On the [delta] 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 ± 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 ± = const deformations of ordinary multiple point, the deformation of a singularity into the collection of ordinary multiple points and the deformation of the type xp + ypk into a collection of Ak's.