Polar exploration of complex surface germs

TL;DR

This paper establishes that the topological type of a normal surface singularity bounds key invariants like multiplicity and polar multiplicity, advancing the understanding of the surface's geometric and topological properties.

Contribution

It introduces a topological bound on Mather discrepancies and relates it to the bounds on resolution complexity and polar varieties of surface germs.

Findings

Bound on multiplicity and polar multiplicity from topology

Finite bounds on resolution blowups related to Nash transform

Advancement in polar exploration of surface singularities

Abstract

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