  We investigate the resolution of a general branched cover $\alpha : C \to
{\bf P}^1$ in its relative canonical embedding $C \subset {\bf P} E$. We
conjecture that the syzygy bundles appearing in the resolution are balanced for
a general cover, provided that the genus is sufficiently large compared to the
degree. We prove this for the Casnati-Ekedahl bundle, or "bundle of quadrics"
$F$ - the first bundle appearing in the resolution of the ideal of the relative
canonical embedding. Furthermore, we prove the conjecture for all syzygy
bundles in the resolution when the genus satisfies $g = 1 \mod d$.
