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Finite Hilbert stability of (bi)canonical curves

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Abstract

We prove that a generic canonically or bicanonically embedded smooth curve has semistable mth Hilbert points for all m≥2. We also prove that a generic bicanonically embedded smooth curve has stable mth Hilbert points for all m≥3. In the canonical case, this is accomplished by proving finite Hilbert semistability of special singular curves with \(\mathbb{G}_{m}\)-action, namely the canonically embedded balanced ribbon and the canonically embedded balanced double A 2k+1-curve. In the bicanonical case, we prove finite Hilbert stability of special hyperelliptic curves, namely Wiman curves. Finally, we give examples of canonically embedded smooth curves whose mth Hilbert points are non-semistable for low values of m, but become semistable past a definite threshold.

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad de los Andes, Cra 1 No. 18A-10, Edificio H, Bogotá, 111711, Colombia

    Jarod Alper

  2. Department of Mathematics, Columbia University, 2990 Broadway, New York, NY, 10027, USA

    Maksym Fedorchuk

  3. Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, MA, 01238, USA

    David Ishii Smyth

Authors
  1. Jarod Alper
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  2. Maksym Fedorchuk
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  3. David Ishii Smyth
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Correspondence to Maksym Fedorchuk.

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To Joe Harris on his sixtieth birthday

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Alper, J., Fedorchuk, M. & Smyth, D.I. Finite Hilbert stability of (bi)canonical curves. Invent. math. 191, 671–718 (2013). https://doi.org/10.1007/s00222-012-0403-6

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  • DOI: https://doi.org/10.1007/s00222-012-0403-6

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