On the canonical ring of curves and surfaces

2011/07/04 by Franciosi, Marco
#14C20 #14H45 #14J29 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1107.0579

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, ω_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 ≤ 3 if there exists a curve C in |K_S| numerically 3-connected and not hyperelliptic.

Related

• The canonical ring of a 3-connected curve
• The degree of the generators of the canonical ring of surfaces with p_g=0
• A remark on the canonical degree of curves on smooth projective surface
• On the Canonical Ring of Covers of Surfaces of Minimal Degree
• Examples of surfaces with canonical maps of degree 12, 13, 15, 16 and 18