Algebraic Surfaces Of General Type Research Articles

Surfaces with p g = q =3

In this paper it is proved that a complete algebraic surface of general type with p g =q=3, without irrational pencil of genus bigger than one is birationally equivalent to the two-symmetric product of a curve of genus 3. This result completes the classification of the surfaces with p g =q=3. The main tools are the Lefschetz theorem and the use of the paracanonical system on the surface.

On Embedding Circle-Bundles in Four-Manifolds

We demonstrate an obstruction to finding certain splittings of four-manifolds along sufficiently twisted circle bundles over Riemann surfaces, arising from Seiberg-Witten theory. These obstructions are used to show a non-splitting result for algebraic surfaces of general type.

The fundamental group of a $\mathcal{CP}^2$ complement of a branch curve as an extension of a solvable group by a symmetric group

The main result in this paper is as follows: Theorem. Let S be the branch curve in CP 2 of a generic projection of a Veronese surface. Then �1(CP 2 S) is an extension of a solvable group by a symmetric group. A group with the property mentioned in the theorem is "almost solvable" in the sense that it contains a solvable normal subgroup of finite index. We pose the following question. Question. For which families of simply connected algebraic surfaces of general type is the fundamental group of the complement of the branch curve of a generic projec- tion to CP 2 an extension of a solvable group by a symmetric group?

Certain algebraic surfaces of general type with irregularity one and their canonical mappings

Certain algebraic surfaces of general type with irregularity one and their canonical mappings

Applications of braid group techniques to the decomposition of moduli spaces, new examples

Every smooth minimal complex algebraic surface of general type, X, may be mapped into a moduli space, M c 1 2 (x), c 2 (x) , of minimal surfaces of general type, all of which have the same Chern numbers. Using the braid group and braid monodromy, we construct infinitely many new examples of pairs of minimal surfaces of general type which have the same Chern numbers and nonisomorphic fundamental groups. Unlike previous examples, our results include X for which ¦π 1 (X)¦ is arbitrarily large. Moreover, the surfaces are of positive signature. This supports our goal of using the braid group and fundamental groups to decompose M c 1 2 (x), c 2 (x) into connected components.

Four-Manifolds without Einstein Metrics

It is shown that there are infinitely many compact simply con- nected smooth 4-manifolds which do not admit Einstein metrics, but nev- ertheless satisfy the strict Hitchin-Thorpe inequality 2 !> 3| |. The exam- ples in question arise as non-minimal complex algebraic surfaces of general type, and the method of proof stems from Seiberg-Witten theory.

Certain algebraic surfaces of general type with irregularity one and their canonical mappings

Certain algebraic surfaces of general type with irregularity one and their canonical mappings

On canonical fibrations of algebraic surfaces

LetS be an algebraic surface of general type. If the canonical system |K S | ofS is a pencil of genusg, we hope to find the largestc(g) such thatK S 2 ≥c(g)p g +constant. We have known thatc(3)≤6. In this paper, we proved thatc(3)≥5.25.

Branched coverings of surfaces with ample cotangent bundle

Let /: X —> Y be a branched covering of compact complex surfaces, where the ramification set in X consists of smooth curves meeting with at most normal crossings and Y has ample cotangent bundle. We further assume that / is locally of form (u,υ) —> (un ,vm). We characterize ampleness of T*X . A class of examples of such X, which are branched covers of degree two, is provided. 1. Introduction. An interesting problem in surface theory is the construction and characterization of surfaces with ample cotangent bundle. They are necessarily algebraic surfaces of general type. Natural examples occur among the complete intersection surfaces of abelian varieties. More subtle examples are those constructed by Hirzebruch [6] using line-arrangements in the plane. The characterization of those of Hirzebruch's line-arrangement surfaces with ample cotangent bundle is due to Sommese [8]. In this article, we will give a characterization of ampleness of the cotangent bundle of a class of surfaces which branch cover another surface with ample cotangent bundle. We will also construct certain branched coverings of explicit line-arrangement surfaces; these constructions will again have ample cotangent bundle. For any vector bundle E over a base manifold M, the projectivization P(E) is a fiber bundle over M, with fiber Pq(E) over q e M given by T?g(E) « (2?*\0)/C*. There is a tautological linebundle ξβ over P(E) satisfying (i) ζE\F ^ « 0(1)P ^ V<? e M, and (ii) the projection pE: P(E) -» M gives /?^ (ξβ) ~ E. In the case that E = T*X we will denote p£ = Pτ*x simply by p. DEFINITION. The vector bundle E is ample if ζβ over ~P(E) is

Even canonical surfaces with small K2, I

Let S be a minimal algebraic surface of general type defined over the complex number field C, and let K denote the canonical bundle. According to [10], we call S a canonical surface if the rational map ФK associated with | K | induces a birational map of S onto the image X. We denote by Q (X) the intersection of all hyperquadrics through X.

