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Finite presentation of the tame fundamental group

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Abstract

Let p be a prime number, and let k be an algebraically closed field of characteristic p. We show that the tame fundamental group of a smooth affine curve over k is a projective profinite group. We prove that the fundamental group of a smooth projective variety over k is finitely presented; more generally, the tame fundamental group of a smooth quasi-projective variety over k, which admits a good compactification, is finitely presented.

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Acknowledgements

We thank Marco D’Addezio for helpful correspondence, and for pointing us to [7]. After posting a first draft of our article, we received various comments, notably on references. We thank Mikhail Borovoi, Luc Illusie and Fabrice Orgogozo for their kind help. We thank the referee for the kind and helpful report.

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Authors and Affiliations

  1. Freie Universität Berlin, Arnimallee 3, 14195, Berlin, Germany

    Hélène Esnault

  2. Mathematics, Harvard University, 1 Oxford Street, Cambridge, MA, 02138, USA

    Mark Shusterman

  3. School of Mathematics, TIFR, Homi Bhabha Road, Mumbai, 400005, India

    Vasudevan Srinivas

Authors
  1. Hélène Esnault
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  2. Mark Shusterman
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  3. Vasudevan Srinivas
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Correspondence to Mark Shusterman.

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The third author was supported during part of the preparation of the article by a J. C. Bose Fellowship of the Department of Science and Technology, India. He also acknowledges support of the Department of Atomic Energy, India under project number RTI4001.

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Esnault, H., Shusterman, M. & Srinivas, V. Finite presentation of the tame fundamental group. Sel. Math. New Ser. 28, 37 (2022). https://doi.org/10.1007/s00029-021-00732-4

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