Normally flat deformations
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 by Bruce Bennett PDF
 Trans. Amer. Math. Soc. 225 (1977), 157 Request permission
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 HilbertSamuel 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 conein 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.References

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Additional Information
 © Copyright 1977 American Mathematical Society
 Journal: Trans. Amer. Math. Soc. 225 (1977), 157
 MSC: Primary 14D15; Secondary 14B05
 DOI: https://doi.org/10.1090/S00029947197704985556
 MathSciNet review: 0498555