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Transactions of the American Mathematical Society

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Analytic sets as branched coverings


Author: John Stutz
Journal: Trans. Amer. Math. Soc. 166 (1972), 241-259
MSC: Primary 32C40
DOI: https://doi.org/10.1090/S0002-9947-1972-0324068-3
MathSciNet review: 0324068
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Abstract: In this paper we study the relation between the tangent structure of an analytic set V at a point p and the local representation of V as a branched covering. A prototype for our type of result is the fact that one obtains a covering of minimal degree by projecting transverse to the Zariski tangent cone $ {C_3}(V,p)$. We show, for instance, that one obtains the smallest possible branch locus for a branched covering if one projects transverse to the cone $ {C_4}(V,p)$. This and similar results show that points where the various tangent cones $ {C_i}(V,p),i = 4,5,6$, have minimal dimension give rise to the simplest branched coverings. This observation leads to the idea of ``Puiseux series normalization", generalizing the situation in one dimension. These Puiseux series allow us to strengthen some results of Hironaka and Whitney on the local structure of certain types of singularities.


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  • [1] R. C. Gunning and H. Rossi, Analytic functions of several complex variables, Prentice-Hall, Englewood Cliffs, N. J., 1965. MR 31 #4927. MR 0180696 (31:4927)
  • [2] H. Whitney, Local properties of analytic varieties, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, N. J., 1965. MR 32 #5924. MR 0188486 (32:5924)
  • [3] -, Tangents to an anlytic variety, Ann. of Math. (2) 81 (1965), 496-549. MR 33 #745. MR 0192520 (33:745)
  • [4] R. Draper, Intersection theory in analytic geometry, Math. Ann. 180 (1969), 175-204. MR 40 #403. MR 0247134 (40:403)
  • [5] R. Narasimhan, Introduction to the theory of analytic spaces, Lecture Notes in Math., no. 25, Springer-Verlag, Berlin, 1966. MR 36 #428. MR 0217337 (36:428)
  • [6] S. Abhyankar, Local analytic geometry, Pure and Appl. Math., vol. 14, Academic Press, New York, 1964. MR 31 #173. MR 0175897 (31:173)
  • [7] O. Zariski, Equisingular points on algebraic varieties, Seminari 1962/63 Anal. Alg. Geom. e Topol., vol. 1, Ist. Naz. Alta Mat., Ediz. Cremonese, Rome, 1965, pp. 164-177. MR 33 #7336. MR 0199187 (33:7336)
  • [8] I. Kimura, On normal analytic sets. I, II, Proc. Japan Acad. 43 (1967), 464-468; ibid. 43 (1967), 719-722. MR 37 #1649. MR 0226059 (37:1649)
  • [9] M. Hervé, Math. Reviews 37 (1968), 309 (article #1649).
  • [10] A. Douady, Flatness and privilege, Enseignement Math. (2) 14 (1968), 47-74. MR 38 #4716. MR 0236420 (38:4716)
  • [11] H. Hironaka, Normal cones in analytic Whitney stratifications, Inst. Hautes Études Sci. Publ. Math., no. 31, 1970. MR 0277759 (43:3492)
  • [12] N. Kuhlmann, Algebraic function fields on complex analytic spaces, Proc. Conf. Complex Analysis (Minneapolis, 1964), Springer, Berlin, 1965, pp. 155-172. MR 30 #4977. MR 0174784 (30:4977)
  • [13] H. Hironaka, Resolution of singularities of algebraic varieties over fields of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109-203; 205-326. MR 33 #7333. MR 0199184 (33:7333)
  • [14] J. Stutz, Equisingularity and equisaturation in codimension 1, preprint. MR 0333240 (48:11565)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0324068-3
Keywords: Branched cover, tangent cones, Puiseux series, normal flatness, saturation
Article copyright: © Copyright 1972 American Mathematical Society

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