Finite Hilbert stability of (bi)canonical curves

Abstract

We prove that a generic canonically or bicanonically embedded smooth curve has semistable m-th Hilbert points for all m. We also prove that a generic bicanonically embedded smooth curve has stable m-th Hilbert points for all m \geq 3. In the canonical case, this is accomplished by proving finite Hilbert semistability of special singular curves with 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 m-th Hilbert points are non-semistable for low values of m, but become semistable past a definite threshold. (This paper subsumes the previous submission and arXiv:1110.5960).

To appear in Inventiones Mathematicae, 2012. The final publication is available at http://www.springerlink.com

References in corpus (3)

• Log minimal model program for the moduli space of stable curves: The first flip
• Alternate Compactifications of Moduli Spaces of Curves
• Singularities with G_m-action and the log minimal model program for ${\bar{M}}_g$

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