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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Abstract:

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.

Keywords:

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
email: romain.bondil@ac-montpellier.fr