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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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References

Arapura, D., Geometry of cohomology support loci for local systems, I. J. Algebraic Geom. 6(3) (1997), 563–597.Google Scholar
Artal, E., Cogolludo-Agustín, J. I. and Matei, D., Characteristic varieties of quasi-projective manifolds and orbifolds, Geom. Topol. 17 (2013), 273–309.CrossRefGoogle Scholar
Debarre, O., Introduction to Mori theory (Universite Paris Diderot, 2016).Google Scholar
Dolgachev, I., Classical algebraic geometry. A modern view (Cambridge University Press, Cambridge, 2012).CrossRefGoogle Scholar
Ellenberg, J. and Venkatesh, A., The number of extensions of a number field with fixed degree and bounded discriminant, Ann. Math. 163(2) (2006), 723–741.CrossRefGoogle Scholar
Falk, M. and Yuzvinsky, S., Multinets, resonance varieties, and pencils of plane curves, Compos. Math. 143(4) (2007), 1069–1088.CrossRefGoogle Scholar
Iversen, B., Critical points of an algebraic function, Invent. Math. 12 (1971), 210–224.CrossRefGoogle Scholar
Kollr, J., Rational curves on algebraic varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 32 (Springer-Verlag, Berlin, 1996).Google Scholar
Lazarsfeld, R., Positivity in algebraic geometry. I. Classical setting: line bundles and linear series. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 48 (Springer-Verlag, Berlin, 2004).Google Scholar
Libgober, A. and Yuzvinsky, S., Cohomology of the Orlik-Solomon algebras and local systems, Compositio Math. 121(3) (2000), 337–361.CrossRefGoogle Scholar
Malle, G., On the distribution of Galois groups, J. Number Theory 92(2) (2002), 315–329.CrossRefGoogle Scholar
Miranda, R., Persson's list of singular fibers for a rational elliptic surface, Math. Z. 205(2) (1990), 191–211.CrossRefGoogle Scholar
Pereira, J. V. and Yuzvinsky, S., Completely reducible hypersurfaces in a pencil, Adv. Math. 219(2) (2008), 672–688.10.1016/j.aim.2008.05.014CrossRefGoogle Scholar
Ruppert, W., Reduzibilität ebener Kurven, J. Reine Angew. Math. 369 (1986), 167–191.Google Scholar
Segre, B., On arithmetic properties of quartic surfaces, Proc. London Math. Soc. 49(2) (1947), 353–395.Google Scholar
Serre, J. P., Groupes algbriques et corps de classes, Vol. VII (Hermann, Paris, Publications de l'institut de mathmatique de l'universit de Nancago, 1959).Google Scholar
Shafarevich, I., et al. Algebraic Surfaces, Proc. Steklov Institute of Math. 75 (1965), 3–215. Translated by American Mathematical Society. Rhode Island, 1967.Google Scholar
Silverman, J., The arithmetic of elliptic curves, 2nd edn. Graduate Texts in Mathematics, 106 (Springer, Dordrecht, 2009).CrossRefGoogle Scholar
Yuzvinsky, S., A new bound on the number of special fibers in a pencil of curves, Proc. Amer. Math. Soc. 137(5) (2009), 1641–1648.CrossRefGoogle Scholar
Zariski, O., Algebraic surfaces. With appendices by S.S. Abhyankar, J. Lipman and D. Mumford. Preface to the appendices by Mumford. Reprint of the second (1971) edition. Classics in Mathematics (Springer-Verlag, Berlin, 1995).Google Scholar