On the canonical ring of curves and surfaces

Abstract

Let C be a curve (possibly non reduced or reducible) lying on a smooth algebraic\ud 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).\ud As a corollary we show that on a smooth algebraic\ud surface of general type with pg(S) ≥ 1 and q(S) = 0 the canonical ring R(S, KS) is generated\ud in degree ≤ 3 if there exists a curve C ∈ |KS| numerically three-connected and not hyperelliptic

The canonical ring of a 3-connected curve

“…We remark that our result implies in particular that the canonical ring of a regular surface of general type is generated in degree ≤ 3 if there exists a curve C ∈ |K S | 3-connected and not honestly hyperelliptic (see [8,Thm. 1.2]).…”

“…In [7], [8], [13] the analysis of the canonical ring is based on the study of the Koszul groups K p,q (C, ω C ) (with p, q small) and their vanishing properties, together with some vanishing results for invertible sheaves of low degree.…”

“…The main result on the canonical ring for singular curves in the literature is the 1-2-3 conjecture, stated by Reid in [15] and proved in [7] and [13], which says that the canonical ring R(C, ω C ) of a connected Gorenstein curve of arithmetic genus p a (C) ≥ 3 is generated in degree 1, 2, 3, with the exception of a small number of cases. More recently in [8] the first author proved that the canonical ring is generated in degree 1 under the strong assumption that C is even (i.e., deg B K C is even on every subcurve B ⊆ C).…”

Gorenstein stable surfaces with KX2=1 and pg>0

“…Proof The main tools to prove statements 1. and 2. are the Riemann–Roch theorem and Serre duality, used to compute , and to determine the base points of ; the base point free pencil trick (see [, Chap. III §3]), used to to show surjectivity of multiplication maps of the form . Since both (a) and (b) hold for an irreducible Gorenstein curve (see for example ) the degree of generators and relations can be determined verbatim as in the case of a smooth curve. For 1. we can thus choose variables such that the unique relation is ; it remains to prove that is not divisible by .…”

“…Since both (a) and (b) hold for an irreducible Gorenstein curve (see for example [16]) the degree of generators and relations can be determined verbatim as in the case of a smooth curve. For 1. we can thus choose variables such that the unique relation is z 2 + h(x 1 , y); it remains to prove that h(x 1 , y) is not divisible by x 2 1 .…”