Quasi-projective varieties whose fundamental group is a free product of cyclic groups

TL;DR

This paper investigates smooth complex quasi-projective surfaces with fundamental groups that are free products of cyclic groups, establishing new structural results and generalizing classical curve configurations.

Contribution

It proves the existence of admissible maps to curves and develops addition-deletion lemmas for fibers, expanding understanding of fundamental groups in algebraic geometry.

Findings

Existence of admissible maps from surfaces to curves.

Addition-deletion lemmas for fibers affecting fundamental groups.

Construction of curves with free product fundamental groups of complements.

Abstract

In this work we study smooth complex quasi-projective surfaces whose fundamental group is a free product of cyclic groups. In particular, we prove the existence of an admissible map from the quasi-projective surface to a smooth complex quasi-projective curve. Associated with this result, we prove addition-deletion Lemmas for fibers of the admissible map which describe how these operations affect the fundamental group of the quasi-projective surface. Our methods also allow us to produce curves in smooth projective surfaces whose fundamental groups of their complements are free products of cyclic groups, generalizing classical results on $C_{p,q}$ curves and torus type projective sextics, and showing how general this phenomenon is.