Finite Hilbert stability of (bi)canonical curves

We prove that a generic canonically or bicanonically embedded smooth curve has semistable mth Hilbert points for all ma parts per thousand yen2. We also prove that a generic bicanonically embedded smooth curve has stable mth Hilbert points for all ma parts per thousand yen3. In the canonical case, this is accomplished by proving finite Hilbert semistability of special singular curves with -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.