On the syzygies of reducible curves

TL;DR

This paper extends Green's theorem on property Np to certain singular, reducible curves, establishing degree bounds that ensure similar syzygy properties as in smooth curves.

Contribution

It generalizes Green's theorem to reducible, singular curves, providing new degree bounds for property Np in these more complex cases.

Findings

Proves property Np for specific reducible curves under degree bounds.

Establishes conditions under which syzygy properties hold for singular, reducible curves.

Extends classical results from smooth to certain singular, reducible cases.

Abstract

A theorem of Green says that a line bundle of degree at least $2g+1+p$ on a smooth curve $X$ of genus $g$ has property $N_p$. We prove a similar conclusion for certain singular, reducible curves $X$ under suitable degree bounds over all irreducible components of $X$.