Polar exploration of complex surface germs

Abstract

We prove that the topological type of a normal surface singularity pX, 0q provides finite bounds for the multiplicity and polar multiplicity of pX, 0q, as well as for the combinatorics of the families of generic hyperplane sections and of polar curves of the generic plane projections of pX, 0q. A key ingredient in our proof is a topological bound of the growth of the Mather discrepancies of pX, 0q, which allows us to bound the number of point blowups necessary to achieve factorization of any resolution of pX, 0q through its Nash transform. This fits in the program of polar explorations, the quest to determine the generic polar variety of a singular surface germ, to which the final part of the paper is devoted.