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Pencils and critical loci on normal surfaces

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Abstract

We study linear pencils of curves on normal surface singularities. Using the minimal good resolution of the pencil, we describe the topological type of generic elements of the pencil and characterize the behaviour of special elements. Furthermore, we show that the critical locus associated to the pencil is linked to the special elements. This gives a decomposition of the critical locus through the minimal good resolution and as a consequence, some information on the topological type of the critical locus.

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Acknowledgements

We thank both referees for their observations that helped us to improve the redaction of the paper. In particular, the remark at the end of the Introduction is mainly due to one of them.

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Authors and Affiliations

  1. IMUVA (Instituto de Investigación en Matemáticas), Universidad de Valladolid, Valladolid, Spain

    F. Delgado

  2. Institut Fourier, Université Grenoble-Alpes, 100 rue des maths, 38610, Gières, France

    H. Maugendre

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  1. F. Delgado
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  2. H. Maugendre
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Correspondence to H. Maugendre.

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F. Delgado: Supported by the Grants MTM2015-65764-C3-1-P and PGC2018-096446-B-C21, both with the help of FEDER Program. The author is thankful to the Institut Fourier, Université de Grenoble-Alpes for hospitality during the stages of this work. H. Maugendre: Supported by the ANR LISA Project ANR-17-CE40-0023. The author is thankful to University of Valladolid for hospitality during the stages of this work.

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Delgado, F., Maugendre, H. Pencils and critical loci on normal surfaces. Rev Mat Complut 34, 691–714 (2021). https://doi.org/10.1007/s13163-020-00366-8

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