Equisingular and equinormalizable deformations of isolated non-normal singularities

Abstract

We present new results on equisingularity and equinormalizability of families with isolated non-normal singularities (INNS) of arbitrary dimension. We define a δ-invariant and a µ-invariant for an INNS and prove necessary and sufficient numerical conditions for equinormalizability and weak equinormalizability using δ and µ. For families of generically reduced curves, we investigate the topological behavior of the Milnor fibre and characterize topological triviality of such families. Finally we state some open problems and conjectures. In addition we give a survey of classical results about equisingularity and equinormalizability so that the article may be useful as a reference source.

Whitney equisingularity in families of generically reduced curves

“…Following [3, p. 248] and [6,Appendix], we recall briefly the notion of a good representative for a family of generically reduced curves p : (X, 0) → (C, 0). If T ⊂ D(0, η) is a closed ball around 0 of radius η( ) and X ⊂ B(0, ) × T is a closed subspace representing (X, 0), and if is sufficiently small and η( ) is sufficiently small with respect to , then p : X → T is called a good representative of p : (X, 0) → (C, 0) (see Figure 1).…”

“…The following Example shows that the hypothesis about the connectivity of the Milnor fiber X t in Theorem 4.3(c) is in fact necessary. where O 6 C{x, y, z, w, v, t}. We have that (X, 0) is a pure dimensional reduced germ of surface with 3 irreducible components (X 1 , 0), (X 2 , 0) and (X 3 , 0) defined by the following ideals…”

“…(a) Suppose first that X is irreducible. Since X is topologically trivial, we have that X admits a weak simultaneous resolution (see [6,Def. 2.1 and Th. 9.3]) n : D × T → X ⊂ B × T which is also the normalization of (X, 0).…”

“…In [9], Công-Trình Lê studied topological triviality of such families comparing it to the constancy of Milnor number; the full statement was proved by Greuel in [6,Th. 9.3].…”

Superfícies com singularidades não isoladas

“…Também estudamos o caso especial de famílias de superfícies em C 3 e o caso de famílias de n-espaços analíticos em C 2n−1 , com n ≥ 3, ambos parametrizados por germes de aplicações A-finitamente determinados f : (C n , 0) → (C 2n−1 , 0). Em [15], Lê Công-Trình descreveu invariantes que controlam a trivialidade topológica de uma família de curvas genericamente reduzidas (veja também [22]). Um dos nossos objetivos é estudar invariantes que controlam a Whitney equisingularidade de famílias de curvas genericamente reduzidas.…”