Whitney equisingularity in families of generically reduced curves

Abstract

In this work we study equisingularity in a one-parameter flat family of generically reduced curves. We consider some equisingular criteria as topological triviality, Whitney equisingularity and strong simultaneous resolution. In this context, we prove that Whitney equisingularity is equivalent to strong simultaneous resolution and it is also equivalent to the constancy of the Milnor number and the multiplicity of the fibers. These results are extensions to the case of flat deformations of generically reduced curves, of known results on reduced curves. When the family (X, 0) is topologically trivial, we also characterize Whitney equisingularity through Cohen-Macaulay property of a certain local ring associated to the parameter space of the family.

On the fifth Whitney cone of a complex analytic curve

“…Consider a flat family of reduced curves p : (X, 0) → (C, 0) and let p : X → T be a good representative (see [2, p. 248], see also [11,Def. 2.2]).…”

On Invariants of Generic Slices of Weighted Homogeneous Corank 1 Map Germs from the Plane to 3-Space

“…If this is the case, we say that f (D( f ) i ) is a generically reduced curve. Recently, Snoussi and the author showed in Silva and Snoussi (2019), Lemma 4.8 that if (C, 0) is a germ of generically reduced curve and (|C|, 0) is its associated reduced curve, then the multiplicities of (C, 0) and (|C|, 0) at 0 are equal. Hence, we also can calculate the multiplicity of f (D( f ) i ) considering its reduced structure and using a corresponding Puiseux parametrization for it.…”

A formula to calculate the invariant $J$ of a quasi-homogeneous map germ

“…If this is the case, we say that f (D(f ) i ) is a generically reduced curve. Recently, the author and Snoussi showed in [16,Lemma 4.8] that if (C, 0) is a germ of generically reduced curve and (|C|, 0) is its associated reduced curve, then the multiplicities of (C, 0) and (|C|, 0) at 0 are equal. Hence, we also can calculate the multiplicity of f (D(f ) i ) considering its reduced structure and using a corresponding Puiseux parametrization for it.…”