On the Koszul cohomology of canonical and Prym-canonical binary curves

TL;DR

This paper investigates Koszul cohomology and related conjectures for canonical and Prym-canonical binary curves, establishing inheritance properties across genera and verifying conjectures for low-genus cases.

Contribution

It proves that property $N_p$ propagates from genus $g$ to $g+1$ for these curves and confirms the Green and Prym-Green conjectures for certain low-genus cases.

Findings

Property $N_p$ holds for genus $g+1$ if it holds for genus $g$.

Green and Prym-Green conjectures verified for low-genus canonical and Prym-canonical binary curves.

Results support conjectures in specific low-genus ranges.

Abstract

In this paper we study Koszul cohomology and the Green and Prym-Green conjectures for canonical and Prym-canonical binary curves. We prove that if property $N_p$ holds for a canonical or a Prym-canonical binary curve of genus $g$ then it holds for a generic canonical or Prym-canonical binary curve of genus $g+1$. We also verify the Green and Prym-Green conjectures for generic canonical and Prym-canonical binary curves of low genus ($6\leq g\leq 15$, $g\neq 8$ for Prym-canonical and $3\leq g\leq 12$ for canonical).