Classification of surfaces of general type with Euler number 3

2013 ◽  
Vol 2013 (679) ◽  
pp. 1-22 ◽  
Author(s):  
Sai-Kee Yeung
Keyword(s):  
Projective Plane ◽  
Recent Work ◽  
Present Article ◽  
Smooth Surface ◽  
General Type ◽  
Euler Number ◽  
Projective Planes ◽  
Surfaces Of General Type ◽  
Surface Of General Type

Abstract The smallest topological Euler–Poincaré characteristic supported on a smooth surface of general type is 3. In this paper, we show that such a surface has to be a fake projective plane unless h1, 0(M) = 1. Together with the classification of fake projective planes given by Prasad and Yeung, the recent work of Cartwright and Steger, and a proof of the arithmeticity of the lattices involved in the present article, this gives a classification of such surfaces.

Algebraic Surfaces and the Miyaoka-Yau Inequality

2017 ◽  
Author(s):  
Paula Tretkoff
Keyword(s):  
Differential Geometry ◽  
Algebraic Surface ◽  
Smooth Surface ◽  
General Type ◽  
Kodaira Dimension ◽  
Algebraic Surfaces ◽  
Surfaces Of General Type ◽  
Smooth Complex ◽  
Dimension 2

This chapter discusses complex algebraic surfaces, with particular emphasis on the Miyaoka-Yau inequality and the rough classification of surfaces. Every complex algebraic surface is birationally equivalent to a smooth surface containing no exceptional curves. The latter is known as a minimal surface. Two related birational invariants, the plurigenus and the Kodaira dimension, play an important role in distinguishing between complex surfaces. The chapter first provides an overview of the rough classification of (smooth complex connected compact algebraic) surfaces before presenting two approaches that, in dimension 2, give the Miyaoka-Yau inequality. The first, due to Miyaoka, uses algebraic geometry, whereas the second, due to Aubin and Yau, uses analysis and differential geometry. The chapter also explains why equality in the Miyaoka-Yau inequality characterizes surfaces of general type that are free quotients of the complex 2-ball.

A Lower Bound on the Euler–Poincaré Characteristic of Certain Surfaces of General Type with a Linear Pencil of Hyperelliptic Curves

2016 ◽  
Vol 68 (1) ◽  
pp. 67-87
Author(s):  
Hirotaka Ishida
Keyword(s):  
Lower Bound ◽  
General Type ◽  
Hyperelliptic Curves ◽  
Surfaces Of General Type ◽  
Linear Pencil ◽  
Surface Of General Type

AbstractLet S be a surface of general type. In this article, when there exists a relatively minimal hyperelliptic fibration whose slope is less than or equal to four, we give a lower bound on the Euler–Poincaré characteristic of S. Furthermore, we prove that our bound is the best possible by giving required hyperelliptic fibrations.

scholarly journals Algebraic Surfaces of General Type with Small c21 in Positive Characteristic

2008 ◽  
Vol 191 ◽  
pp. 111-134 ◽  
Author(s):  
Christian Liedtke
Keyword(s):  
Characteristic Zero ◽  
General Type ◽  
Positive Characteristic ◽  
Algebraic Surfaces ◽  
Surfaces Of General Type ◽  
Arbitrary Characteristic ◽  
Numerical Invariants ◽  
Characteristic 2

AbstractWe establish Noether’s inequality for surfaces of general type in positive characteristic. Then we extend Enriques’ and Horikawa’s classification of surfaces on the Noether line, the so-called Horikawa surfaces. We construct examples for all possible numerical invariants and in arbitrary characteristic, where we need foliations and deformation techniques to handle characteristic 2. Finally, we show that Horikawa surfaces lift to characteristic zero.

scholarly journals Real Hypersurfaces of Nonflat Complex Projective Planes Whose Jacobi Structure Operator Satisfies a Generalized Commutative Condition

2016 ◽  
Vol 2016 ◽  
pp. 1-6
Author(s):  
Theocharis Theofanidis
Keyword(s):  
Projective Plane ◽  
Projective Planes ◽  
Additional Restriction ◽  
Real Hypersurfaces ◽  
Specific Function ◽  
Complex Projective Plane ◽  
Jacobi Structure ◽  
Complex Projective

Real hypersurfaces satisfying the conditionϕl=lϕ(l=R(·,ξ)ξ)have been studied by many authors under at least one more condition, since the class of these hypersurfaces is quite tough to be classified. The aim of the present paper is the classification of real hypersurfaces in complex projective planeCP2satisfying a generalization ofϕl=lϕunder an additional restriction on a specific function.

scholarly journals THE BICANONICAL MAP OF SURFACES WITH pg = 0 AND K2 [ges ] 7

2001 ◽  
Vol 33 (3) ◽  
pp. 265-274 ◽  
Author(s):  
MARGARIDA MENDES LOPES ◽  
RITA PARDINI
Keyword(s):  
Minimal Surface ◽  
General Type ◽  
Surfaces Of General Type ◽  
Bicanonical Map ◽  
Surface Of General Type

