Conic Bundles Research Articles

A conic bundle description of Moishezon twistor spaces without effective divisors of degree one

In [K2] Moishezon twistor spaces over the connected sum \(n{\mathbb{CP}}^2\) (\(n\geq 4\)), which do not contain effective divisors of degree one, were constructed as deformations of the twistor spaces introduced in [LeB]. We study their structure for \(n\geq 4\) by constructing a modification which is a conic bundle over \({\mathbb P}^2\). We show that they are rational. In case n = 4 we give explicit equations for such conic bundles and use them to construct explicit birational maps between these conic bundles and \({\mathbb P}^3\).

Birational automorphisms of a three-dimensional double cone

Birational rigidity is proved for a three-dimensional double cone, that is, a double cover of a quadric cone in with branching in a smooth section of a quartic, and its non-rationality and the absence of conic bundle structures are also proved. The group of birational automorphisms of this variety is computed.

Fibrations en variétés de Severi-Brauer au-dessus de la droite projective sur le corps des fonctions d'une courbe réelle

Let R be an Archimedean real closed field and K be the function field of an integral, projective, smooth R-curve. We show, by generalizing some demonstrations of [3], relative to conic bundles, that the “reciprocity” obstruction to the Hasse principle is the only one for Severi-Brauer varieties bundles over ℙ K 1.

Birational automorphisms of algebraic threefolds with a pencil of Del Pezzo surfaces

In this paper it is proved that there is only one pencil of rational surfaces on a smooth three-dimensional variety fibred into Del Pezzo surfaces of degree 1, 2 or 3 by a morphism (Iskovskikh's conjecture), provided that the class of 1-cycles (where is the canonical class and is the class of a line in a fibre) is not effective for any . This condition is satisfied if is sufficiently “twisted” over the base of the pencil. This implies that these varieties admit no conic bundle structures, and, in particular, that they are nonrational. Certain higher-dimensional generalizations of this theorem are considered: similar statements are true for varieties with a pencil of thee-dimensional quartics of general position, and analogous results are obtained in these cases.

On the existence of complements of the canonical divisor for Mori conic bundles

This paper continues the author's study of extremal contractions in the sense of Mori from three-dimensional varieties onto surfaces. Such contractions occur in a natural way in the birational classification theory of three-dimensional algebraic varieties. Reid's “general elephant” conjecture of the complementedness of the canonical divisor and also the conjecture about singularities of the base surface are discussed. The situation is studied locally near a singular fibre.

On Standard Projective Plane Bundles

For aK-formVKofP2over the function fieldKover an algebraically closed fieldkof char(k)=0, we will construct a proper flat surjective morphism τ:V→XwithVandXsmooth projective varieties overksuch that (i) the function field ofXisK, (ii) the generic fibre of τ is isomorphic toVK, (iii) the relative Picard number is equal to one. This is a generalization of a result of V. G. Sarkisov about standard conic bundles.

A rationality criterion for conic bundles

It is that a three-dimensional variety that is a conic bundle in the Mori sense has a base with at most double rational singularities of type . A rationality criterion is proved subject to this assumption in the case when the discriminant curve is large enough, for example, for the case when .

On the non existence of some rational 3-folds in ℙ5

In this paper we show that there are no smooth rational 3-folds in ℙ5 (C) which are rational conic bundles, over minimal surfaces, whose generic fibre is embedded as a rational curve of degreeh≥3, (ifh=2 there is a complete classification for these 3-folds as well as for the case of ℙ1-bundles). Except for conic bundles, we also give the complete list of rational 3-folds in ℙ5 which are minimal according to Mori’s theory. These are little steps towards the classification of all smooth 3-folds in ℙ5 not of general type.

Algebraic threefolds with two extremal morphisms

In [3] Mori gives a description of all extremal rays (extremal morphisms) arising on a smooth projective threefold with a numerically non-effective canonical bundle. Generally speaking, every smooth projective threefold V with a numerically non-effective canonical class Kv admits an extremal morphism π: V— → Y. The assumption that V admits a non-trivial pair of extremal morph-ismsimposes strong conditions on V. This is the essence of the Theorem 1.5 of the present work. In particular, we obtain a description of the threefolds which admit two biregular structures of conic bundles over non-singular surfaces S1 = Y1 and S2 = Y2. By the results of §3 the surfaces Sl and S2 must be either ruled surfaces with isomorphic basic curves, or S1 ≈ S2 ≈ P2.

