Quasi-projective varieties whose fundamental group is a free product of cyclic groups
José I. Cogolludo-Agustín
Universidad de Zaragoza, Zaragoza, SpainEva Elduque
Universidad Autónoma de Madrid, Madrid, Spain

Abstract
In the context of Serre’s question, we study smooth complex quasi-projective varieties whose fundamental group is a free product of cyclic groups. In particular, we focus on the case of surfaces and prove the existence of an admissible map from such a quasi-projective surface to a smooth complex quasi-projective curve. Associated with this result, we prove addition-deletion lemmas which describe a natural operation correlating this family of quasi-projective surfaces and groups. Our methods also allow us to produce examples of curves in smooth projective surfaces whose complements have free products of cyclic groups as fundamental groups, generalizing classical results on curves and torus-type projective sextics, and describing the conditions under which this phenomenon occurs.
Cite this article
José I. Cogolludo-Agustín, Eva Elduque, Quasi-projective varieties whose fundamental group is a free product of cyclic groups. Rev. Mat. Iberoam. (2025), published online first
DOI 10.4171/RMI/1550