On the δ=const Collisions of Singularities of Complex Plane Curves

Abstract

We study a specific class of deformations of curve singularities: the case when the singular point splits to several ones, such that the total $\delta$ 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 new invariant of the singular type is introduced: the dual graph. It imposes severe restrictions on the possible collisions/deformations. And allows to prove some bounds on the variation of classical invariants in collisions. We consider in details the $\delta=const$ deformations of ordinary multiple point, the deformation of a singularity into the collection of ordinary multiple points and the deformation of the type $x^p+y^{pk}$ into a collection of $A_k$'s.