Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Normally flat deformations
HTML articles powered by AMS MathViewer

by Bruce Bennett PDF
Trans. Amer. Math. Soc. 225 (1977), 1-57 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 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.
References
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC: 14D15, 14B05
  • Retrieve articles in all journals with MSC: 14D15, 14B05
Additional 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