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Free quotients of fundamental groups of smooth quasi-projective varieties

Part of: (Co)homology theory Curves Cycles and subschemes

Published online by Cambridge University Press:  27 October 2021

Jose I. Cogolludo  and
Anatoly Libgober
Jose I. Cogolludo
Affiliation:
Departamento de Matemáticas, IUMA Universidad de Zaragoza, C. Pedro Cerbuna 12, 50009Zaragoza, Spain(jicogo@unizar.es)
Anatoly Libgober
Affiliation:
Department of Mathematics, University of Illinois, Chicago, IL60607, USA(libgober@uic.edu)
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Abstract

We study the fundamental groups of the complements to curves on simply connected surfaces, admitting non-abelian free groups as their quotients. We show that given a subset of the Néron–Severi group of such a surface, there are only finitely many classes of equisingular isotopy of curves with irreducible components belonging to this subset for which the fundamental groups of the complement admit surjections onto a free group of a given sufficiently large rank. Examples of subsets of the Néron–Severi group are given with infinitely many isotopy classes of curves with irreducible components from such a subset and fundamental groups of the complements admitting surjections on a free group only of a small rank.

Keywords

quasi-projective groupspencils of divisorseffective cone

MSC classification

Primary: 14F17: Vanishing theorems 14H30: Coverings, fundamental group 14H50: Plane and space curves
Secondary: 14C21: Pencils, nets, webs
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 64 , Issue 4 , November 2021 , pp. 924 - 946
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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