On the canonical ring of curves and surfaces

Original · EN

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