Deformations of formal embeddings of schemes
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 by Miriam P. Halperin PDF
 Trans. Amer. Math. Soc. 221 (1976), 303321 Request permission
Abstract:
A family of isolated singularities of kvarieties will be here called equisingular if it can be simultaneously resolved to a family of hypersurfaces embedded in nonsingular spaces which induce only locally trivial deformations of pairs of schemes over local artin kalgebras. The functor of locally trivial deformations of the formal embedding of an exceptional set has a versal object in the sense of Schlessinger. When the exceptional set ${X_0}$ is a collection of nonsingular curves meeting normally in a nonsingular surface X, the moduli correspond to Laufer’s moduli of thick curves. When X is a nonsingular scheme of finite type over an algebraically closed field k and ${X_0}$ is a reduced closed subscheme of X, every deformation of $(X,{X_0})$ to $k[\varepsilon ]$ such that the deformation of ${X_0}$ is locally trivial, is in fact a locally trivial deformation of pairs.References

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Additional Information
 © Copyright 1976 American Mathematical Society
 Journal: Trans. Amer. Math. Soc. 221 (1976), 303321
 MSC: Primary 14D15; Secondary 14E15
 DOI: https://doi.org/10.1090/S00029947197604070160
 MathSciNet review: 0407016