The Harder-Narasimhan Filtration of a Trigonal Canonical Curve

A trigonal canonical curve lies on a rational normal surface scroll $Q\subset {\mathbb{P}}^{g-1}$. In this note we use this fact to compute the Harder-Narasimhan filtration of the normal bundle of a general such curve $C$ in ${\mathbb{P}}^{g-1}$. We also compute the Harder-Narasimhan filtration of the Normal bundle of a general canonical curve of genus $6$.