Analytic Singularities

Abstract

1. let X be a “space” and $\left({{\rm{X}}_{\rm{t}}}\right)_{{\rm{t\varepsilon T}}}$ a “continuous family of subSpaces of X ”; in general that means that one has a “total space” X̃, a “moduli (or parameter) space” T and morphisms $\widetilde{\rm{X}}\xrightarrow{\pi}{\rm{T}}$, $\widetilde{\rm{X}}\xrightarrow{\varphi}{\rm{X}}$ where the family $\left({{\rm{X}}_{\rm{t}}}\right)_{{\rm{t\varepsilon T}}}$ is the family of the fibres (π⁻¹(t))_t∈t of π and for each t ∈ T, the restriction of φ to X_t. is an embedding of X_t in X. Obviously the meaning of the words subspace, fibre, embedding has to be specified depending on the geometric context (algebraic or analytic geometry, differential topology etc…) in which one is working. What happens in general is that there exists a closed subset Δ of T (the “discriminant locus”) s.th. locally on T − Δ, π can be viewed as the projection map of a product space; in particular on each connected component of T − Δ, the fibres X_t, are equivalent to each other.