Syzygies of canonical ribbons on higher genus curves

TL;DR

This paper investigates the syzygies of canonical ribbons on higher genus curves, establishing conditions under which Clifford indices coincide and exploring implications for Green's conjecture and gonality conjecture.

Contribution

It demonstrates the equality of linear series Clifford index and resolution Clifford index for general ribbons on high genus curves and links split ribbons to major conjectures in algebraic geometry.

Findings

Equality of Clifford indices for general ribbons when p_a ≥ max{3g+7, 6g-4}

Relation between split ribbons and gonality conjecture

Reduction of Green's conjecture to Koszul cohomology vanishing

Abstract

We study the syzygies of the canonical embedding of a ribbon $\widetilde{C}$ on a curve $C$ of genus $g \geq 1$. 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 $p_{a} \geq \operatorname{max}\{3g+7, 6g-4\}$. 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 $C'$ of $C$ with $g(C') \geq \textrm{max}\{3g+2, 6g-4\}$. 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.