scholarly journals Every algebraic Kummer surface is the K3-cover of an Enriques surface

1990 ◽  
Vol 118 ◽  
pp. 99-110 ◽  
Author(s):  
Jong Hae Keum
Keyword(s):  
Crucial Role ◽  
K3 Surfaces ◽  
Abelian Surface ◽  
Enriques Surface ◽  
Kummer Surface ◽  
3 Dimensional ◽  
Complex Torus ◽  
Kummer Surfaces ◽  
Dimension 2 ◽  
Simply Connected

A Kummer surface is the minimal desingularization of the surface T/i, where T is a complex torus of dimension 2 and i the involution automorphism on T. T is an abelian surface if and only if its associated Kummer surface is algebraic. Kummer surfaces are among classical examples of K3-surfaces (which are simply-connected smooth surfaces with a nowhere-vanishing holomorphic 2-form), and play a crucial role in the theory of K3-surfaces. In a sense, all Kummer surfaces (resp. algebraic Kummer surfaces) form a 4 (resp. 3)-dimensional subset in the 20 (resp. 19)-dimensional family of K3-surfaces (resp. algebraic K3 surfaces).

scholarly journals A Reduction of the Batyrev-Manin Conjecture for Kummer Surfaces

2004 ◽  
Vol 47 (3) ◽  
pp. 398-406
Author(s):  
David McKinnon
Keyword(s):  
Number Field ◽  
Open Subset ◽  
Rational Points ◽  
Rational Curves ◽  
Abelian Surface ◽  
Kummer Surface ◽  
Geometric Genus ◽  
Finite Set ◽  
Genus 2 ◽  
Kummer Surfaces

AbstractLet V be a K3 surface defined over a number field k. The Batyrev-Manin conjecture for V states that for every nonempty open subset U of V, there exists a finite set ZU of accumulating rational curves such that the density of rational points on U − ZU is strictly less than the density of rational points on ZU. Thus, the set of rational points of V conjecturally admits a stratification corresponding to the sets ZU for successively smaller sets U.In this paper, in the case that V is a Kummer surface, we prove that the Batyrev-Manin conjecture for V can be reduced to the Batyrev-Manin conjecture for V modulo the endomorphisms of V induced by multiplication by m on the associated abelian surface A. As an application, we use this to show that given some restrictions on A, the set of rational points of V which lie on rational curves whose preimages have geometric genus 2 admits a stratification of Batyrev-Manin type.

scholarly journals Kummer surfaces associated to (1, 2)-polarized abelian surfaces

2011 ◽  
Vol 202 ◽  
pp. 127-143
Author(s):  
Afsaneh Mehran
Keyword(s):  
Double Cover ◽  
Pezzo Surface ◽  
Abelian Surface ◽  
Kummer Surface ◽  
Abelian Surfaces ◽  
Del Pezzo Surface ◽  
Singular Fibers ◽  
Kummer Surfaces ◽  
Del Pezzo

AbstractThe aim of this paper is to describe the geometry of the generic Kummer surface associated to a (1, 2)-polarized abelian surface. We show that it is the double cover of a weak del Pezzo surface and that it inherits from the del Pezzo surface an interesting elliptic fibration with twelve singular fibers of typeI2.

scholarly journals Degeneration of Kummer surfaces

2020 ◽  
pp. 1-33
Author(s):  
OTTO OVERKAMP
Keyword(s):  
Galois Representation ◽  
Base Change ◽  
Abelian Surface ◽  
Kummer Surface ◽  
Semistable Reduction ◽  
Valued Field ◽  
Monodromy Conjecture ◽  
Finite Base ◽  
Kummer Surfaces ◽  
Special Fibre

Abstract We prove that a Kummer surface defined over a complete strictly Henselian discretely valued field K of residue characteristic different from 2 admits a strict Kulikov model after finite base change. The Kulikov models we construct will be schemes, so our results imply that the semistable reduction conjecture is true for Kummer surfaces in this setup, even in the category of schemes. Our construction of Kulikov models is closely related to an earlier construction of Künnemann, which produces semistable models of Abelian varieties. It is well known that the special fibre of a strict Kulikov model belongs to one of three types, and we shall prove that the type of the special fibre of a strict Kulikov model of a Kummer surface and the toric rank of a corresponding Abelian surface are determined by each other. We also study the relationship between this invariant and the Galois representation on the second ℓ-adic cohomology of the Kummer surface. Finally, we apply our results, together with earlier work of Halle–Nicaise, to give a proof of the monodromy conjecture for Kummer surfaces in equal characteristic zero.

scholarly journals Poincare's conjecture and the homeotopy group of a closed, orientable 2-manifold

