Conic Bundles Research Articles

Discriminant Divisors for Conic Bundles

ABSTRACT We study some foundational properties on discriminant divisors for generically smooth conic bundles. In particular, we extend the formula $\Delta_f \equiv -f_*K_{X/T}^2$ to arbitrary characteristics.

Curve classes on conic bundle threefolds and applications to rationality

Curve classes on conic bundle threefolds and applications to rationality

On the birational geometry of conic bundles over the projective space

AbstractLet be a general minimal ‐fold conic bundle with a hypersurface of degree as discriminant. We prove that if , then is not pseudo‐effective, and that if , then none of the integral multiples of is effective. Finally, we provide examples of smooth unirational ‐fold conic bundles with a discriminant of arbitrarily high degree.

The second fundamental form of the moduli space of cubic threefolds in $\mathcal A_5$

We study the second fundamental form of the Siegel metric in $\mathcal A_5$ restricted to the locus of intermediate Jacobians of cubic threefolds. We prove that the image of this second fundamental form, which is known to be non-trivial, is contained in the kernel of a suitable multiplication map. Some ingredients are: the conic bundle structure of cubic threefolds, Prym theory, Gaussian maps and Jacobian ideals.

Equivariant pliability of the projective space

We classify finite subgroups Gsubset {textrm{PGL}}_4({mathbb {C}}) such that {mathbb {P}}^3 is not G-birational to conic bundles and del Pezzo fibrations, and explicitly describe all G-Mori fibre spaces that are G-birational to {mathbb {P}}^3 for these subgroups.

A preconditioned iterative interior point approach to the conic bundle subproblem

AbstractThe conic bundle implementation of the spectral bundle method for large scale semidefinite programming solves in each iteration a semidefinite quadratic subproblem by an interior point approach. For larger cutting model sizes the limiting operation is collecting and factorizing a Schur complement of the primal-dual KKT system. We explore possibilities to improve on this by an iterative approach that exploits structural low rank properties. Two preconditioning approaches are proposed and analyzed. Both might be of interest for rank structured positive definite systems in general. The first employs projections onto random subspaces, the second projects onto a subspace that is chosen deterministically based on structural interior point properties. For both approaches theoretic bounds are derived for the associated condition number. In the instances tested the deterministic preconditioner provides surprisingly efficient control on the actual condition number. The results suggest that for large scale instances the iterative solver is usually the better choice if precision requirements are moderate or if the size of the Schur complemented system clearly exceeds the active dimension within the subspace giving rise to the cutting model of the bundle method.

Rationality of Real Conic Bundles With Quartic Discriminant Curve

Abstract We study real double covers of $\mathbb P^{1}\times \mathbb P^{2}$ branched over a $(2,2)$-divisor, which are conic bundles with smooth quartic discriminant curve by the second projection. In each isotopy class of smooth plane quartics, we construct examples where the total space is $\mathbb R$-rational. For five of the six isotopy classes, we construct $\mathbb C$-rational examples with obstructions to rationality over $\mathbb R$, and for the sixth class, we show that the models we consider are all rational. Moreover, for three of the five classes with irrational members, we characterize rationality using the real locus and the intermediate Jacobian torsor obstruction of Hassett–Tschinkel and Benoist–Wittenberg. These double cover models were introduced by Frei, Sankar, Viray, Vogt, and the first author, who determined explicit descriptions for their intermediate Jacobian torsors.

Contractions of hyper-Kähler fourfolds and the Brauer group

We study the geometry of exceptional loci of birational contractions of hyper-Kähler fourfolds that are of K3[2]-type. These loci are conic bundles over K3 surfaces and we determine their classes in the Brauer group. For this we use the results on twisted sheaves on K3 surfaces, on contractions and on the corresponding Heegner divisors. For a general K3 surface of fixed degree there are three (T-equivalence) classes of order two Brauer group elements. The elements in exactly two of these classes are represented by conic bundles on such fourfolds. We also discuss various examples of such conic bundles.

Real-Fibered Morphisms of del Pezzo Surfaces and Conic Bundles

It goes back to Ahlfors that a real algebraic curve admits a real-fibered morphism to the projective line if and only if the real part of the curve disconnects its complex part. Inspired by this result, we are interested in characterising real algebraic varieties of dimension n admitting real-fibered morphisms to the n-dimensional projective space. We present a criterion to classify real-fibered morphisms that arise as finite surjective linear projections from an embedded variety which relies on topological linking numbers. We address special attention to real algebraic surfaces. We classify all real-fibered morphisms from real del Pezzo surfaces to the projective plane and determine which such morphisms arise as the composition of a projective embedding with a linear projection. Furthermore, we give some insights in the case of real conic bundles.

Schinzel Hypothesis on average and rational points

We resolve Schinzel’s Hypothesis (H) for 100% of polynomials of arbitrary degrees. We deduce that a positive proportion of diagonal conic bundles over {mathbb {Q}} with any given number of degenerate fibres have a rational point, and obtain similar results for generalised Châtelet equations.

Amplified endomorphisms of Fano fourfolds

Let X be a smooth Fano fourfold admitting a conic bundle structure. We show that X is toric if and only if X admits an amplified endomorphism; in this case, X is a rational variety.

