Syzygies of 5-gonal Canonical Curves

TL;DR

This paper investigates the syzygies of 5-gonal canonical curves, revealing that for certain genera, the syzygy modules are not determined by the scrolls associated with their degree 5 pencils.

Contribution

It demonstrates that for specific odd and even genera, the higher syzygy modules of 5-gonal curves are not governed by the scrolls from their degree 5 pencils, providing new insights into their algebraic structure.

Findings

Syzygy modules are not determined by scrolls for certain genera.

Results apply to odd genus g ≥ 13 and even genus g ≥ 28.

Highlights limitations of scroll-based descriptions of syzygies.

Abstract

We show that for $5$-gonal curves of odd genus $g\geq 13$ and even genus $g\geq 28$ the $\lceil \frac{g-1}{2}\rceil$-th syzygy module of the curve is not determined by the syzygies of the scroll swept out by the special pencil of degree $5$.