scholarly journals Conic bundles that are not birational to numerical Calabi--Yau pairs

2017 ◽  
Vol Volume 1 ◽  
Author(s):  
János Kollár
Keyword(s):  
Projective Plane ◽  
Number Field ◽  
Projective Variety ◽  
Conic Bundle ◽  
Anticanonical Divisor ◽  
Conic Bundles ◽  
Branch Curve

Let $X$ be a general conic bundle over the projective plane with branch curve of degree at least 19. We prove that there is no normal projective variety $Y$ that is birational to $X$ and such that some multiple of its anticanonical divisor is effective. We also give such examples for 2-dimensional conic bundles defined over a number field.

COUNTING MULTISECTIONS IN CONIC BUNDLES OVER A CURVE DEFINED OVER 𝔽q

2011 ◽  
Vol 07 (06) ◽  
pp. 1663-1680
Author(s):  
SEYFI TÜRKELLI
Keyword(s):  
Function Field ◽  
Algebraic Numbers ◽  
Conic Bundle ◽  
Conic Bundles ◽  
Special Case

For a given conic bundle X over a curve C defined over 𝔽q, we count irreducible branch covers of C in X of degree d and height e ≫ 1. As a special case, we get the number of algebraic numbers of degree d and height e over the function field 𝔽q(C).

scholarly journals STABILITY OF CONIC BUNDLES: WITH AN APPENDIX BY MUNDET I RIERA

2000 ◽  
Vol 11 (08) ◽  
pp. 1027-1055 ◽  
Author(s):  
TOMÁS L. GÓMEZ ◽  
IGNACIO SOLS
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Conic Bundle ◽  
Conic Bundles ◽  
Definition Of

Roughly speaking, a conic bundle is a surface, fibered over a curve, such that the fibers are conics (not necessarily smooth). We define stability for conic bundles and construct a moduli space. We prove that (after fixing some invariants) these moduli spaces are irreducible (under some conditions). Conic bundles can be thought of as generalizations of orthogonal bundles on curves. We show that in this particular case our definition of stability agrees with the definition of stability for orthogonal bundles. Finally, in an appendix by I. Mundet i Riera, a Hitchin-Kobayashi correspondence is stated for conic bundles.

Cyclic homology, Serre's local factors and λ-operations

2014 ◽  
Vol 14 (1) ◽  
pp. 1-45 ◽  
Author(s):  
Alain Connes ◽  
Caterina Consani
Keyword(s):  
Number Field ◽  
Projective Variety ◽  
Smooth Projective Variety ◽  
Cyclic Homology ◽  
Local Factors ◽  
The Right

AbstractWe show that for a smooth, projective variety X defined over a number field K, cyclic homology with coefficients in the ring , provides the right theory to obtain, using λ-operations, Serre's archimedean local factors of the complex L-function of X as regularized determinants.

Descente galoisienne sur le groupe de Brauer

2012 ◽  
Vol 2013 (682) ◽  
pp. 141-165
Author(s):  
Jean-Louis Colliot-Thélène ◽  
Alexei N. Skorobogatov
Keyword(s):  
Number Field ◽  
Characteristic Zero ◽  
Projective Variety ◽  
Algebraic Closure ◽  
Invariant Subgroup ◽  
Brauer Group ◽  
Intersection Pairing ◽  
Nous Montrons ◽  
Numerical Equivalence

Abstract. Soit X une variété projective et lisse sur un corps k de caractéristique zéro. Le groupe de Brauer de X s'envoie dans les invariants, sous le groupe de Galois absolu de k, du groupe de Brauer de la même variété considérée sur une clôture algébrique de k. Nous montrons que le quotient est fini. Sous des hypothèses supplémentaires, par exemple sur un corps de nombres, nous donnons des estimations sur l'ordre de ce quotient. L'accouplement d'intersection entre les groupes de diviseurs et de 1-cycles modulo équivalence numérique joue ici un rôle important. For a smooth and projective variety X over a field k of characteristic zero we prove the finiteness of the cokernel of the natural map from the Brauer group of X to the Galois-invariant subgroup of the Brauer group of the same variety over an algebraic closure of k. Under further conditions, e.g., over a number field, we give estimates for the order of this cokernel. We emphasise the rôle played by the exponent of the discriminant groups of the intersection pairing between the groups of divisors and curves modulo numerical equivalence.

scholarly journals Cohomology bounds for sheaves of dimension one

2014 ◽  
Vol 25 (11) ◽  
pp. 1450103 ◽  
Author(s):  
Jinwon Choi ◽  
Kiryong Chung
Keyword(s):  
Projective Plane ◽  
Moduli Space ◽  
Hilbert Scheme ◽  
Closed Subset ◽  
Projective Variety ◽  
Global Section ◽  
Sharp Bounds ◽  
One Dimensional ◽  
Dimension One

We find sharp bounds on h0(F) for one-dimensional semistable sheaves F on a projective variety X. When X is the projective plane ℙ2, we study the stratification of the moduli space by the spectrum of sheaves. We show that the deepest stratum is isomorphic to a closed subset of a relative Hilbert scheme. This provides an example of a family of semistable sheaves having the biggest dimensional global section space.

scholarly journals A Characterization of Some Fano 4-folds Through Conic Fibrations

2019 ◽  
Author(s):  
Pedro Montero ◽  
Eleonora Anna Romano
Keyword(s):  
Del Pezzo Surfaces ◽  
Conic Bundle ◽  
Conic Bundles ◽  
Del Pezzo ◽  
Bundle Structure

