The second syzygy schemes of curves of large degree

TL;DR

This paper investigates the second syzygy schemes of high-degree algebraic curves, establishing conditions under which these schemes coincide with the curves themselves, extending previous work on canonical curves.

Contribution

It provides new sufficient conditions for the second syzygy scheme of a genus-g curve of degree at least 2g+2 to match the curve, generalizing prior results.

Findings

Second syzygy scheme equals the curve under certain conditions

Property (N_2) ensures the scheme coincides with the curve

Analysis extends to cases where (N_2) fails

Abstract

The present paper is a natural continuation of a previous work where we studied the second syzygy scheme of canonical curves. We find sufficient conditions ensuring that the second syzygy scheme of a genus--$g$ curve of degree at least $2g+2$ coincide with the curve. If the property $(N_2)$ is satisfied, the equality is ensured by a more general fact. If $(N_2)$ fails, then the analysis uses the known case of canonical curves.