Inventiones mathematicae

, Volume 191, Issue 3, pp 671–718

Finite Hilbert stability of (bi)canonical curves

Authors

  • Jarod Alper
    • Departamento de MatemáticasUniversidad de los Andes
    • Department of MathematicsColumbia University
  • David Ishii Smyth
    • Department of MathematicsHarvard University
Article

DOI: 10.1007/s00222-012-0403-6

Cite this article as:
Alper, J., Fedorchuk, M. & Smyth, D.I. Invent. math. (2013) 191: 671. doi:10.1007/s00222-012-0403-6

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 doubleA2k+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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© Springer-Verlag 2012