Fine polar invariants of minimal singularities of surface

We consider the polar curves $\text{PSO}$ arising from generic projections of a germ $(S,0)$ of complex surface singularity onto ${\text{C}}^2$. Taking $(S,0)$ to be a minimal singularity of normal surface (i.e. a rational singularity with reduced tangent cone), we give the $\delta$-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 $\text{PSO}$ 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.