Resolutions of general canonical curves on rational normal scrolls

Let C subset of Pg-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) >= 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(k)(1) = vertical bar L vertical bar with L is an element of W-k(1)(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 - 7/2)(2) - 2k + 23/4 > 0.