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Cremona Transformations of Weighted Projective Planes, Zariski Pairs, and Rational Cuspidal Curves

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Singularities and Their Interaction with Geometry and Low Dimensional Topology

Part of the book series: Trends in Mathematics ((TM))

Abstract

In this work, we study a family of Cremona transformations of weighted projective planes which generalize the standard Cremona transformation of the projective plane. Starting from special plane projective curves we construct families of curves in weighted projective planes with special properties. We explain how to compute the fundamental groups of their complements, using the blow-up-down decompositions of the Cremona transformations, we find examples of Zariski pairs in weighted projective planes (distinguished by the Alexander polynomial). As another application of this machinery we study a family of singularities called weighted Lê–Yomdin, which provide infinitely many examples of surface singularities with a rational homology sphere link. To end this paper we also study a family of surface singularities generalizing Brieskorn–Pham singularities in a different direction. This family contains infinitely many examples of integral homology sphere links, answering a question by Némethi.

To András Némethi, source of inspiration in singularity theory

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Acknowledgements

We thank the anonymous referee for their valuable suggestions which have helped to improve the manuscript.

Partially supported by MTM2016-76868-C2-2-P and Gobierno de Aragón (Grupo de referencia “Álgebra y Geometría”), E22_17R and E22_20R, cofunded by Feder 2014–2020 “Construyendo Europa desde Aragón”. The third author is also partially supported by FQM-333 “Junta de Andalucía”.

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Authors and Affiliations

  1. Departamento de Matemáticas, IUMA, Universidad de Zaragoza, Zaragoza, Spain

    Enrique Artal Bartolo & José I. Cogolludo-Agustín

  2. Centro Universitario de la Defensa, IUMA, Academia General Militar, Zaragoza, Spain

    Jorge Martín-Morales

Authors
  1. Enrique Artal Bartolo
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  2. José I. Cogolludo-Agustín
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  3. Jorge Martín-Morales
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Corresponding author

Correspondence to Enrique Artal Bartolo .

Editor information

Editors and Affiliations

  1. Basque Center for Applied Mathematics Ikerbasque (Basque Foundation for Science), Bilbao, Spain

    Javier Fernández de Bobadilla

  2. Faculty of Mathematics and Computer Science, Babeş-Bolyai University, Cluj-Napoca, Romania

    Tamás László

  3. Alfréd Rényi Institute of Mathematics, Budapest, Hungary

    András Stipsicz

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Bartolo, E.A., Cogolludo-Agustín, J.I., Martín-Morales, J. (2021). Cremona Transformations of Weighted Projective Planes, Zariski Pairs, and Rational Cuspidal Curves. In: Fernández de Bobadilla, J., László, T., Stipsicz, A. (eds) Singularities and Their Interaction with Geometry and Low Dimensional Topology . Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-61958-9_7

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