A minimal surface of general type with pg(S) = 0 satisfies 1 [les ] K2 [les ] 9, and it is known that the image of the bicanonical map φ is a surface for K2S [ges ] 2, whilst for K2S [ges ] 5, the bicanonical map is always a morphism. In this paper it is shown that φ is birational if K2S = 9, and that the degree of φ is at most 2 if K2S = 7 or K2S = 8.By presenting two examples of surfaces S with K2S = 7 and 8 and bicanonical map of degree 2, it is also shown that this result is sharp. The example with K2S = 8 is, to our knowledge, a new example of a surface of general type with pg = 0.The degree of φ is also calculated for two other known surfaces of general type with pg = 0 and K2S = 8. In both cases, the bicanonical map turns out to be birational.

scholarly journals Automorphisms of surfaces of general type with acting trivially in cohomology

2013 ◽  
Vol 149 (10) ◽  
pp. 1667-1684 ◽  
Author(s):  
Jin-Xing Cai ◽  
Wenfei Liu ◽  
Lei Zhang
Keyword(s):  
Automorphism Group ◽  
Minimal Surfaces ◽  
General Type ◽  
Cohomology Ring ◽  
Complete Classification ◽  
Surfaces Of General Type

AbstractIn this paper we prove that surfaces of general type with irregularity $q\geq 3$ are rationally cohomologically rigidified, and so are minimal surfaces $S$ with $q(S)= 2$ unless ${ K}_{S}^{2} = 8\chi ({ \mathcal{O} }_{S} )$. Here a surface $S$ is said to be rationally cohomologically rigidified if its automorphism group $\mathrm{Aut} (S)$ acts faithfully on the cohomology ring ${H}^{\ast } (S, \mathbb{Q} )$. As examples we give a complete classification of surfaces isogenous to a product with $q(S)= 2$ that are not rationally cohomologically rigidified.

scholarly journals Double Kodaira fibrations with small signature

2020 ◽  
Vol 31 (07) ◽  
pp. 2050052 ◽  
Author(s):  
Ju A Lee ◽  
Michael Lönne ◽  
Sönke Rollenske
Keyword(s):  
General Type ◽  
Fiber Bundles ◽  
Surfaces Of General Type ◽  
Positive Signature ◽  
Fix Point ◽  
Ramified Covers

Kodaira fibrations are surfaces of general type with a non-isotrivial fibration, which are differentiable fiber bundles. They are known to have positive signature divisible by [Formula: see text]. Examples are known only with signature 16 and more. We review approaches to construct examples of low signature which admit two independent fibrations. Special attention is paid to ramified covers of product of curves which we analyze by studying the monodromy action for bundles of punctured curves. As a by-product, we obtain a classification of all fix-point-free automorphisms on curves of genus at most [Formula: see text].

scholarly journals Embeddings of quaternion space in S4

1998 ◽  
Vol 65 (3) ◽  
pp. 313-325 ◽  
Author(s):  
Atsuko Katanaga ◽  
Osamu Saeki
Keyword(s):  
Projective Plane ◽  
Fundamental Group ◽  
Euler Number ◽  
Projective Planes ◽  
Connected Components ◽  
Connected Component ◽  
Real Projective Plane ◽  
Quaternion Group ◽  
Topological Embedding ◽  
Quaternion Space

AbstractConsider a (real) projective plane which is topologically locally flatly embedded in S4. It is known that it always admits a 2-disk bundle neighborhood, whose boundary is homeomorphic to the quaternion space Q, the total space of the nonorientable S1-bundle over RP2 with Euler number ± 2, with fundamental group isomorphic to the quaternion group of order eight. Conversely let f: Q → S4 be an arbitrary locally flat topological embedding. Then we show that the closure of each connected component of S4 − f(Q) is always homeomorphic to the exterior of a topologically locally flatly embedded projective plane in S4. We also show that, for a large class of embedded projective planes in S4, a pair of exteriors of such embedded projective planes is always realized as the closures of the connected components of S4 − f(Q) for some locally flat topological embedding f: Q → S4.

scholarly journals The classification of minimal irregular surfaces of general type with K^2=2p_g

2014 ◽  
Vol 1 (4) ◽  
pp. 479-488 ◽  
Author(s):  
Ciro Ciliberto ◽  
Margarida Mendes Lopes ◽  
Rita Pardini
Keyword(s):  
General Type ◽  
Surfaces Of General Type ◽  
Irregular Surfaces

scholarly journals MIXED QUASI-ÉTALE QUOTIENTS WITH ARBITRARY SINGULARITIES

2014 ◽  
Vol 57 (1) ◽  
pp. 143-165 ◽  
Author(s):  
DAVIDE FRAPPORTI ◽  
ROBERTO PIGNATELLI
Keyword(s):  
Minimal Surface ◽  
Finite Subset ◽  
General Type ◽  
Minimal Resolution ◽  
Surfaces Of General Type ◽  
Geometric Genus ◽  
Canonical Class ◽  
Two Factors ◽  
Surface Of General Type ◽  
Étale Quotients

AbstractA mixed quasi-étale quotient is the quotient of the product of a curve of genus at least 2 with itself by the action of a group which exchanges the two factors and acts freely outside a finite subset. A mixed quasi-étale surface is the minimal resolution of its singularities. We produce an algorithm computing all mixed quasi-étale surfaces with given geometric genus, irregularity and self-intersection of the canonical class. We prove that all irregular mixed quasi-étale surfaces of general type are minimal. As an application, we classify all irregular mixed quasi-étale surfaces of general type with genus equal to the irregularity, and all the regular ones with K2 > 0, thus constructing new examples of surfaces of general type with χ = 1. We mention the first example of a minimal surface of general type with pg = q = 1 and Albanese fibre of genus bigger than K2.

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