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.

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 APPROXIMATIONS FONCTIONNELLES DES COURBES DES ESPACES PROJECTIFS

2011 ◽  
Vol 07 (05) ◽  
pp. 1195-1215 ◽  
Author(s):  
PATRICE PHILIPPON
Keyword(s):  
Diophantine Approximation ◽  
Function Field ◽  
Algebraic Closure ◽  
Algebraic Independence ◽  
Projective Spaces ◽  
Higher Dimension ◽  
Algebraic Numbers ◽  
Flexible Approach ◽  
Algebraic Approximation ◽  
Multiplicity Estimates

Algebraic approximation to points in projective spaces offers a new and more flexible approach to algebraic independence theory. When working over the field of algebraic numbers, it leads to open conjectures in higher dimension extending known results in Diophantine approximation. We show here that over the algebraic closure of a function field in one variable, the analog of these conjectures is true. We also derive transfer lemmas which have applications in the study of multiplicity estimates, for example.

scholarly journals A subspace theorem for ordinary linear differential equations

1991 ◽  
Vol 50 (2) ◽  
pp. 320-332
Author(s):  
Alice Ann Miller
Keyword(s):  
Differential Equations ◽  
Function Field ◽  
Algebraic Numbers ◽  
Finite Point ◽  
Subspace Theorem ◽  
Power Series Solutions ◽  
Linear Differential ◽  
Image Position ◽  
Formal Power ◽  
Ordinary Linear Differential Equations

AbstractThe study of the S-unit equation for algebraic numbers rests very heavily on Schmidt's Subspace Theorem. Here we prove an effective subspace theorem for the differential function field case, which should be valuable in the proof of results concerning the S-unit equation for function fields. Theorem 1 states that either has a given upper bound where are linearly independent linear forms in the polynomials with coefficients that are formal power series solutions about x = 0 of non-zero differential equations and where Orda denotes the order of vanishing about a regular (finite) point of functions ƒk, i: (k = 1, n; i = 1, n) or lies inside one of a finite number of proper subspaces of (K(x))n. The proof of the theorem is based on the wroskian methods and graded sub-rings of Picard-Vessiot extensions developed by D. V. Chudnovsky and G. V. Chudnovsky in their function field analogues of the Roth and Schmidt theorems. A brief discussion concerning the possibility of a subspace theorem for a product of valuations including the infinite one is also included.

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.

scholarly journals A note on Baker's method

1969 ◽  
Vol 10 (1-2) ◽  
pp. 197-203
Author(s):  
K. Ramachandra
Keyword(s):  
Natural Number ◽  
General Result ◽  
Algebraic Numbers ◽  
Baker’S Method ◽  
Arbitrary Natural Number ◽  
Explicit Constant ◽  
Special Case

Let …, αn, (n ≧ 2) be (fixed) multiplicatively independent non zero algebraic numbers and set M(H) = min |β1log α1+…+βn log αn| the minimum taken over all algebraic numbers bgr;1,…βn not all equal to zero, of degrees not exceeding a fixed natural number d0, and heights not exceeding an arbitrary natural number H. Then an important result [1] of Baker states that for every fixed ε > 0 and an explicit constant . It may be remarked that Baker deduces his general result from the special case where βn is fixed to be —1. The following straight forward generalization might be of some interest since it shows that the exponent n+1+ε need not be the best, and that the best exponent obtainable by his method has some chance of being 1 + ε (see the corollary to the Theorem).

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.

scholarly journals Stable rationality of higher dimensional conic bundles

2018 ◽  
Vol Volume 2 ◽  
Author(s):  
Hamid Ahmadinezhad ◽  
Takuzo Okada
Keyword(s):  
Vector Bundle ◽  
Canonical Divisor ◽  
Conic Bundle ◽  
Final Version ◽  
Stable Rationality ◽  
Higher Dimensional ◽  
Projective Vector ◽  
Conic Bundles

We prove that a very general nonsingular conic bundle $X\rightarrow\mathbb{P}^{n-1}$ embedded in a projective vector bundle of rank $3$ over $\mathbb{P}^{n-1}$ is not stably rational if the anti-canonical divisor of $X$ is not ample and $n\geq 3$. Comment: Final version. To appear in Epijournal de Geometrie Algebrique

Comparison theorems for conic bundles

1981 ◽  
Vol 90 (1) ◽  
pp. 21-31 ◽  
Author(s):  
P. E. Newstead
Keyword(s):  
Vector Bundle ◽  
Comparison Theorems ◽  
Singular Case ◽  
Conic Bundle ◽  
Proposition 8 ◽  
Conic Bundles ◽  
Quadratic Complex

In (10), M. S. Narasimhan and S. Ramanan proved a theorem to the effect that a certain conic bundle associated with a non-singular quadratic complex does not come from a vector bundle ((10), proposition 8·1); a similar topological result was proved in (12). In the course of attempting to extend these results to the singular case, I found that I wanted to use some results on conic bundles which were not readily available in the literature. The object of this note is to give proofs of these results; the work on quadratic complexes is still in progress and the first part will appear shortly (13). A further application will appear in (14).

scholarly journals Derived categories and rationality of conic bundles

2013 ◽  
Vol 149 (11) ◽  
pp. 1789-1817 ◽  
Author(s):  
Marcello Bernardara ◽  
Michele Bolognesi
Keyword(s):  
Rational Surface ◽  
Derived Categories ◽  
Derived Category ◽  
Conic Bundle ◽  
Direct Sum ◽  
Conic Bundles ◽  
Intermediate Jacobian

AbstractWe show that a standard conic bundle over a minimal rational surface is rational and its Jacobian splits as the direct sum of Jacobians of curves if and only if its derived category admits a semiorthogonal decomposition by exceptional objects and the derived categories of those curves. Moreover, such a decomposition gives the splitting of the intermediate Jacobian also when the surface is not minimal.

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