On the equi-normalizable deformations of singularities of complex plane curves

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 imposes severe restrictions on the possible collisions/deformations. And allows to prove some bounds on the variation of classical invariants in equi-normalizable families. We consider in details deformations of ordinary multiple point, the deformations of a singularity into the collections of ordinary multiple points and deformations of the type x ^p  + y ^pk into the collections of A _k ’s. The research was constantly supported by the Skirball postdoctoral fellowship of the Center of Advanced Studies in Mathematics (Mathematics Department of Ben Gurion University, Israel). Part of the work was done in Mathematische Forschungsinsitute Oberwolfach, during the author’s stay as an OWL-fellow. Some results were published in the preprint 17].