Conic Bundles Research Articles

New point compression method for elliptic [formula omitted]-curves of j-invariant 0

In the article we propose a new compression method (to 2⌈log2⁡(q)⌉+3 bits) for the Fq2-points of an elliptic curve Eb:y2=x3+b (for b∈Fq2⁎) of j-invariant 0. It is based on Fq-rationality of some generalized Kummer surface GKb. This is the geometric quotient of the Weil restriction Rb:=RFq2/Fq(Eb) under the order 3 automorphism restricted from Eb. More precisely, we apply the theory of conic bundles (i.e., conics over the function field Fq(t)) to obtain explicit and quite simple formulas of a birational Fq-isomorphism between GKb and A2. Our point compression method consists in computation of these formulas. To recover (in the decompression stage) the original point from Eb(Fq2)=Rb(Fq) we find an inverse image of the natural map Rb→GKb of degree 3, i.e., we extract a cubic root in Fq. For q≢1(mod27) this is just a single exponentiation in Fq, hence the new method seems to be much faster than the classical one with x-coordinate, which requires two exponentiations in Fq.

Elliptic fibrations on K3 surfaces with a non-symplectic involution fixing rational curves and a curve of positive genus

In this paper we complete the classification of the elliptic fibrations on K3 surfaces which admit a non-symplectic involution acting trivially on the Néron–Severi group. We use the geometric method introduced by Oguiso and moreover we provide a geometric construction of the fibrations classified. If the non-symplectic involution fixes at least one curve of genus 1, we relate all the elliptic fibrations on the K3 surface with either elliptic fibrations or generalized conic bundles on rational elliptic surfaces. This description allows us to write the Weierstrass equations of the elliptic fibrations on the K3 surfaces explicitly and to study their specializations.

Coble fourfold, 𝔖6-invariant quartic threefolds, and Wiman–Edge sextics

We construct two small resolutions of singularities of the Coble fourfold (the double cover of the four-dimensional projective space branched over the Igusa quartic). We use them to show that all $S_6$-invariant three-dimensional quartics are birational to conic bundles over the quintic del Pezzo surface with the discriminant curves from the Wiman-Edge pencil. As an application, we check that $S_6$-invariant three-dimensional quartics are unirational, obtain new proofs of rationality of four special quartics among them and irrationality of the others, and describe their Weil divisor class groups as $S_6$-representations.

On the integral Hodge conjecture for real varieties, II

We establish the real integral Hodge conjecture for 1-cycles on various classes of uniruled threefolds (conic bundles, Fano threefolds with no real point, some del Pezzo fibrations) and on conic bundles over higher-dimensional bases which themselves satisfy the real integral Hodge conjecture for 1-cycles. In addition, we show that rationally connected threefolds over non-archimedean real closed fields do not satisfy the real integral Hodge conjecture in general and that over such fields, Br\"ocker's EPT theorem remains true for simply connected surfaces of geometric genus zero but fails for some K3 surfaces.

Geometry of ℙ2 blown up at seven points

Abstract In this paper, we prove that blown up at seven general points admits a conic bundle structure over ℙ1 and it can be embedded as (2, 2) divisor in ℙ1 × ℙ2. Conversely, any smooth surface in the complete linear system |(2, 2)| of ℙ1 × ℙ2 can be obtained as an embedding of blowing up ℙ2 at seven points. We also show that smooth surface linearly equivalent to (2, 2) in ℙ1 × ℙ2 has at most four (−2) curves.

A Characterization of Some Fano 4-folds Through Conic Fibrations

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 $\delta _{X}=3$.

A note on flatness of some fiber type contractions

We discuss the flatness property of some fiber type contractions of complex smooth projective varieties of arbitrary dimensions. We relate the flatness of some morphisms having one-dimensional fibers with their conic bundles structures, also in the general case in which some mild singularities of the varieties are admitted.

Conic bundle fourfolds with nontrivial unramified Brauer group

We derive a formula for the unramified Brauer group of a general class of rationally connected fourfolds birational to conic bundles over smooth threefolds. We produce new examples of conic bundles over P^3 where this formula applies and which have nontrivial unramified Brauer group. The construction uses the theory of contact surfaces and, at least implicitly, matrix factorizations and symmetric arithmetic Cohen--Macaulay sheaves, as well as the geometry of special arrangements of rational curves in P^2. We also prove the existence of universally CH_0-trivial resolutions for the general class of conic bundle fourfolds we consider. Using the degeneration method, we thus produce new families of rationally connected fourfolds whose very general member is not stably rational.

