Conic Bundles Research Articles

The Sen limit

F-theory compactifications on elliptic Calabi-Yau manifolds may be related to IIb compactifications by taking a certain limit in complex structure moduli space, introduced by A. Sen. The limit has been characterized on the basis of SL(2,Z) monodromies of the elliptic fibration. Instead, we introduce a stable version of the Sen limit. In this picture the elliptic Calabi-Yau splits into two pieces, a P 1 -bundle and a conic bundle, and the intersection yields the IIb space-time. We get a precise match between F-theory and perturbative type IIb. The correspondence is holographic, in the sense that physical quantities seemingly spread in the bulk of the F-theory Calabi-Yau may be rewritten as expressions on the log boundary. Smoothing the F-theory Calabi-Yau corresponds to summing up the D(−1)-instanton corrections to the IIb theory.

Counting rational points on del Pezzo surfaces with a conic bundle structure

For any number field $k$, upper bounds are established for the number of $k$-rational points of bounded height on non-singular del Pezzo surfaces defined over $k$, which are equipped with suitable conic bundle structures over $k$.

Derived categories and rationality of conic bundles

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.

Scheme of lines on a family of 2-dimensional quadrics: geometry and derived category

Given a generic family $$Q$$ of 2-dimensional quadrics over a smooth 3-dimensional base $$Y$$ we consider the relative Fano scheme $$M$$ of lines of it. The scheme $$M$$ has a structure of a generically conic bundle $$M \rightarrow X$$ over a double covering $$X \rightarrow Y$$ ramified in the degeneration locus of $$Q \rightarrow Y$$ . The double covering $$X \rightarrow Y$$ is singular in a finite number of points (corresponding to the points $$y \in Y$$ such that the quadric $$Q_y$$ degenerates to a union of two planes), the fibers of $$M$$ over such points are unions of two planes intersecting in a point. The main result of the paper is a construction of a semiorthogonal decomposition for the derived category of coherent sheaves on $$M$$ . This decomposition has three components, the first is the derived category of a small resolution $$X^+$$ of singularities of the double covering $$X \rightarrow Y$$ , the second is a twisted resolution of singularities of $$X$$ (given by the sheaf of even parts of Clifford algebras on $$Y$$ ), and the third is generated by a completely orthogonal exceptional collection.

Manin's conjecture for a singular quartic del Pezzo surface

We prove Manin's conjecture for a split singular quartic del Pezzo surface with singularity type 2A1 and eight lines. This is achieved by equipping the surface with a conic bundle structure. To handle the sum over the family of conics, we prove a result of independent interest on a certain restricted divisor problem for four binary linear forms.

Algebraic approximation of Kähler threefolds

AbstractIn the present work, we investigate existence of deformations and algebraic approximability for certain uniruled Kähler threefolds. In the first part, we establish existence of infinitesimal deformations for all conic bundles with relative Picard number one over a non‐algebraic compact Kähler surface S and existence of positive‐dimensional families of deformations in all but some special cases. In the second part, we study the question of algebraic approximability for projective bundles over S and threefolds bimeromorphic to \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {P}_1\times S$\end{document}.

Equations of G-minimal conic bundles

We develop a method for obtaining equations for G-minimal conic bundles with an arbitrary number of singular fibres. When the number of singular fibres is equal to 4, 6, or 7, a detailed classification is given, which includes obtaining the equations for minimal conic bundles (S,G) and an explicit indication of the action of the group G on the Picard group Pic(S) and on the surface S itself.Bibliography: 19 titles.

Manin’s conjecture for quartic Del Pezzo surfaces with a conic fibration

An asymptotic formula is established for the number of rational points of bounded height on a non-singular quartic del Pezzo surface with a conic bundle structure.

COUNTING MULTISECTIONS IN CONIC BUNDLES OVER A CURVE DEFINED OVER 𝔽q

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).

Geometrically rational real conic bundles and very transitive actions

AbstractIn this article we study the transitivity of the group of automorphisms of real algebraic surfaces. We characterize real algebraic surfaces with very transitive automorphism groups. We give applications to the classification of real algebraic models of compact surfaces: these applications yield new insight into the geometry of the real locus, proving several surprising facts on this geometry. This geometry can be thought of as a half-way point between the biregular and birational geometries.

Equations of G-minimal conic bundles with degree 4

We consider the first nontrivial case of relatively G-minimal conic bundles which are G-minimal and have r = 4 singular fibers. Classification obtained gives explicit equations of minimal conic bundles (S, G) and an explicit action of the group G on the Picard group Pic(S) and on the surface S.

