Inventiones mathematicae

, Volume 191, Issue 3, pp 671-718

First online:

Finite Hilbert stability of (bi)canonical curves

  • Jarod AlperAffiliated withDepartamento de Matemáticas, Universidad de los Andes
  • , Maksym FedorchukAffiliated withDepartment of Mathematics, Columbia University Email author 
  • , David Ishii SmythAffiliated withDepartment of Mathematics, Harvard University

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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.