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

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.