The Milnor number and deformations of complex curve singularities

Milnor Number of Weighted-Lê–Yomdin Singularities

On the Freeness of Equisingular Deformations of Plane Curve Singularities

We consider surface singularities in C3 arising as the total space of an equisingular deformation of an isolated curve singularity of the form f(x, y)+zg(x, y) with f and g weighted homogeneous. We give a criterion that such a surface is a free divisor in the sense of Saito. We deduce that the Hessian deformation defines a free divisor for nonsimple weighted homogeneous singularities, and that ...

Equisingular Deformations of Plane Curve and of Sandwiched Singularities

Let C be an isolated plane curve singularity, and (C, l) be a decorated curve. In this article we compare the equisingular deformations of C and the sandwiched singularity X(C, l). We will prove that for l ≫ 0 the functor of equisingular deformations of C and (C, l) are equivalent. From this we deduce a proof of a formula for the dimension of the equisingular stratum. Furthermore we will show h...

On the Equi - Normalizable Deformations of Singularities of Complex Plane

We study a specific class of deformations of curve singularities: the case when the singular point splits to several ones, such that the total δ invariant is preserved. These are also known as equi-normalizable or equi-generic deformations. We restrict primarily to the deformations of singularities with smooth branches. A natural invariant of the singular type is introduced: the dual graph. It ...

Bridge Number and the Curve Complex

We show that there are hyperbolic tunnel-number one knots with arbitrarily high bridge number and that “most” tunnelnumber one knots are not one-bridge with respect to an unknotted torus. The proof relies on a connection between bridge number and a certain distance in the curve complex of a genus-two surface.