276240