Fiberwise bimeromorphic maps of 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.

Rational points and non-anticanonical height functions

A conjecture of Batyrev and Manin predicts the asymptotic behaviour of rational points of bounded height on smooth projective varieties over number fields. We prove some new cases of this conjecture for conic bundle surfaces equipped with some non-anticanonical height functions. As a special case, we verify these conjectures for the first time for some smooth cubic surfaces for height functions associated to certain ample line bundles.

Positivity of anticanonical divisors from the viewpoint of Fano conic bundles

We give the first examples of flat fiber type contractions of Fano manifolds onto varieties that are not weak Fano, and we prove that these morphisms are Fano conic bundles. We also review some known results about the interaction between the positivity properties of anticanonical divisors of varieties of contractions.

Degree and the Brauer–Manin obstruction

Let X be a smooth variety over a number field k embedded as a degree d subvariety of $\mathbb{P}^n$ and suppose that X is a counterexample to the Hasse principle explained by the Brauer-Manin obstruction. We consider the question of whether the obstruction is given by the d-primary subgroup of the Brauer group, which would have both theoretic and algorithmic implications. We prove that this question has a positive answer in the case of torsors under abelian varieties, Kummer surfaces and (conditional on finiteness of Tate-Shafarevich groups) bielliptic surfaces. For Kummer varieties we show that the obstruction is already given by the 2-primary torsion. We construct a conic bundle over an elliptic curve that shows that, in general, the answer is no.

Stable rationality of higher dimensional 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 $n\geq 3$.

Geometric Brauer residue via root stacks

We reinterpret the residue map for the Brauer group of a smooth variety using a root stack construction and Weil restriction for algebraic stacks, and apply the result to find a geometric representative of for the residue of a Brauer class associated to a conic bundle, or more generally a Brauer-Severi bundle.

The rationality problem for conic bundles

This expository paper is concerned with the rationality problem for three-dimensional algebraic varieties with a conic bundle structure. The main methods of this theory are discussed, proofs of certain principal results are sketched, and some recent achievements are presented. Many open problems are also stated. Bibliography: 209 titles.

Conic bundles and iterated root stacks

We generalize a classical result by V. G. Sarkisov about standard models for conic bundles to the case of a not necessarily algebraically closed perfect field, using iterated root stacks, destackification, and resolution of singularities.

Stable rationality of quadric and cubic surface bundle fourfolds

We study the stable rationality problem for quadric and cubic surface bundles over surfaces from the point of view of the degeneration method for the Chow group of 0-cycles. Our main result is that a very general hypersurface X of bidegree (2,3) in P^2 x P^3 is not stably rational. Via projections onto the two factors, X is a cubic surface bundle over P^2 and a conic bundle over P^3, and we analyze the stable rationality problem from both these points of view. This provides another example of a smooth family of rationally connected fourfolds with rational and nonrational fibers. Finally, we introduce new quadric surface bundle fourfolds over P^2 with discriminant curve of any even degree at least 8, having nontrivial unramified Brauer group and admitting a universally CH_0-trivial resolution.

Rational points of bounded height on general conic bundle surfaces

A conjecture of Manin predicts the asymptotic distribution of rational points of bounded height on Fano varieties. In this paper we use conic bundles to obtain correct lower bounds or a wide class of surfaces over number fields for which the conjecture is still far from being proved. For example, we obtain the conjectured lower bound of Manin's conjecture for any del Pezzo surface whose Picard rank is sufficiently large, or for arbitrary del Pezzo surfaces after possibly an extension of the ground field of small degree.

Linear systems on rational elliptic surfaces and elliptic fibrations on K3 surfaces

We consider K3 surfaces which are double covers of rational elliptic surfaces. The former are endowed with a natural elliptic fibration, which is induced by the latter. There are also other elliptic fibrations on such K3 surfaces, which are necessarily induced by special linear systems on the rational elliptic surfaces. We describe these linear systems. In particular, we observe that every conic bundle on the rational surface induces a genus 1 fibration on the K3 surface and we classify the singular fibers of the genus 1 fibration on the K3 surface it terms of singular fibers and special curves on the conic bundle on the rational surface.

Degenerations of Gushel\u2013Mukai fourfolds, with a view towards irrationality proofs

We study a certain class of degenerations of Gushel–Mukai fourfolds as conic bundles, which we call tame degenerations and which are natural if one wants to prove that very general Gushel–Mukai fourfolds are irrational using the degeneration method due to Voisin, Colliot-Thélène–Pirutka, Totaro et al. However, we prove that no such tame degenerations exist.