# The canonical ring of a 3-connected curve

### Marco Franciosi

Università di Pisa, Italy### Elisa Tenni

Università di Firenze, Italy

## 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 $R(C, \omega_C) = \bigoplus_{k\geq 0} H^0(C, {\omega_C}^{\otimes k})$ is generated in degree 1 if $C$ is 3-connected and not (honestly) hyperelliptic; we show moreover that $R(C, L)=\bigoplus_{k\geq 0} H^0(C,L^{\otimes k})$ is generated in degree 1 if $C$ is reduced with planar singularities and $L$ is an invertible sheaf such that $\deg L_{|B} \geq 2p_a(B)+1$ for every $B\subseteq C$.

## Cite this article

Marco Franciosi, Elisa Tenni, The canonical ring of a 3-connected curve. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 25 (2014), no. 1, pp. 37–51

DOI 10.4171/RLM/667