BIRATIONAL RIGIDITY OF ORBIFOLD DEGREE 2 DEL PEZZO FIBRATIONS

AbstractVarieties fibered into del Pezzo surfaces form a class of possible outputs of the minimal model program. It is known that del Pezzo fibrations of degrees$1$and$2$over the projective line with smooth total space satisfying the so-called$K^2$-condition are birationally rigid: their Mori fiber space structure is unique. This implies that they are not birational to any Fano varieties, conic bundles, or other del Pezzo fibrations. In particular, they are irrational. The families of del Pezzo fibrations with smooth total space of degree$2$are rather special, as formostfamilies a general del Pezzo fibration has the simplest orbifold singularities. We prove that orbifold del Pezzo fibrations of degree$2$over the projective line satisfying explicit generality conditions as well as a generalized$K^2$-condition are birationally rigid.

Shokurov’s conjecture on conic bundles with canonical singularities

AbstractA conic bundle is a contraction$X\to Z$between normal varieties of relative dimension$1$such that$-K_X$is relatively ample. We prove a conjecture of Shokurov that predicts that if$X\to Z$is a conic bundle such thatXhas canonical singularities andZis$\mathbb {Q}$-Gorenstein, thenZis always$\frac {1}{2}$-lc, and the multiplicities of the fibres over codimension$1$points are bounded from above by$2$. Both values$\frac {1}{2}$and$2$are sharp. This is achieved by solving a more general conjecture of Shokurov on singularities of bases of lc-trivial fibrations of relative dimension$1$with canonical singularities.

Birational self-maps of threefolds of (un)-bounded genus or gonality

We study the complexity of birational self-maps of a projective threefold $X$ by looking at the birational type of surfaces contracted. These surfaces are birational to the product of the projective line with a smooth projective curve. We prove that the genus of the curves occuring is unbounded if and only if $X$ is birational to a conic bundle or a fibration into cubic surfaces. Similarly, we prove that the gonality of the curves is unbounded if and only if $X$ is birational to a conic bundle.

The density of fibres with a rational point for a fibration over hypersurfaces of low degree

We prove asymptotics for the proportion of fibres with a rational point in a conic bundle fibration. The base of the fibration is a general hypersurface of low degree.

Rational points on conic bundles over elliptic curves

We study rational points on conic bundles over elliptic curves with positive rank over a number field. We show that the étale Brauer–Manin obstruction is insufficient to explain failures of the Hasse principle for such varieties. We then further consider properties of the distribution of the set of rational points with respect to its image in the rational points of the elliptic curve. In the process, we prove results on a local-to-global principle for torsion points on elliptic curves over \({{{\mathbb {Q}}}}\).

Symplectic rational $G$-surfaces and equivariant symplectic cones

We give characterizations of a finite group $G$ acting symplectically on a rational surface ($\mathbb{CP}^2$ blown up at two or more points). In particular, we obtain a symplectic version of the dichotomy of $G$-conic bundles versus $G$-del Pezzo surfaces for the corresponding $G$-rational surfaces, analogous to a classical result in algebraic geometry. Besides the characterizations of the group $G$ (which is completely determined for the case of $\mathbb{CP}^2 \# N \overline{\mathbb{CP}^2}, N={2,3,4}$), we also investigate the equivariant symplectic minimality and equivariant symplectic cone of a given $G$-rational surface.

A log PSS morphism with applications to Lagrangian embeddings

Let $M$ be a smooth projective variety and $\mathbf{D}$ an ample normal crossings divisor. From topological data associated to the pair $(M, \mathbf{D})$, we construct, under assumptions on Gromov-Witten invariants, a series of distinguished classes in symplectic cohomology of the complement $X = M \backslash \mathbf{D}$. Under further "topological" assumptions on the pair, these classes can be organized into a Log(arithmic) PSS morphism, from a vector space which we term the logarithmic cohomology of $(M, \mathbf{D})$ to symplectic cohomology. Turning to applications, we show that these methods and some knowledge of Gromov-Witten invariants can be used to produce dilations and quasi-dilations (in the sense of Seidel-Solomon [SS]) in examples such as conic bundles. In turn, the existence of such elements imposes strong restrictions on exact Lagrangian embeddings, especially in dimension 3. For instance, we prove that any exact Lagrangian in a complex 3-dimensional conic bundle over $(\mathbb{C}^*)^2$ must be diffeomorphic to $T^3$ or a connect sum $\#^n S^1 \times S^2$.

Unramified Brauer groups of conic bundle threefolds in characteristic two

We establish a formula for computing the unramified Brauer group of tame conic bundle threefolds in characteristic 2. The formula depends on the arrangement and residue double covers of the discriminant components, the latter being governed by Artin-Schreier theory (instead of Kummer theory in characteristic not 2). We use this to give new examples of threefold conic bundles defined over the integers that are not stably rational over the complex numbers.

Quotients of higher-dimensional Cremona groups

We study large groups of birational transformations Bir(X), where X is a variety of dimension at least 3, defined over C or a subfield of C. Two prominent cases are when X is the projective space, in which case Bir(X) is the Cremona group of rank n, or when X is a smooth cubic hypersurface. In both cases, and more generally when X is birational to a conic bundle, we produce infinitely many distinct group homomorphisms from Bir(X) to Z/2, showing in particular that the group Bir(X) is not perfect and thus not simple. As a consequence we also obtain that the Cremona group of rank n at least 3 is not generated by linear and Jonqui\`eres elements.