Syzygies of canonical ribbons on higher genus curves

We study the syzygies of the canonical embedding of a ribbon $\tilde{C}$ on a curve $C$ of genus $g\ge 1$. We show that the linear series Clifford index and the resolution Clifford index are equal for a general ribbon of arithmetic genus $p_a$ on a general curve of genus $g$ with $p_a\ge \max \{3g+7,6g-4\}$. Among non-general ribbons, the case of split ribbons is particularly interesting. Equality of the two Clifford indices for a split ribbon is related to the gonality conjecture for $C$ and it implies Green’s conjecture for all double covers $C^{\mathrm{\prime }}$ of $C$ with $g(C^{\mathrm{\prime }})\ge \text{max}\{3g+2,6g-4\}$. We reduce it to the vanishing of certain Koszul cohomology groups of an auxiliary module of syzygies associated to $C$, which may be of independent interest.