Abstract We find a characterization for Fano 4-folds $X$ with Lefschetz defect $\delta _{X}=3$: besides the product of two del Pezzo surfaces, they correspond to varieties admitting a conic bundle structure $f\colon X\to Y$ with $\rho _{X}-\rho _{Y}=3$. Moreover, we observe that all of these varieties are rational. We give the list of all possible targets of such contractions. Combining our results with the classification of toric Fano $4$-folds due to Batyrev and Sato we provide explicit examples of Fano conic bundles from toric $4$-folds with $\delta _{X}=3$.

scholarly journals Arithmetic hyperbolicity: automorphisms and persistence

2021 ◽  
Author(s):  
Ariyan Javanpeykar
Keyword(s):  
Automorphism Group ◽  
Number Field ◽  
Projective Variety ◽  
Rational Points ◽  
General Criterion ◽  
Finitely Generated ◽  
Integral Points ◽  
Lang's Conjecture ◽  
Lang’S Conjecture

AbstractWe show that if the automorphism group of a projective variety is torsion, then it is finite. Motivated by Lang’s conjecture on rational points of hyperbolic varieties, we use this to prove that a projective variety with only finitely many rational points has only finitely many automorphisms. Moreover, we investigate to what extent finiteness of S-integral points on a variety over a number field persists over finitely generated fields. To this end, we introduce the class of mildly bounded varieties and prove a general criterion for proving this persistence.

scholarly journals Dynamical Uniform Bounds for Fibers and a Gap Conjecture

2019 ◽  
Author(s):  
Jason Bell ◽  
Dragos Ghioca ◽  
Matthew Satriano
Keyword(s):  
Growth Rate ◽  
Positive Integer ◽  
Number Field ◽  
Projective Variety ◽  
Rational Map ◽  
Uniform Bounds ◽  
Weil Height ◽  
Forward Orbit

Abstract We prove a uniform version of the Dynamical Mordell–Lang Conjecture for étale maps; also, we obtain a gap result for the growth rate of heights of points in an orbit along an arbitrary endomorphism of a quasiprojective variety defined over a number field. More precisely, for our 1st result, we assume $X$ is a quasi-projective variety defined over a field $K$ of characteristic $0$, endowed with the action of an étale endomorphism $\Phi $, and $f\colon X\longrightarrow Y$ is a morphism with $Y$ a quasi-projective variety defined over $K$. Then for any $x\in X(K)$, if for each $y\in Y(K)$, the set $S_{x,y}:=\{n\in{\mathbb{N}}\colon f(\Phi ^n(x))=y\}$ is finite, then there exists a positive integer $N_x$ such that $\sharp S_{x,y}\le N_x$ for each $y\in Y(K)$. For our 2nd result, we let $K$ be a number field, $f:X\dashrightarrow{\mathbb{P}}^1$ is a rational map, and $\Phi $ is an arbitrary endomorphism of $X$. If ${\mathcal{O}}_{\Phi }(x)$ denotes the forward orbit of $x$ under the action of $\Phi $, then either $f({\mathcal{O}}_{\Phi }(x))$ is finite, or $\limsup _{n\to \infty } h(f(\Phi ^n(x)))/\log (n)>0$, where $h(\cdot )$ represents the usual logarithmic Weil height for algebraic points.

ON RAMIFIED COVERS OF THE PROJECTIVE PLANE I: INTERPRETING SEGRE'S THEORY (WITH AN APPENDIX BY EUGENII SHUSTIN)

2011 ◽  
Vol 22 (05) ◽  
pp. 619-653 ◽  
Author(s):  
MICHAEL FRIEDMAN ◽  
MAXIM LEYENSON ◽  
EUGENII SHUSTIN
Keyword(s):  
Projective Plane ◽  
Smooth Surface ◽  
Plane Curve ◽  
Main Question ◽  
Generic Projection ◽  
Complete Answer ◽  
Classical Work ◽  
Numerical Invariants ◽  
Ramified Covers ◽  
Branch Curve

We study ramified covers of the projective plane ℙ2. Given a smooth surface S in ℙN and a generic enough projection ℙN → ℙ2, we get a cover π: S → ℙ2, which is ramified over a plane curve B. The curve B is usually singular, but is classically known to have only cusps and nodes as singularities for a generic projection. The main question that arises is with respect to the geometry of branch curves; i.e. how can one distinguish a branch curve from a non-branch curve with the same numerical invariants? For example, a plane sextic with six cusps is known to be a branch curve of a generic projection iff its six cusps lie on a conic curve, i.e. form a special 0-cycle on the plane. The classical work of Beniamino Segre gives a complete answer to the second question in the case when S is a smooth surface in ℙ3. We give an interpretation of the work of Segre in terms of relation between Picard and Chow groups of 0-cycles on a singular plane curve B. In addition, the appendix written by E. Shustin shows the existence of new Zariski pairs.

scholarly journals Fiberwise bimeromorphic maps of conic bundles

2019 ◽  
Vol 30 (11) ◽  
pp. 1950059 ◽  
Author(s):  
Constantin Shramov
Keyword(s):  
Finite Groups ◽  
Conic Bundle ◽  
Conic Bundles

Given a holomorphic conic bundle without sections, we show that the orders of finite groups acting by its fiberwise bimeromorphic transformations are bounded. This provides an analog of a similar result obtained by Bandman and Zarhin for quasi-projective conic bundles.

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