Del Pezzo Fibrations Research Articles

Birational geometry of del Pezzo fibrations with terminal quotient singularities

Del Pezzo fibrations appear as minimal models of rationally connected varieties. The rationality of smooth del Pezzo fibrations is a well studied question but smooth fibrations are not dense in moduli. Little is known about the rationality of the singular models. We prove birational rigidity, hence non-rationality, of del Pezzo fibrations with simple non-Gorenstein singularities satisfying the famous $K^2$-condition. We then apply this result to study embeddings of $\operatorname{PSL}_2(7)$ into the Cremona group.

Standard Models of Degree 1 del Pezzo Fibrations

We construct a standard birational model (a model that has Gorenstein canonical singularities) for the three-dimensional del Pezzo fibrations ${ \pi \colon X \longrightarrow C }$ of degree $1$ and relative Picard number $1$. We also embed the standard model into the relative weighted projective space $\mathbb{P}_C (1,1,2,3)$. Our construction works in the $G$-equivariant category where $G$ is a finite group.

On stable rationality of Fano threefolds and del Pezzo fibrations

Abstract We prove that very general non-rational Fano threefolds which are not birational to cubic threefolds are not stably rational.

On the existence of almost Fano threefolds with del Pezzo fibrations

By Jahnke–Peternell–Radloff and Takeuchi, almost Fano threefolds with del Pezzo fibrations were classified. Among them, there exist 10 classes such that the existence of members of these was not proved. In this paper, we construct such examples belonging to each of 10 classes.

Automorphisms of threefolds that can be represented as an intersection of two quadrics

We prove that any -del Pezzo threefold of degree , except for a one-parameter family and four distinguished cases, can be equivariantly reconstructed to the projective space , a quadric , a -conic bundle or a del Pezzo fibration. We also show that one of these four distinguished varieties is birationally rigid with respect to an index subgroup of its automorphism group. Bibliography: 15 titles.

Nonrational del Pezzo fibrations admitting an action of the Klein simple group

We present a series of del Pezzo fibrations of degree 2 admitting an action of the Klein simple group and prove their nonrationality by the reduction modulo p method of Kollar. This is relevant to the embedding of the Klein simple group into the Cremona group of rank 3.

On pliability of del Pezzo fibrations and Cox rings

Abstract We develop some concrete methods to build Sarkisov links, starting from Mori fibre spaces. This is done by studying low rank Cox rings and their properties. As part of this development, we give an algorithm to construct explicitly the coarse moduli space of a toric Deligne–Mumford stack. This can be viewed as the generalisation of the notion of well-formedness for weighted projective spaces to homogeneous coordinate ring of toric varieties. As an illustration, we apply these methods to study birational transformations of certain fibrations of del Pezzo surfaces over ℙ 1 ${\mathbb{P}^{1}}$ , into other Mori fibre spaces, using Cox rings and variation of geometric invariant theory. We show that the pliability of these Mori fibre spaces is at least three and they are not rational.

Embedding pointed curves in K3 surfaces

We analyze morphisms from pointed curves to K3 surfaces with a distinguished rational curve, such that the marked points are taken to the rational curve, perhaps with specified cross ratios. This builds on work of Mukai and others characterizing embeddings of curves into K3 surfaces via non-abelian Brill–Noether theory. Our study leads naturally to enumerative problems, which we solve in several specific cases. These have applications to the existence of sections of del Pezzo fibrations with prescribed invariants.

On del Pezzo fibrations that are not birationally rigid

We consider $\mathbb{P}(1,1,1,2)$ bundles over $\mathbb{P}^1$ and construct hypersurfaces of these bundles which form a degree 2 del Pezzo fibration over $\mathbb{P}^1$ as a Mori fibre space. We classify all such hypersurfaces whose type $\III$ or $\IV$ Sarkisov links pass to a different Mori fibre space. A similar result for cubic surface fibrations over $\mathbb{P}^2$ is also presented.

Do Uniruled Six-manifolds Contain Sol Lagrangian Submanifolds?

