Green's canonical syzygy conjecture for ribbons

TL;DR

This paper proves an analogue of Green's canonical syzygy conjecture for a special class of non-reduced curves called ribbons, extending the understanding of syzygies beyond smooth curves using advanced algebraic geometry techniques.

Contribution

It establishes the validity of Green's conjecture for ribbons, a class of non-reduced curves, using methods from Voisin and Hirschowitz-Ramanan.

Findings

Green's conjecture holds for ribbons.

The proof leverages results from smooth curve cases.

Extends syzygy theory to non-reduced curves.

Abstract

Green's canonical syzygy conjecture asserts a simple relationship between the Clifford index of a smooth projective curve and the shape of the minimal free resolution of its homogeneous ideal in the canonical embedding. We prove the analogue of this conjecture formulated by Bayer and Eisenbud for a class of non-reduced curves called ribbons. Our proof uses the results of Voisin and Hirschowitz-Ramanan on Green's conjecture for general smooth curves.