On the canonical ring of curves and surfaces

Let C be a curve (possibly non reduced or reducible) lying on a smooth algebraic surface. We show that the canonical ring R(C, \omega_C) is generated in degree 1 if C is numerically 4-connected, not hyperelliptic and even (i.e. with K_C of even degree on every component). As a corollary we show that on a smooth algebraic surface of general type with p_g(S)>0 and q(S)=0 the canonical ring R(S, K_S) is generated in degree \leq 3 if there exists a curve C in |K_S| numerically 3-connected and not hyperelliptic.