On the canonical rings of some Horikawa surfaces. I (original) (raw)
On the Canonical Rings of Some Horikawa Surfaces. Part 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 K2 = 2pg-3, pg > 3, and such that its canonical linear system has one base point. 1. Known results. 1.1. Let 5 be a minimal surface of general type with K2 = 2pg-3, pg > 3. We assume that the canonical linear system \K\ of S has one base point P. Following Horikawa [4, Lemma 2] let ir: S'-► 5 be the quadratic transformation with center P and exceptional curve E.
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 .
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).
A note on surfaces of general type with pg=0p_g=0pg=0 and K2ge7K^2\ge 7K2ge7
1999
A minimal surface of general type with pg(S) = 0 satisfies 1 ≤ K 2 ≤ 9 and it is known that the image of the bicanonical map φ is a surface for K2 S ≥ 2, whilst for K 2 S ≥ 5, the bicanonical map is always a morphism. Here we prove that if K2 = 7 or K2 = 8 then the degree of φ is at most 2 and that if K2 = 9, then φ is birational.
A new family of surfaces with pg=0p_g=0pg=0 and K2=3K^2=3K2=3
arXiv (Cornell University), 2003
Let S be a minimal complex surface of general type with p g = 0 such that the bicanonical map ϕ of S is not birational and let Z be the bicanonical image. In [M. Mendes Lopes, R. Pardini, Enriques surfaces with eight nodes, Math. Zeit. 241 4 (2002), 673-683] it is shown that either: i) Z is a rational surface, or ii) K 2 S = 3, ϕ is a degree two morphism and Z is birational to an Enriques surface. Up to now no example of case ii) was known. Here an explicit construction of all such surfaces is given. Furthermore it is shown that the corresponding subset of the moduli space of surfaces of general type is irreducible and uniruled of dimension 6.
arXiv (Cornell University), 2008
We study minimal complex surfaces S of general type with q(S) = q and p g (S) = 2q -3, q ≥ 5. We give a complete classification in case that S has a fibration onto a curve of genus ≥ 2. For these surfaces K 2 = 8χ. In general we prove that K 2 ≥ 7χ -1 and that the stronger inequality K 2 ≥ 8χ holds under extra assumptions (e.g., if the canonical system has no fixed part or the canonical map has even degree). We also describe the Albanese map of S.
Canonical surfaces of higher degree
Rendiconti del Circolo Matematico di Palermo (1952 -), 2016
We consider a family of surfaces of general type S with K S ample, having K 2 S = 24, p g (S) = 6, q(S) = 0. We prove that for these surfaces the canonical system is base point free and yields an embedding Φ 1 : S → P 5 . This result answers a question posed by G. and M. Kapustka [Kap-Kap15]. We discuss some related open problems, concerning also the case p g (S) = 5, where one requires the canonical map to be birational onto its image.
New surfaces with canonical map of high degree
arXiv: Algebraic Geometry, 2018
We give an algorithm that, for a given value of the geometric genus pg,p_g,pg, computes all regular product-quotient surfaces with abelian group that have at most canonical singularities and have canonical system with at most isolated base points. We use it to show that there are exactly two families of such surfaces with canonical map of degree 323232. We also construct a surface with q=1q=1q=1 and canonical map of degree 242424. These are regular surfaces with pg=3p_g=3pg=3 and base point free canonical system. We discuss the case of regular surfaces with pg=4p_g=4pg=4 and base point free canonical system.
Surfaces with pg=q=3p_g=q=3pg=q=3
2001
We classify minimal complex surfaces of general type with pg=q=3p_g=q=3pg=q=3. More precisely, we show that such a surface is either the symmetric product of a curve of genus 3 or a free Z_2−\Z_2-Z_2−quotient of the product of a curve of genus 2 and a curve of genus 3. Our main tools are the generic vanishing theorems of Green and Lazarsfeld and Fourier--Mukai transforms. The same result has been obtained independently at the same time by G. Pirola using different methods.
A note on a theorem of Horikawa
Revista Matemática Complutense, 1997
In this paper we classify the algebraic surfaces on with 4 = 4, p 9 = 3 and canonical map of degree d = 3. By our resu]t atid tIte previous ono of Horikawa (10] we obtain tIte complete detennination of surfaces with ¡<2 = 4 atid p9 = 3. Intreductien TIte aim of this paper is te classif>' ¡ninimal surfacos Sen C with K 2 = 4, = 3 and canonical map of degree! = 3. TIto ex15tence of such surfaces is claimod witItout proof un ¡10] [Soction 2, p. 1101. Lix tIte same papor Horikawa sItewed that surfaces with Rj = 4 and p 9 = 3 Itave o! = 2,3,4 and he classified tIte casos o! = 2 atid o! = 4. Wo hayo aIread>' considered surfacos witIt K~st 4, p9 = 3 ando! = 3 in [14], but in this article we adopt a different point of view. We will explicitol>' censtnict a birational medol X CE IP 3 of 5 where X is a quintic witIt only a singular point whicIt is an elliptie Gorenstein singularity of type E 8 (cf. [12] and tIte flrst soction below). Main theorem Let A., = {(io,ií,i2) E Z~jio + 11 + 22 = 5-s,3io + 2i~+~2 =6} ano! lot .4 be tite aublinear ayatem of tite quintica X CE IP 3 initit tite fo¿lowing oquation: 3 1) X = fr E r3¡>3 >3 a¡a" 0} s=O ¡eA 8
Some new surfaces with pg=q=0p_g = q = 0pg=q=0
2003
Motivated by a question by D. Mumford : can a computer classify all surfaces with pg=0p_g = 0pg=0 ? we try to show the complexity of the problem. We restrict it to the classification of the minimal surfaces of general type with pg=0,K2=8p_g = 0, K^2 = 8pg=0,K2=8 which are constructed by the Beauville construction, namely, which are quotients of