A new series of compact minitwistor spaces and Moishezon twistor spaces over them

In recent papers [Honda, J. Diff. Geom 82: 411–444, 2009], [Honda, Invent. Math 174, 463–504, 2008], we gave explicit description of some new Moishezon twistor spaces. In this paper, developing the method in the papers much further, we explicitly give projective models of a number of new Moishezon twistor spaces, as conic bundles over some rational surfaces (called minitwistor spaces). These include the twistor spaces studied in the papers as very special cases.

On $ℚ$-Conic Bundles, III

A ℚ -conic bundle germ is a proper morphism from a threefold with only terminal singularities to the germ (Z\ni o) of a normal surface such that fibers are connected and the anti-canonical divisor is relatively ample. Building upon our previous paper [MP08a], we prove the existence of a Du Val anti-canonical member under the assumption that the central fiber is irreducible.

Explicit construction of new Moishezon twistor spaces

In this paper we explicitly construct Moishezon twistor spaces on nCP^2 for arbitrary n>1 which admit a holomorphic C*-action. When n=2, they coincide with Y. Poon's twistor spaces. When n=3, they coincide with the one studied by the author in math.DG/0403528. When n>3, they are new twistor spaces, to the best of the author's knowledge. By investigating the anticanonical system, we show that our twistor spaces are bimeromorphic to conic bundles over certain rational surfaces. The latter surfaces can be regarded as orbit spaces of the C*-action on the twistor spaces. Namely they are minitwistor spaces. We explicitly determine their defining equations in CP^4. It turns out that the structure of the minitwistor space is independent of n. Further we concrelely construct a CP^2-bundle over the resolution of this surface, and provide an explicit defining equation of the conic bundles. It shows that the number of irreducible components of the discriminant locus for the conic bundles increases as n does. Thus our twistor spaces have a lot of similarities with the famous LeBrun twistor spaces, where the minitwistor space CP^1 x CP^1 in LeBrun's case is replaced by our minitwistor spaces found in math.DG/0508088.

Relative Chow–Künneth Decompositions for Conic Bundles and Prym Varieties

We construct a relative Chow-Kunneth decomposition for a conic bundle over a surface such that the middle projector gives the Prym variety of the associated double covering of the discriminant of the conic bundle. This gives a refinement (up to an isogeny) of Beauville's theorem on the relation between the intermediate Jacobian of the conic bundle and the Prym variety of the double covering.

The set of nonsquares in a number field is diophantine

Fix a number field k. We prove that k* - k*^2 is diophantine over k. This is deduced from a theorem that for a nonconstant separable polynomial P(x) in k[x], there are at most finitely many a in k* modulo squares such that there is a Brauer-Manin obstruction to the Hasse principle for the conic bundle X given by y^2 - az^2 = P(x).

On $ℚ$-conic Bundles, II

A ℚ -conic bundle germ is a proper morphism from a threefold with only terminal singularities to the germ (Z\ni o) of a normal surface such that fibers are connected and the anti-canonical divisor is relatively ample. We obtain the complete classification of ℚ -conic bundle germs when the base surface germ is singular. This is a generalization of [MP08], which further assumed that the fiber over o is irreducible.

Linearisation of finite Abelian subgroups of the Cremona group of the plane

Given a finite Abelian subgroup of the Cremona group of the plane, we provide a way to decide whether it is birationally conjugate to a group of automorphisms of a minimal surface. In particular, we prove that a finite cyclic group of birational transformations of the plane is linearisable if and only if none of its non-trivial elements fix a curve of positive genus. For finite Abelian groups, there exists only one surprising exception, a group isomorphic to ℤ/2ℤ × ℤ/4ℤ , whose non-trivial elements do not fix a curve of positive genus but which is not conjugate to a group of automorphisms of a minimal rational surface. We also give some descriptions of automorphisms (not necessarily of finite order) of del Pezzo surfaces and conic bundles.

Note on Conic Bundles over Henselian Discrete Valued Fields with Real Closed Residue Field

Let K be the fraction field of a Henselian discrete valuation ring. In Bazyleu et al. (2007) an explicit description of the central simple algebras of exponent 2 in the kernel of the natural map 2 Br(K(x)) → ∏ω∈Ω 2 Br(K(x)ω), (the product taken over all orderings ω and K(x)ω the real closure of K(x) at ω), is given. This allows to describe a class of conic bundle surfaces over K in terms of their local data (i.e., in terms of their degenerate fibres). Such a description is given in Theorem 3.1, the main result of this note.

On $ℚ$-conic Bundles

A ℚ -conic bundle is a proper morphism from a threefold with only terminal singularities to a normal surface such that fibers are connected and the anti-canonical divisor is relatively ample. We study the structure of ℚ -conic bundles near their singular fibers. One corollary to our main results is that the base surface of every ℚ -conic bundle has only Du Val singularities of type A (a positive solution of a conjecture by Iskovskikh). We obtain the complete classification of ℚ -conic bundles under the additional assumption that the singular fiber is irreducible and the base surface is singular.