Stability of 2nd Hilbert points of canonical curves

TL;DR

This paper proves the GIT semistability of the second Hilbert point for general canonical curves, providing bounds on slopes of Gorenstein curves and exploring the role of hyperelliptic curves in GIT quotients.

Contribution

It establishes GIT semistability for 2nd Hilbert points of general canonical curves using a simple geometric approach.

Findings

GIT semistability of 2nd Hilbert points for general canonical curves

Upper bounds on slopes of Gorenstein curve families

Insights into the role of hyperelliptic curves in GIT quotients

Abstract

We establish 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 the GIT quotients of the Hilbert scheme of canonical curves.