Green's conjecture for general covers

TL;DR

This paper proves Green's conjecture for certain classes of covers of curves with high Clifford dimension and explores syzygies of curves with involutions, providing new cases where Green's conjecture holds.

Contribution

It establishes Green's conjecture for covers of higher Clifford dimension and analyzes syzygies of curves with involutions, including Nikulin surfaces, expanding known cases.

Findings

Green's conjecture holds for specific curve covers with high Clifford dimension.

Curves with involutions on Nikulin surfaces satisfy Green's conjecture but not Prym-Green conjecture.

Identifies explicit loci in moduli space where Green's conjecture is verified.

Abstract

We establish Green's syzygy conjecture for classes of covers of curves of higher Clifford dimension. These curves have an infinite number of minimal pencils, in particular they do not verify a well-known Brill-Noether theoretic sufficient condition that implies Green's conjecture. Secondly, we study syzygies of curves with a fixed point free involution and prove that sections of Nikulin surfaces of minimal Picard number 9, verify the classical Green Conjecture but fail the Prym-Green Conjecture on syzygies of Prym-canonical curves. This provides an explicit locus in the moduli space R_g where Green's Conjecture is known to hold.