Linear syzygies on curves with prescribed gonality

TL;DR

This paper proves new results about the structure of linear syzygies of algebraic curves with prescribed gonality, confirming conjectures and extending previous work in algebraic geometry.

Contribution

It establishes that general curves of fixed gonality satisfy Schreyer's Conjecture and provides an optimal effective version of the Gonality Conjecture for general k-gonal curves.

Findings

General curves of non-maximal gonality satisfy Schreyer's Conjecture.

All highest order linear syzygies are of Eagon-Northcott type.

Proves an optimal effective version of the Gonality Conjecture for general k-gonal curves.

Abstract

We prove two statements concerning the linear strand of the minimal free resolution of a curve of fixed gonality. Firstly, we show that a general curve C of genus g of non-maximal gonality k\leq (g+1)/2 satisfies Schreyer's Conjecture, that is, b_{g-k,1}(C,K_C)=g-k, and all its highest order linear syzygies are of Eagon-Northcott type. This is a statement going beyond Green's Conjecture and predicts that all highest order linear syzygies in the canonical embedding of C are determined by the syzygies of the (k-1)-dimensional scroll containing C. Secondly, we formulate an optimal effective version of the Gonality Conjecture and prove it for general k-gonal curves. This generalizes the asymptotic Gonality Conjecture proved by Ein-Lazarsfeld and improves results of Rathmann in the case where C is a general curve of fixed gonality.