Fine polar invariants of minimal singularities of surfaces

Abstract

We consider the polar curves PS,0 arising from generic projections of a germ (S, 0) of complex surface singularity onto C. Taking (S, 0) to be a minimal singularity of normal surface (i.e. a rational singularity with reduced tangent cone), we give the δ-invariant of these polar curves, as well as the equisingularity-type of their generic plane projections, which are also the discriminants of generic projections of (S, 0). These two (equisingularity)-data for PS,0 are described in term, on the one side of the geometry of the tangent cone of (S, 0) and on the other side of the limit-trees introduced by T. de Jong and D. van Straten for the deformation theory of these minimal singularities. These trees give a combinatorial device for the description of the polar curve which makes it much clearer than in our previous Note on the subject. This previous work mainly relied on a result of M. Spivakovsky. Here we give a geometrical proof via deformations (on the tangent cone, and what we call Scott deformations) and blow-ups, although we need Spivakovsky’s result at some point, extracting some other consequences of it along the way.