## Normally flat deformations

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- by Bruce Bennett
- Trans. Amer. Math. Soc.
**225**(1977), 1-57 - DOI: https://doi.org/10.1090/S0002-9947-1977-0498555-6
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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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## Bibliographic Information

- © Copyright 1977 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**225**(1977), 1-57 - MSC: Primary 14D15; Secondary 14B05
- DOI: https://doi.org/10.1090/S0002-9947-1977-0498555-6
- MathSciNet review: 0498555