1974 ◽  
Vol 17 (2) ◽  
pp. 214-221 ◽  
Author(s):  
Joan S. Birman
Keyword(s):  
Dimensional Manifold ◽  
3 Dimensional ◽  
Dimension 2 ◽  
Poincare Conjecture ◽  
Simply Connected

In 1904 Poincaré [11] conjectured that every compact, simply-connected closed 3-dimensional manifold is homeomorphic to a 3-sphere. The corresponding result for dimension 2 is classical; for dimension ≧ 5 it was proved by Smale [12] and Stallings [13], but for dimensions 3 and 4 the question remains open. It has been discovered in recent years that the 3-dimensional Poincaré conjecture could be reformulated in purely algebraic terms [6, 10, 14, 15] however the algebraic problems which are posed in the references cited above have not, to date, proved tractable.

scholarly journals Kummer surfaces associated to (1, 2)-polarized abelian surfaces

2011 ◽  
Vol 202 ◽  
pp. 127-143 ◽  
Author(s):  
Afsaneh Mehran
Keyword(s):  
Double Cover ◽  
Pezzo Surface ◽  
Abelian Surface ◽  
Kummer Surface ◽  
Abelian Surfaces ◽  
Del Pezzo Surface ◽  
Singular Fibers ◽  
Kummer Surfaces ◽  
Del Pezzo

AbstractThe aim of this paper is to describe the geometry of the generic Kummer surface associated to a (1, 2)-polarized abelian surface. We show that it is the double cover of a weak del Pezzo surface and that it inherits from the del Pezzo surface an interesting elliptic fibration with twelve singular fibers of type I2.

scholarly journals Contact metric manifolds with large automorphism group and (κ, µ)-spaces

2019 ◽  
Vol 6 (1) ◽  
pp. 294-302 ◽  
Author(s):  
Antonio Lotta
Keyword(s):  
Automorphism Group ◽  
Symmetric Operator ◽  
Homogeneous Spaces ◽  
Point Of View ◽  
Maximum Dimension ◽  
Contact Metric Manifold ◽  
Contact Metric Manifolds ◽  
Large Automorphism Group ◽  
Dimension 2 ◽  
Simply Connected

AbstractWe discuss the classifiation of simply connected, complete (κ, µ)-spaces from the point of view of homogeneous spaces. In particular, we exhibit new models of (κ, µ)-spaces having Boeckx invariant -1. Finally, we prove that the number ${{(n + 1)(n + 2)} \over 2}$ is the maximum dimension of the automorphism group of a contact metric manifold of dimension 2n +1, n ≥ 2, whose symmetric operator h has rank at least 3 at some point; if this dimension is attained, and the dimension of the manifold is not 7, it must be a (κ, µ)-space. The same conclusion holds also in dimension 7 provided the almost CR structure of the contact metric manifold under consideration is integrable.

scholarly journals On Characterizing a Three-Dimensional Sphere

Mathematics ◽  
2021 ◽  
Vol 9 (24) ◽  
pp. 3311
Author(s):  
Nasser Bin Turki ◽  
Sharief Deshmukh ◽  
Gabriel-Eduard Vîlcu
Keyword(s):  
Three Dimensional ◽  
Dimensional Sphere ◽  
Sasakian Manifolds ◽  
3 Dimensional ◽  
Simply Connected

In this paper, we find a characterization of the 3-sphere using 3-dimensional compact and simply connected trans-Sasakian manifolds of type (α,β).

scholarly journals Elliptic Fibrations and the Hilbert Property

2019 ◽  
Author(s):  
Julian Lawrence Demeio
Keyword(s):  
Number Field ◽  
Algebraic Variety ◽  
Rational Point ◽  
K3 Surfaces ◽  
Sufficient Condition ◽  
Algebraic Varieties ◽  
Elliptic Fibrations ◽  
Kummer Surfaces ◽  
Cubic Hypersurfaces

Abstract For a number field $K$, an algebraic variety $X/K$ is said to have the Hilbert Property if $X(K)$ is not thin. We are going to describe some examples of algebraic varieties, for which the Hilbert Property is a new result. The first class of examples is that of smooth cubic hypersurfaces with a $K$-rational point in ${\mathbb{P}}_n/K$, for $n \geq 3$. These fall in the class of unirational varieties, for which the Hilbert Property was conjectured by Colliot-Thélène and Sansuc. We then provide a sufficient condition for which a surface endowed with multiple elliptic fibrations has the Hilbert Property. As an application, we prove the Hilbert Property of a class of K3 surfaces, and some Kummer surfaces.

Sign in / Sign up

Export Citation Format

Share Document