Polar exploration of complex surface germs

Abstract

We prove that the topological type of a normal surface singularity ( X , 0 ) (X,0) provides finite bounds for the multiplicity and polar multiplicity of ( X , 0 ) (X,0) , as well as for the combinatorics of the families of generic hyperplane sections and of polar curves of the generic plane projections of ( X , 0 ) (X,0) . A key ingredient in our proof is a topological bound of the growth of the Mather discrepancies of ( X , 0 ) (X,0) , which allows us to bound the number of point blowups necessary to achieve factorization of any resolution of ( X , 0 ) (X,0) 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.