Cylinders in del Pezzo fibrations

Abstract

We show that a del Pezzo fibration π : V → W of degre d contains a vertical open cylinder, that is, an open subset whose intersection with the generic fiber of π is isomorphic to Z × A 1 K 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 → P 1 of degree d ≤ 4.

Super-rigid affine Fano varieties

“…Combined with the above results on -polar cylinders of singular del Pezzo surfaces and Example 1.7, this leads to anticipate that the complement of a del Pezzo surface with a -polar cylinder contains an open -cylinder. Ideas evolving from the proof of [CPW16a, Theorem 4.1] and the study of open ‘vertical’ -cylinders in del Pezzo fibrations [DK18] turn out to confirm this expectation, as follows.…”

“…Combined with the above results on (−K S )-polar cylinders of singular del Pezzo surfaces and Example 1.7, this leads to anticipate that the complement of a surface with a (−K S )-polar cylinder contains an open A 1 -cylinder. Ideas evolving from the proof of [11,Theorem 4.1] and the study of open "vertical" A 1 -cylinders in del Pezzo fibrations [21] turn out to confirm this expectation, namely: Before we proceed the proof, let us first prepare setups for the proof. Due to Theorem 4.3 above the surface S contains a singular point P .…”

“…This pencil gives a rational map ρ : P P 1 . Its generic fiber S η is the weighted surface defined by equation As in the case of S \ (H ∪ Q), the complement S η \ (H η ∪ Q η ) is an A 1 -cylinder in S η , defined over the field k(λ), and hence P \ S contains an open A 1 -cylinder (see [21,Lemma 3]). Now we consider the surface S defined by a quasi-homogenous equation of type (4.9).…”

“…Its generic fiber is the weighted surface defined by equation in the corresponding weighted projective space over the field , where is a parameter. It is singular at the point , and the intersection of with the surface in given by consists of the -rational hyperplane section and the -rational (weighted) hypersurface section As in the case of , the complement is an -cylinder in , defined over the field , and hence contains an open -cylinder (see [DK18, Lemma 3]).…”

Deformations of $$\mathbb {A}^1$$-cylindrical varieties

“…a Zariski open subset of the form Z × A 1 for some algebraic variety Z. Such A 1 -cylinders appear naturally in many recent problems and questions related to the geometry of algebraic varieties, both affine and projective [16,5,6,7,8,1,2,3,17,18,19,24,25]. Clearly, there are only two A 1 -cylindrical smooth complex curves: the affine line A 1 and the projective line P 1 .…”