We prove using symplectic field theory combined with some Gromov–Witten theory that if the suspension of a hyperbolic diffeomorphism of the two-torus Lagrangian embeds in a closed uniruled symplectic six-manifold, then its first rational homology is generated by the boundary of a symplectic disk. This prevents any Sol manifold from being in the real locus of an orientable real Del Pezzo fibration over a curve, confirming an expectation of Kollár. It also constrains Hamiltonian diffeomorphisms of uniruled symplectic four-manifolds.

Smoothable del Pezzo surfaces with quotient singularities

AbstractWe classify del Pezzo surfaces with quotient singularities and Picard rank one which admit a ℚ-Gorenstein smoothing. These surfaces arise as singular fibres of del Pezzo fibrations in the 3-fold minimal model program and also in moduli problems.

On the continuous part of codimension 2 algebraic cycles on three-dimensional varieties

Let be a nonsingular projective threefold over an algebraically closed field and let be the group of algebraically trivial codimension 2 algebraic cycles on modulo rational equivalence with coefficients in . Assume that is birationally equivalent to a threefold fibered over an integral curve with generic fiber satisfying the following three conditions: the motive is finite-dimensional; ; is spanned by divisors on . We prove that under these three assumptions the group is weakly representable: there exist a curve and a correspondence on such that induces an epimorphism , where is isomorphic to tensored with . In particular, this result holds for threefolds birationally equivalent to three-dimensional del Pezzo fibrations over a curve.Bibliography: 12 titles.

Multiple fibers of del Pezzo fibrations

We prove that a terminal three-dimensional del Pezzo fibration has no fibers of multiplicity >6. We also obtain a rough classification of possible configurations of singular points on multiple fibers and give some examples.

On the rationality of non-singular threefolds with a pencil of Del Pezzo surfaces of degree 4

A criterion for the non-singularity of a complete intersection of two fibrewise quadrics in is obtained. The following addition to Alexeev's theorem on the rationality of standard Del Pezzo fibrations of degree 4 over is deduced as a consequence: each fibration of this kind with topological Euler characteristic is proved to be rational.

Birationally rigid del Pezzo fibrations

In this paper we study del Pezzo fibrations z:X→ℙ1 of degree 1 and 2 such that X is smooth, rk Pic(X)=2 and Open image in new window These are examples of smooth birationally rigid 3-fold Mori fibre spaces. We describe all birational transformations of the 3-fold X into elliptic fibrations, fibrations of surfaces of Kodaira dimension zero, and canonical Fano 3-folds.

Stable rank two bundles on Del-Pezzo fibrations of degree 1 or 2

In this paper we describe stable rank two bundles on Del-Pezzo fibrations of degree 1 and 2 exploiting the correspondence with the lines in fibers. We show that the their moduli space has an open subset birational to a certain projective bundle over the Hilbert scheme of lines.

A Note on Del Pezzo Fibrations of Degree 1

The purpose of this paper is to extend the result in Park (Park, J. (2001). Birational maps of del Pezzo fibrations. J. Reine Angew. Math. 538:213–221) in the case of del Pezzo fibrations of degree 1. To this end we investigate the anticanonical linear systems of del Pezzo surfaces of degree 1. We then classify all possible effective anticanonical divisors on Gorenstein del Pezzo surfaces of degree 1 with canonical singularities.

On birational morphisms between pencils of Del Pezzo surfaces

Let X / S X/S and X ′ / S ′ X’/S’ be two Del Pezzo fibrations of degrees d d , d ′ d’ respectively. Assume that X X and X ′ X’ differ by a flop. Then we prove that d = d ′ d=d’ and give a short list of values of other basic numerical invariants of X X and X ′ X’ .

Birational maps of del Pezzo fibrations

In this paper, we study rigidity of nonsingular del Pezzo fibrations over a germ of smooth curve.

On the birational rigidity of some pencils of Del Pezzo surfaces

In this note, we discuss birational properties of some three-dimensional Del Pezzo fibrations of degree two.