Fundamental group for the complement of the Cayley's singularities

TL;DR

This paper demonstrates how to compute the fundamental group of the complement of singularities on a surface using braid monodromy techniques, exemplified through the Cayley cubic, marking a pioneering application of these methods.

Contribution

It introduces a novel application of braid monodromy techniques to compute the fundamental group of a singular surface's complement, specifically for the Cayley cubic.

Findings

Successfully computed the fundamental group for the Cayley cubic.

Established braid monodromy as a viable tool for fundamental group calculations.

Provided one of the first examples of this technique's application to such problems.

Abstract

Given a singular surface X, one can extract information on it by investigating the fundamental group $\pi_1(X - Sing_X)$. However, calculation of this group is non-trivial, but it can be simplified if a certain invariant of the branch curve of X - called the braid monodromy factorization - is known. This paper shows, taking the Cayley cubic as an example, how this fundamental group can be computed by using braid monodromy techniques and their liftings. This is one of the first examples that uses these techniques to calculate this sort of fundamental group.