Cylinders in Mori Fiber Spaces: forms of the quintic del Pezzo threefold

TL;DR

This paper investigates the existence of certain affine cylinders within Mori Fiber Spaces whose fibers are quintic del Pezzo threefolds, providing conditions for the presence of higher-dimensional cylinders.

Contribution

It demonstrates that these Mori Fiber Spaces always contain relative A2-cylinders and characterizes when they admit relative A3-cylinders based on special lines in fibers.

Findings

Total spaces always contain relative A2-cylinders.

Existence of relative A3-cylinders depends on special lines in fibers.

Provides geometric criteria for cylinder existence.

Abstract

Motivated by the general question of existence of open A1-cylinders in higher dimensional pro-jective varieties, we consider the case of Mori Fiber Spaces of relative dimension three, whose general closed fibers are isomorphic to the quintic del Pezzo threefold V5 , the smooth Fano threefold of index two and degree five. We show that the total spaces of these Mori Fiber Spaces always contain relative A2-cylinders, and we characterize those admitting relative A3-cylinders in terms of the existence of certain special lines in their generic fibers.