On the canonical ring of curves and surfaces

Abstract

Let C be a curve (possibly non reduced or reducible) lying on a smooth algebraic surface. We show that the canonical ring $${ R(C, \omega_C)=\bigoplus_{k\geq 0} H^0(C, {\omega_C}^{\otimes k})}$$ 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 p g (S) ≥ 1 and q(S) = 0 the canonical ring R(S, K S ) is generated in degree ≤ 3 if there exists a curve $${C \in |K_S|}$$ numerically three-connected and not hyperelliptic.