Noether's Inequality for Non-complete Algebraic Surfaces of General Type, II

Noether's Inequality for Non-complete Algebraic Surfaces of General Type, II

Noether's Inequality for Non-complete Algebraic Surfaces of General Type

Let V be a nonsingular projective surface of Kodaira dimension K(V) > 0. Let D be a reduced, effective, nonzero divisor on V with only normal crossings. In the present article, a pair (F, D) is said to be a minimal logarithmic surface of general type, if, by definition, Kv + D is a numerically effective divisor of self intersection number (Kv + D) 2 > 0 and if Kv + D has positive intersection with every exceptional curve of the first kind on V. Here Kv is the canonical divisor of V. In the case, on the one hand, Sakai [8; Theorem 7.6] proved a Miyaoka — Yau type inequality (cf ) := (Kv + D) 2 — c2 -by making use of [8; Theorem 5.5]. In the present article, we shall prove that (cf ) > -c2 — 2 provided that the rational map &\Ky+D defined by the complete linear system \KV + D| has a surface as the image of V. Moreover, if the equality holds, then the logarithmic geometric genus pg := h°(V, Kv + D) = -(cf ) + 2 = 3, D is an elliptic curve and V is the canonical resolution in the sense of Horikawa associated with a double covering h: Y -> P. In addition, the branch locus B of h is a reduced curve of degree eight and the singular locus Sing B consists of points of multiplicity < 3 except for at most one simple quadruple point. Introduction This is a succession of the previous paper [9]. We work over an algebraically closed field fe of characteristic zero. Let V be a nonsingular projective surface defined over fe. If V is a minimal surface of general type, we have the following inequality due to M. Noether: This inequality, together with the Noether formula 12%((9V) = c±(V) 2 + c2(V Communicated by K. Saito, May 18, 1990. 1991 Mathematics Subject Classification: 14J29 Department of Mathematics, the National University of Singapore, Singapore.

Minimal models of covering spaces of P2 branched along two nonsingular curves with normal crossings

In this paper, we study minimal models of finite covering spaces of P 2 branched along two nonsingular curves with only normal crossings. We examine the structure of them and investigate some properties of surfaces of general type which are obtained from the above covering spaces. We show that every algebraic surface of general type has negative index if it is birational to a covering space of P 2 branched along two nonsingular curves with only normal crossings.

Algebraic surfaces of general type with $c^2_1=3p_g-7$

Algebraic surfaces of general type with $c^2_1=3p_g-7$

On the number of non-equivalent differentiable structures on 4-manifolds

By using results from [7], [8] we show that for any positive integer k there exist k simply-conected algebraic surfaces of general type which are pairwise homeomorphic but not diffeomorphic.

Existence of simply connected algebraic surfaces of general type with positive and zero indices

In the classification problem of algebraic surfaces of general type, an important conjecture states that for simply connected such surfaces Chern numbers satisfy the inequality c(1) (2) </= 2c(2) (or equivalently, the index tau </= 0). We disprove this conjecture by computing fundamental groups of Galois coverings corresponding to generic CP(2) projections of projective embeddings of CP(1) x CP(1) related to linear systems [unk]al(1) + bl(2)[unk], a >/= 3, b >/= 2. Also, we proved the existence of simply connected minimal surfaces of general type with zero index (e.g., c(1) (2) = 2c(2)). Previously, it was conjectured that these are exactly the surfaces uniformizable in the polydisk. So this conjecture is also disproved.

A remark on the Ricci curvature of algebraic surfaces of general type

A remark on the Ricci curvature of algebraic surfaces of general type

Algebraic surfaces of general type with smallc 1 2 . III

Algebraic surfaces of general type with smallc 1 2 . III

Algebraic surfaces of general type with smallc 1 2 . IV

Algebraic surfaces of general type with smallc 1 2 . IV

A class of algebraic surfaces of general type constructed from quaternion algebras

A class of algebraic surfaces of general type constructed from quaternion algebras