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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Normally flat deformations

Author: Bruce Bennett
Journal: Trans. Amer. Math. Soc. 225 (1977), 1-57
MSC: Primary 14D15; Secondary 14B05
MathSciNet review: 0498555
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Abstract: We study flat families $ Z/T$, together with a section $ \sigma :T \to Z$ such that the normal cone to the image of $ \sigma $ in Z is flat over T. Such a family is called a ``normally flat deformation (along $ \sigma $)"; it corresponds intuitively to a deformation of a singularity which preserves the Hilbert-Samuel function. We construct the versal normally flat deformation of an isolated singularity (X,x) in terms of the flat strata of the relative jets of the ``usual'' versal deformation of X. We give explicit criteria, in terms of equations, for a flat family to be normally flat along a given section. These criteria are applied to demonstrate the smoothness of normally flat deformation theoryand of the canonical map from it to the cone deformation theory of the tangent cone-in the case of strict complete intersections. Finally we study the tangent space to the normally flat deformation theory, expressing it as the sum of two spaces: The first is a piece of a certain filtration of the tangent space to the usual deformation theory of X; the second is the tangent space to the special fibre of the canonical map $ N \to S$, where N (resp. S) is the parameter space for the versal normally flat deformation of (X, x) (resp. for the versal deformation of X). We discuss the relation of this second space to infinitesimal properties of sections.

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  • [1] A. Grothendieck and J. Dieudonné, Éléments de géométrie algébrique, Inst. Hautes Études Sci. Publ. Math. No. 32 (1967). MR 39 #220.
  • [2] D. Mumford, Lectures on curves on an algebraic surface, Ann. of Math. Studies, no. 59, Princeton Univ. Press, Princeton, N. J., 1966. MR 35 #187. MR 0209285 (35:187)
  • [3] -, Further pathologies in algebraic geometry, Amer. J. Math. 84 (1962), 642-648. MR 26 #6177. MR 0148670 (26:6177)
  • [4] M. Schlessinger, Functors of Artin rings, Trans. Amer. Math. Soc. 130 (1968), 208-222. MR 36 #184. MR 0217093 (36:184)
  • [5] -, Thesis, Harvard Univ., Cambridge, Mass., 1964.
  • [6] B. M. Bennett, On the characteristic functions of a local ring, Ann. of Math. (2) 91 (1970), 25-87. MR 40 #5608. MR 0252388 (40:5608)
  • [7] M. Lejeune-Jalabert and B. Teissier, Normal cones and sheaves of relative jets, Thèses, École Polytechnique.
  • [8] H. Hironaka, Resolution of singularities of an algebraic variety over afield of characteristic 0. I, II, Ann. of Math. (2) 79 (1964), 109-203, 205-326. MR 33 #7333. MR 0199184 (33:7333)
  • [9] B. Teissier, Deformations à type topologique constant. I, II, Séminaire Douady-Verdier, Secrétariat Mathématique, Paris, 1972.
  • [10] J. Wahl, Deformations of branched covers and equisingularity, Thesis, Harvard Univ., Cambridge, Mass., 1971.
  • [11] F. Pham, Classification des singularités, Université de Nice (mimeographed notes).
  • [12] H. Hironaka, Additive groups associated with points of a projective space, Ann. of Math. (2) 92 (1970), 327-334. MR 42 #4553. MR 0269658 (42:4553)
  • [13] S. Lichtenbaum and M. Schlessinger, The cotangent complex of a morphism, Trans. Amer. Math. Soc. 128 (1967), 41-70. MR 35 #237. MR 0209339 (35:237)
  • [14] B. Bennett, Normalization theorems for certain modular discriminantal loci, Compositio Math. 32 (1976), fasc. 1, 13-32. MR 0399096 (53:2947)
  • [15] J. Giraud, Sur la théorie du contact maximal. Math. Z. 137 (1974), 285-310. MR 0460712 (57:705)
  • [16] -, Contact maximal en caracteristique positive (preprint).

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Article copyright: © Copyright 1977 American Mathematical Society

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