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We establish GIT semistability of the 2nd Hilbert point of everyGieseker-Petri general canonical curve by a simple geometric argument. As aconsequence, we obtain an upper bound on slopes of general families ofGorenstein curves. We also explore the question of what replaces hyperellipticcurves in the GIT quotients of the Hilbert scheme of canonical curves.

Stability of 2nd Hilbert points of canonical curves