Cylinders in canonical del Pezzo fibrations

In this article, we shall look into the existence of vertical cylinders contained in a weak del Pezzo fibration as a generalization of the former work due to Dubouloz and Kishimoto in which they observed that of vertical cylinders found in del Pezzo fibrations. The essence lying in the existence of a cylinder in the generic fiber, we devote mainly ourselves into a geometry of minimal weak del Pezzo surfaces defined over a field of characteristic zero from the point of view of cylinders. As a result, we give the classification of minimal weak del Pezzo surfaces defined over a field of characteristic zero, moreover, we show that weak del Pezzo fibrations containing vertical cylinders are quite restrictive.

We discuss various aspects of affine space fibrations \(f : X \rightarrow Y\) including the generic fiber, singular fibers and the case with a unipotent group action on X. The generic fiber \(X_\eta \) is a form of \({\mathbb A}^n\) defined over the function field k(Y) of the base variety. Singular fibers in the case where X is a smooth (or normal) surface or a smooth threefold have been studied, but we do not know what they look like even in the case where X is a singular surface. The propagation of properties of a given smooth fiber to nearby fibers will be studied in the equivariant case of Abhyankar-Sathaye Conjecture in dimension three. We also treat the triviality of a form of \({\mathbb A}^n\) if it has a unipotent group action. Treated subjects are classified into the following four themes 1. Singular fibers of \({\mathbb A}^1\)- and \({\mathbb P}^1\)-fibrations, 2. Equivariant Abhyankar-Sathaye Conjecture in dimension three, 3. Forms of \({\mathbb A}^3\) with unipotent group actions, 4. Cancellation problem in dimension three.

Varieties 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.

Let R be a real closed field and let X be an affine algebraic variety over R. We say that X is universally map rigid (UMR for short) if, for each irreducible affine algebraic variety Z over R, the set of nonconstant rational maps from Z to X is finite. A bijective map Open image in new window from an affine algebraic variety over R to X is called weak change of the algebraic structure of X if it is regular and φ−1 is a Nash map, which preserves nonsingular points. We prove the following rigidity theorem: every affine algebraic variety over R is UMR up to a weak change of its algebraic structure. Let us introduce another notion. Let Y be an affine algebraic variety over R. We say that X and Y are algebraically unfriendly if all the rational maps from X to Y and from Y to X are trivial, i.e., Zariski locally constant. From the preceding theorem, we infer that, if dim (X)≥1, then there exists a set Open image in new window of weak changes of the algebraic structure of X such that, for each t,s ∈ R with t≠s, Open image in new window and Open image in new window are algebraically unfriendly. This result implies the following expected fact: For each (nonsingular) affine algebraic variety X over R of positive dimension, the natural Nash structure of X does not determine the algebraic structure of X. In fact, the moduli space of birationally nonisomorphic (nonsingular) affine algebraic varieties over R, which are Nash isomorphic to X, has the same cardinality of R. This result was already known under the special assumption that R is the field of real numbers and X is compact and nonsingular.

We study certain pencils of del Pezzo surfaces generated by a smooth del Pezzo surface $S$ of degree less or equal to 3 anti-canonically embedded into a weighted projective space P and an appropriate multiple of a hyperplane $H$. Our main observation is that every minimal model program relative to the morphism lifting such pencil on a suitable resolution of its indeterminacies preserves the open subset $P H ≃ A^3$. As an application, we obtain projective completions of $A^3$ into del Pezzo fibrations over $P^1$ of every degree less or equal to 4. We also obtain completions of $A^3$ into Mori conic bundles, whose restrictions to $A^3$ are twisted $C*$-fibrations over $A^2$.

We show that a del Pezzo fibration π: V → W of degree d contains a vertical open cylinder, that is, an open subset whose intersection with the generic fiber of π is isomorphic to Z × AK1 for some quasi-projective variety Z defined over the function field K of W, if and only if d ≥ 5 and π: V → W admits a rational section. We also construct twisted cylinders in total spaces of threefold del Pezzo fibrations π: V → P1 of degree d ≤ 4.

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.

We consider threefold del Pezzo fibrations over a curve germ whose central fiber is nonrational. Under the additional assumption that the singularities of the total space are at worst ordinary double points, we apply a suitable base change and show that there is a one-to-one correspondence between such fibrations and certain nonsingular del Pezzo fibrations equipped with a cyclic group action.

We consider free affine actions of unipotent complex algebraic groups on Cn and prove that such actions admit an analytic geometric quotient if their degree is at most 2. Moreover, we classify free affine C2-actions on Cn of degree n - 1 and n - 2. For every n > 4, an action of degree n - 2 appears in the classification whose quotient topology is not Hausdorff.

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.

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.

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

We study linear actions of algebraic groups on smooth projective varieties X. A guiding goal for us is to understand the cohomology of quotients under such actions, by generalizing (from reductive to non-reductive group actions) existing methods involving Mumford's geometric invariant theory (GIT). We concentrate on actions of unipotent groups H, and define sets of stable points X^s and semistable points X^{ss}, often explicitly computable via the methods of reductive GIT, which reduce to the standard definitions due to Mumford in the case of reductive actions. We compare these with definitions in the literature. Results include (1) a geometric criterion determining whether or not a ring of invariants is finitely generated, (2) the existence of a geometric of X^s, and (3) the existence of a canonical enveloping quotient variety of X^{ss}, denoted X//H, which (4) has a projective completion given by a reductive GIT and (5) is itself projective and isomorphic to Proj(k[X]^H) when k[X]^H is finitely generated.

Unipotent group actions on affine varieties

Let $X$ be a del Pezzo surface over the function field of a complex curve. We study the behavior of rational points on $X$ leading to bounds on the counting function in Geometric Manin's Conjecture. A key tool is the Movable Bend and Break Lemma which yields an inductive approach to classifying relatively free sections for a del Pezzo fibration over a curve. Using this lemma we prove Geometric Manin's Conjecture for certain split del Pezzo surfaces of degree $\geq 2$ admitting a birational morphism to $\mathbb P^2$ over the ground field.