Deformations of formal embeddings of schemes

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.