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A case in which irreducibility of an analytic germ implies irreducibility of the tangent cone

Author: Richard Draper
Journal: Proc. Amer. Math. Soc. 39 (1973), 443-449
MSC: Primary 32B10; Secondary 32C40
MathSciNet review: 0316744
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Abstract: There are simple examples in which a variety is irreducible at a point but has a reducible tangent cone. The following theorem is proved. If $ {X_p}$ is an irreducible analytic germ and if the Jacobian ideal beomes principal on the normalization then the tangent cone of $ X$ at $ p$ is irreducible. If, moreover, the singular set of $ X$ is a manifold at $ p$ then $ X$ is Whitney $ a,b$-regular along the singular set at $ p$.

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  • [1] S. S. Abhyankar, Local analytic geometry, Pure and Appl. Math., vol. 14, Academic Press, New York, 1964. MR 31 #173. MR 0175897 (31:173)
  • [2] H. Hironaka, Normal cones in analytic Whitney stratifications, Inst. Hautes Études Sci. Publ. Math. No. 36 (1969), 127-138. MR 43 #3492. MR 0277759 (43:3492)
  • [3] -, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109-326. MR 33 #7333. MR 0199184 (33:7333)
  • [4] -, Seminar on singularities. I, Woods Hole Lecture Notes, 1964.
  • [5] John Stutz, Analytic sets as branched coverings, Trans. Amer. Math. Soc. 166 (1972), 241-261. MR 0324068 (48:2420)
  • [6] H. Whitney, Local properties of analytic varieties, Differential and Combinatorial Topology, Princeton Univ. Press, Princeton, N.J., 1965. MR 0188486 (32:5924)
  • [7] O. Zariski, Studies in equisingularity. II. Equisingularity in codimension 1 (and characteristic zero), Amer. J. Math. 87 (1965), 972-1006. MR 33 #125. MR 0191898 (33:125)

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Keywords: Tangent cone, Jacobian ideal, normalization, Whitney $ a,b$-regularity
Article copyright: © Copyright 1973 American Mathematical Society

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