Fine polar invariants of minimal singularities of surfaces

Romain Bondil

Journal of Singularities
volume 14 (2016), 91-112

Received: 18 August 2015. Received in revised form: 1 August 2016.

DOI: 10.5427/jsing.2016.14f

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We consider the polar curves P_{S,0} arising from generic projections of a germ (S,0) of a complex surface singularity onto 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 pieces of equisingularity data for P_{S,0} are described on the one hand by the geometry of the tangent cone of (S,0), and on the other hand by 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.


rational surface singularity, minimal singularity, polar curve, discriminant, limit tree, deformation, tangent cone, Scott deformation

Mathematical Subject Classification:

Primary: 32S15, 32S25, Secondary: 14H20, 14B07

Author(s) information:

Romain Bondil
Lycée Joffre
150 Allée de la citadelle
Université de Montpellier II
34060 Montpellier Cedex 02, FRANCE