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Transactions of the American Mathematical Society

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Deformations of formal embeddings of schemes

Author: Miriam P. Halperin
Journal: Trans. Amer. Math. Soc. 221 (1976), 303-321
MSC: Primary 14D15; Secondary 14E15
MathSciNet review: 0407016
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Abstract: A family of isolated singularities of k-varieties 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 k-algebras. 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.

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Keywords: Equisingular deformation, deformation of a pair, formal moduli, formally embedded exceptional set, locally complete intersection
Article copyright: © Copyright 1976 American Mathematical Society