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

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 ($6\le g\le 15$, $g\mathrm{\ne }8$ for Prym-canonical and $3\le g\le 12$ for canonical).