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, ωC) is generated in degree 1 if C is numerically four-connected, not hyperelliptic and even (i.e. with ω_C of even degree on every component). As a corollary we show that on a smooth algebraic surface of general type with pg(S) ≥ 1 and q(S) = 0 the canonical ring R(S, KS) is generated in degree ≤ 3 if there exists a curve C ∈ |KS| numerically three-connected and not hyperelliptic.