Finite Hilbert stability of (bi)canonical curves

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 Gm-action, namely the canonically embedded balanced ribbon and the canonically embedded balanced double A2k+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.