Canonical projections of irregular algebraic surfaces

Journal of Geometry and Symmetry in Physics, 2010

Advances in Geometry, 2000

Bulletin of the London Mathematical Society, 1984

Mathematical proceedings of the Cambridge Philosophical Society, 1987

Nagoya Mathematical Journal, 1998

Mathematische Zeitschrift, 2008

In this article we present a 3-dimensional analogue of a well-known theorem of E. Bombieri (in 1973) which characterizes the bi-canonical birationality of surfaces of general type. Let X be a projective minimal 3-fold of general type with Q-factorial terminal singularities and the geometric genus p g (X) ≥ 5. We show that the 4-canonical map ϕ 4 is not birational onto its image if and only if X is birationally fibred by a family C of irreducible curves of geometric genus 2 with K X • C 0 = 1 where C 0 is a general irreducible member in C .

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 .

Advances in Mathematics, 2007

Given a birational normal extension O of a two-dimensional local regular ring (R, m), we describe all the equisingularity types of the complete m-primary ideals J in R whose blowing-up X = Bl J (R) has some point Q whose local ring O X,Q is analytically isomorphic to O. * 1 fixed a birational normal extension O of a local regular ring (R, m O ), we describe the equisingularity type of any complete m O -primary ideal J ⊂ R such that its blowing-up X = Bl J (R) has some point Q whose local ring O X,Q is analytically isomorphic to O. In this case, we will say that the surface X contains the singularity O for short, making a slight abuse of language. This is done by describing the Enriques diagram of the cluster of base points of any such ideal J: such a diagram will be called an Enriques diagram for the singularity O. Recall that an Enriques diagram is a tree together with a binary relation (proximity) representing the topological equivalence classes of clusters of points in the plane (see §1.3). Previous works by Spivakovsky and Möhring [12] describe a type of Enriques diagram that exists for any given sandwiched surface singularity (detailed in §2) and provide other types mostly in the case of cyclic quotients (see [12] 2.7) and minimal singularities (see 2.5).

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.