On the Geometry of Conic Bundles Arising in Adjunction Theory

On the Geometry of Conic Bundles Arising in Adjunction Theory

ON THE RATIONALITY PROBLEM FOR CONIC BUNDLES

For a standard conic bundle over a rational surface the equivalence of two different pairs of conditions sufficient for rationality is proved, along with the necessity of certain weaker conditions.

Projective surfaces with bi-elliptic hyperplane sections

We study projective surfaces X which have a bi-elliptic curve (i.e. 2∶1 covering of an elliptic curve) among their hyperplane sections . We give a complete characterization of those surfaces when their degree d is d≥17 (only conic bundles and scrolls if d≥19, three possible exception otherwise) and when d≤8. A conjecture is given for the remaining cases. The main tool we use is the study of the adjunction mapping on X.

Vertically of (-1)-Lines in Scrolls Over Smooth Surfaces

AbstractLet S be a smooth surface contained as an ample divisor in a smooth complex projective threefold X, which is a P1 -bundle, and assume that induces OP1 (1) on the fibres of X. The following fact is proven. The restriction to S of the bundle projection of X is exactly the reduction morphism of the pair provided that this one is not a conic bundle. The proof is very simple and does not involve any consideration on the nefness of the adjoint bundle Some applications of the proof are given.

K-unirationality of conic bundles, the Kneser-Tits conjecture for spinor groups and central simple algebras

K-unirationality of conic bundles, the Kneser-Tits conjecture for spinor groups and central simple algebras

On the adjunction process over a surface in char.p

Let Ks be the canonical bundle on a non singular projective surface S (over an algebraically closed field F, char F=p) and L be a very ample line bundle on S. Suppose (S,L) is not one of the following pairs: (P2,O(e)), e=1,2, a quadric, a scroll, a Del Pezzo surface, a conic bundle. Then 1) (Ks⊗L)2 is spanned at each point by global sections. Let\(\phi :S \to P^N _F \) be the map given by the sections Γ(Ks⊗L)2, and let φ=s o r its Stein factorization. 2) r:S→S′=r(S) is the contraction of a finite number of lines, Ei for i=1,...r, such that Ei·Ei=KS·Ei=−L·Ei=−1. 3) If h°(L)≥6 and L·L≥9, then s is an embedding.

Topology of Conic Bundles

Topology of Conic Bundles

On the rationality problem for conic bundles

For a standard conic bundle over a rational surface the equivalence of two different pairs of conditions sufficient for rationality is proved, along with the necessity of certain weaker conditions.

THE INTERMEDIATE JACOBIAN OF A THREE-DIMENSIONAL CONIC BUNDLE IS A PRYM VARIETY

The paper contains the computation of the intermediate Jacobian of the general conic bundle. This computation is performed in three steps: a) computation of the Betti number (§2), b) computation of the Jacobian to within isogeny (§4), and c) computation of the polarization (§4).The author's results conclude the study of the intermediate Jacobian of conic bundles and give an efficient general answer.Bibliography: 6 titles.

PRYM VARIETIES: THEORY AND APPLICATIONS

In this paper the author determines when the principally polarized Prymian of a Beauville pair satisfying a certain stability type condition is isomorphic to the Jacobian of a nonsingular curve. As an application, he points out new components in the Andreotti-Mayer variety of principally polarized abelian varieties of dimension whose theta-divisors have singular locus of dimension ; he also proves a rationality criterion for conic bundles over a minimal rational surface in terms of the intermediate Jacobian. The first part of the paper contains the necessary preliminary material introducing the reader to the modern theory of Prym varieties.Figures: 10. Bibliography: 32 titles.

ON CONIC BUNDLE STRUCTURES

This paper determines under which conditions an algebraic variety can be uniquely (up to equivalence) represented in the form of a conic bundle. The results are used to show that many conic bundles over rational varieties are nonrational, and to construct examples of nonrational algebraic threefolds whose three-dimensional integral cohomology group is trivial. Bibliography: 16 titles.