Analytic sets as branched coverings

Stutz, John

doi:10.1090/S0002-9947-1972-0324068-3

Analytic sets as branched coverings
HTML articles powered by AMS MathViewer

by John Stutz

Trans. Amer. Math. Soc. 166 (1972), 241-259

DOI: https://doi.org/10.1090/S0002-9947-1972-0324068-3

PDF | Request permission

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.

References

Robert C. Gunning and Hugo Rossi, Analytic functions of several complex variables, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. MR 0180696
Hassler Whitney, Local properties of analytic varieties, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, N.J., 1965, pp. 205–244. MR 0188486
Hassler Whitney, Tangents to an analytic variety, Ann. of Math. (2) 81 (1965), 496–549. MR 192520, DOI 10.2307/1970400
Richard N. Draper, Intersection theory in analytic geometry, Math. Ann. 180 (1969), 175–204. MR 247134, DOI 10.1007/BF01350737
Raghavan Narasimhan, Introduction to the theory of analytic spaces, Lecture Notes in Mathematics, No. 25, Springer-Verlag, Berlin-New York, 1966. MR 0217337, DOI 10.1007/BFb0077071
Shreeram Shankar Abhyankar, Local analytic geometry, Pure and Applied Mathematics, Vol. XIV, Academic Press, New York-London, 1964. MR 0175897
Oscar 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 0199187
Ikuo Kimura, On normal analytic sets. I, II, Proc. Japan Acad. 43 (1967), 464–468; 43 (1967), ibid. 719–722. MR 0226059

37

A. Douady, Flatness and privilege, Enseign. Math. (2) 14 (1968), 47–74. MR 236420
Heisuke Hironaka, Normal cones in analytic Whitney stratifications, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 127–138. MR 277759, DOI 10.1007/BF02684601
N. Kuhlmann, Algebraic function fields on complex analytic spaces, Proc. Conf. Complex Analysis (Minneapolis, 1964) Springer, Berlin, 1965, pp. 155–172. MR 0174784
Heisuke Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109–203; 79 (1964), 205–326. MR 0199184, DOI 10.2307/1970547
John Stutz, Equisingularity and equisaturation in codimension $1$, Amer. J. Math. 94 (1972), 1245–1268. MR 333240, DOI 10.2307/2373573