Quasi-projective varieties with orbifold fundamental groups

José Ignacio Cogolludo-Agustín, Eva Elduque

Published 2022-10-31Version 1

In this work we study smooth complex quasi-projective varieties whose fundamental group is a free product of cyclic groups. In particular, we prove the existence of a (maybe irrational) pencil from the quasi-projective variety to a Riemann surface (endowed with an orbifold structure) which induces an isomorphism of (orbifold) fundamental groups. The orbifold structure is given by the multiple fibers of the pencil. Associated with this result, we prove addition-deletion Lemmas of fibers in the pencil which explain how these operations affect the fundamental group of the quasi-projective variety. Our methods also allow us to produce curves whose fundamental group of their complement is a free product of cyclic groups, generalizing classical results on $C_{p,q}$ curves and torus type projective sextics, and showing how general this phenomenon is.