Abstract

A complex plane curve singularity determines, by taking the boundary of a small neighborhood, an iterated torus link in the three-sphere. The algebraic geometry of the singularity and the topology of the link are intimately related; for instance, Zariski showed that the series of blowups needed to resolve the singularity carries data equivalent to the isotopy class of the link. More recently, Campillo, Delgado, and Gusein-Zade gave a formula equating the Alexander polynomial of the link to a generating series populated by Euler characteristics of spaces of functions defined at the singularity. I will state a conjectural generalization of their formula: the HOMFLY invariant of the link of a plane curve singularity is a generating function of Euler characteristics of moduli spaces of schemes supported at the singularity. I will discuss the evidence for the conjecture, and show that it holds for torus knots. The talk presents joint work with Alexei Oblomkov.