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


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.


Modulus Space Monomial Basis Stable Curf Geometric Invariant Theory Irreducible Decomposition 
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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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