The canonical ring of a 3-connected curve

Abstract

Let $C$ be a projective curve either reduced with planar singularities or contained in a smooth algebraic surface. We show that the canonical ring $R(C,{\omega }_C)={⨁}_{k\ge 0}{H}^0(C,{{\omega }_C}^{\otimes k})$ is generated in degree 1 if $C$ is 3-connected and not (honestly) hyperelliptic; we show moreover that $R(C,L)={⨁}_{k\ge 0}{H}^0(C,L^{\otimes k})$ is generated in degree 1 if $C$ is reduced with planar singularities and $L$ is an invertible sheaf such that $\mathrm{deg}{L}_{|B}\ge 2p_a(B)+1$ for every $B\subseteq C$.