Stability of 2nd hilbert points of canonical curves

We establish Geometric Invariant Theory (GIT) semistability of the 2nd Hilbert point of every Gieseker-Petri general canonical curve by a simple geometric argument. As a consequence, we obtain an upper bound on slopes of general families of Gorenstein curves. We also explore the question of what replaces hyperelliptic curves in GIT quotients of the Hilbert scheme of canonical curves.