On the Canonical Rings of Some Horikawa Surfaces. Part I (original) (raw)

On the canonical rings of some Horikawa surfaces. I

Transactions of the American Mathematical Society, 1988

This paper is devoted to finding necessary and sufficient conditions for a graded ring to be the canonical ring of a minimal surface of general type with K 2 = 2 p g − 3 {K^2} = 2{p_g} - 3 , p g ⩾ 3 {p_g} \geqslant 3 , and such that its canonical linear system has one base point.

On the canonical ring of curves and surfaces

2011

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 ) = k≥0 H 0 (C, ω C ⊗k ) is generated in degree 1 if C is numerically 4-connected, not hyperelliptic and even (i.e. with ω C of even degree on every component).

The degree of the generators of the canonical ring of surfaces of general type with p g = 0

Archiv der Mathematik, 1997

Upper bounds for the degree of the generators of the canonical rings of surfaces of general type were found by Ciliberto [C]. In particular it was established that the canonical ring of a minimal surface of general type with p g = 0 is generated by its elements of degree lesser or equal to 6, ([C], th. (3.6)). This was the best bound possible to obtain at the time, since Reider's results, [R], were not yet available. In this note, this bound is improved in some cases (theorems (3.1), (3.2)). In particular it is shown that if K 2 ≥ 5, or if K 2 ≥ 2 and |2K S | is base point free this bound can be lowered to 4. This result is proved by showing first that, under the same hypothesis, the degree of the bicanonical map is lesser or equal to 4 if K 2 ≥ 3, (theorem (2.1)), implying that the hyperplane sections of the bicanonical image have not arithmetic genus 0. The result on the generation of the canonical ring then follows by the techniques utilized in [C]. Notation and conventions. We will denote by S a projective algebraic surface over the complex field. Usually S will be smooth, minimal, of general type. We denote by K S , or simply by K if there is no possibility of confusion, a canonical divisor on S. As usual, for any sheaf F on S, we denote by h i (S, F) the dimension of the cohomology space H i (S, F), and by p g and q the geometric genus and the irregularity of S. By a curve on S we mean an effective, non zero divisor on S. We will denote the intersection number of the divisors C, D on S by C • D and by C 2 the self-intersection of the divisor C. We denote by ≡ the linear equivalence for divisors on S. |D| will be the complete linear system of the effective divisors D ′ ≡ D, and φ D : S → P(H 0 (S, O S (D) ∨) = |D| ∨ the natural rational map defined by |D|. We will denote by Σ d the rational ruled surface P(O P 1 ⊕ O P 1 (d)), for d ≥ 0. ∆ ∞ will denote the section of Σ d with minimum self-intersection −d and Γ will be a fibre of the projection to P 1 .