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Inventiones mathematicae

, Volume 191, Issue 3, pp 671–718 | Cite as

Finite Hilbert stability of (bi)canonical curves

  • Jarod Alper
  • Maksym FedorchukEmail author
  • David Ishii Smyth
Article

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.

Keywords

Modulus Space Monomial Basis Stable Curf Geometric Invariant Theory Irreducible Decomposition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Jarod Alper
    • 1
  • Maksym Fedorchuk
    • 2
    Email author
  • David Ishii Smyth
    • 3
  1. 1.Departamento de MatemáticasUniversidad de los AndesBogotáColombia
  2. 2.Department of MathematicsColumbia UniversityNew YorkUSA
  3. 3.Department of MathematicsHarvard UniversityCambridgeUSA

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