Notes on cylinders in smooth projective surfaces

TL;DR

This paper investigates the conditions under which cylinders exist in smooth minimal geometrically rational surfaces and establishes an equivalence criterion for cylinders under birational maps between such surfaces.

Contribution

It characterizes the existence of cylinders in smooth minimal geometrically rational surfaces and proves their invariance under birational transformations.

Findings

Cylinders exist under specific conditions in these surfaces.

Existence of a cylinder is preserved under birational maps.

Provides criteria for cylinders in smooth projective surfaces.

Abstract

In this article, we 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, one contains a cylinder if and only if so does the other.