Open algebraic surfaces of logarithmic Kodaira dimension one

2013 ◽  
Author(s):  
Hideo Kojima
Keyword(s):  
Kodaira Dimension ◽  
Algebraic Surfaces ◽  
Dimension One

scholarly journals Algebraic models of the Euclidean plane

2018 ◽  
Vol Volume 2 ◽  
Author(s):  
Jérémy Blanc ◽  
Adrien Dubouloz
Keyword(s):  
Euclidean Plane ◽  
Kodaira Dimension ◽  
Algebraic Surfaces ◽  
Algebraic Models ◽  
The Real ◽  
Homology Groups ◽  
Real Algebraic Surfaces ◽  
Compact Case ◽  
Birational Diffeomorphism

We introduce a new invariant, the real (logarithmic)-Kodaira dimension, that allows to distinguish smooth real algebraic surfaces up to birational diffeomorphism. As an application, we construct infinite families of smooth rational real algebraic surfaces with trivial homology groups, whose real loci are diffeomorphic to $\mathbb{R}^2$, but which are pairwise not birationally diffeomorphic. There are thus infinitely many non-trivial models of the euclidean plane, contrary to the compact case. Comment: 16 pages

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.

Complex Geometry of Iwasawa Manifolds

2018 ◽  
Vol 2020 (23) ◽  
pp. 9420-9439
Author(s):  
Vasily Rogov
Keyword(s):  
Complex Geometry ◽  
Complex Multiplication ◽  
Complex Surface ◽  
Kodaira Dimension ◽  
Elliptic Surface ◽  
Abelian Surface ◽  
Complex Curve ◽  
Cocompact Lattice ◽  
Complex Homogeneous Manifold ◽  
Dimension One

Abstract An Iwasawa manifold is a compact complex homogeneous manifold isomorphic to a quotient $G/\Lambda $, where $G$ is the group of complex unipotent $3 \times 3$ matrices and $\Lambda \subset G$ is a cocompact lattice. In this work, we study holomorphic submanifolds in Iwasawa manifolds. We prove that any compact complex curve in an Iwasawa manifold is contained in a holomorphic subtorus. We also prove that any complex surface in an Iwasawa manifold is either an abelian surface or a Kähler non-projective isotrivial elliptic surface of Kodaira dimension one. In the Appendix, we show that any subtorus in Iwasawa manifold carries complex multiplication.

ENRIQUES’ CLASSIFICATION IN CHARACTERISTIC 0$" height="12pt">: THE -THEOREM

2018 ◽  
Vol 235 ◽  
pp. 201-226
Author(s):  
FABRIZIO CATANESE ◽  
BINRU LI
Keyword(s):  
Positive Characteristic ◽  
Kodaira Dimension ◽  
Algebraic Surfaces ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Precise Version

The main goal of this paper is to show that Castelnuovo–Enriques’ $P_{12}$ - theorem (a precise version of the rough classification of algebraic surfaces) also holds for algebraic surfaces $S$ defined over an algebraically closed field $k$ of positive characteristic ( $\text{char}(k)=p>0$ ). The result relies on a main theorem describing the growth of the plurigenera for properly elliptic or properly quasielliptic surfaces (surfaces with Kodaira dimension equal to 1). We also discuss the limit cases, i.e., the families of surfaces which show that the result of the main theorem is sharp.

Normal quintic surfaces with Kodaira dimension one

2015 ◽  
Vol 26 (02) ◽  
pp. 1550015
Author(s):  
Yumiko Umezu
Keyword(s):  
Projective Space ◽  
Three Dimensional ◽  
Kodaira Dimension ◽  
Elliptic Surfaces ◽  
Base Curve ◽  
Dimensional Projective Space ◽  
Dimension One

We study normal quintic surfaces in the three-dimensional projective space whose nonsingular models are surfaces of Kodaira dimension one. It turns out that the genus of the base curve of their elliptic fibration is equal to 0 or 1, and the possible values of other invariants of these surfaces and the singularities on them are obtained. We give several examples to show the existence of such surfaces. Moreover we determine the defining equations of general quintic surfaces whose nonsingular models are irregular elliptic surfaces of Kodaira dimension one.

scholarly journals Projective normality and syzygies of algebraic surfaces

1999 ◽  
Vol 1999 (506) ◽  
pp. 145-180 ◽  
Author(s):  
F. J Gallego ◽  
B. P Purnaprajna
Keyword(s):  
General Type ◽  
Kodaira Dimension ◽  
Algebraic Surfaces ◽  
List Type ◽  
Singular Surfaces ◽  
Surfaces Of General Type ◽  
New Techniques ◽  
Uniform Bounds ◽  
Projective Normality ◽  
Koszul Cohomology

Abstract In this work we develop new techniques to compute Koszul cohomology groups for several classes of varieties. As applications we prove results on projective normality and syzygies for algebraic surfaces. From more general results we obtain in particular the following: Mukai's conjecture (and stronger variants of it) regarding projective normality and normal presentation for surfaces with Kodaira dimension 0, and uniform bounds for higher syzygies associated to adjoint linear series,effective bounds along the lines of Mukai's conjecture regarding projective normality and normal presentation for surfaces of positive Kodaira dimension, and,results on projective normality for pluricanonical models of surfaces of general type (recovering and strengthening results by Ciliberto) and generalizations of them to higher syzygies. In addition, we also extend the above results to singular surfaces.

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