Planar curve singularities having constant intersection multiplicity with a smooth boundary

Abstract

We study deformations of curve singularities in a smooth surface having a constant local intersection multiplicity $w$ with a smooth boundary curve. In particular, we consider the codimension of the equisingular locus in the semiuniversal deformation space and prove results on the inclusion relations between ideals describing different kinds of deformations. A classification is given of singularities expected to appear in codimension $\le 3$ in a general family.

v3: Title changed. Reference added to literature on relative singularities. Full classification of $C$ with ${\tau }_{\log }^{\mathrm{es}}(C)\le 3$