  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)= \oplus_{k \geq 0} H^0(C, \omega_C^k is generated in degree 1 if C is
3-connected and not (honestly) hyperelliptic; we show moreover that R(C,
L)=\oplus_{k \geq 0} H^0(C,L^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.
