Notes on cylinders in smooth projective surfaces

A Zariski open subset of an algebraic variety is called a cylinder if it is isomorphic to the direct product of the affine line and an algebraic variety. We consider the existing condition of relative cylinders with respect to a projective dominant morphism of relative dimension two. Since this consideration is essentially a determination of the existence of cylinders in the generic fiber, we study smooth projective surfaces defined over a perfect field from the point of view of cylinders. In previous work, the existing condition of cylinders in smooth minimal del Pezzo surfaces over a field of characteristic zero is known. In this article, we completely determine the existing condition of cylinders in smooth minimal geometrically rational surfaces over a perfect field. Furthermore, we show that for any birational map between smooth projective surfaces over a perfect field, one contains a cylinder if and only if so does the other.