Resolutions of General Canonical Curves on Rational Normal Scrolls

Let $C\subset \mathbb{P}^{g-1}$ be a general curve of genus $g$ and let $k$ be a positive integer such that the Brill-Noether number $\rho(g,k,1)\geq 0$ and $g > k+1$. The aim of this short note is to study the relative canonical resolution of $C$ on a rational normal scroll swept out by a $g^1_k=|L|$ with $L\in W^1_k(C)$ general. We show that the bundle of quadrics appearing in the relative canonical resolution is unbalanced if and only if $\rho>0$ and $(k-\rho-\frac{7}{2})^2-2k+\frac{23}{4}>0$.