Polarized cylinders in Du Val del Pezzo surfaces of degree two

TL;DR

This paper studies del Pezzo surfaces of degree 2 with Du Val singularities and constructs polar cylinders for any ample divisor, expanding understanding of their geometric structures.

Contribution

It introduces a method to construct an $H$-polar cylinder for any ample $Q$-divisor on such surfaces, generalizing previous results.

Findings

Constructed an $H$-polar cylinder for any ample $Q$-divisor on the surface.

Extended the class of del Pezzo surfaces known to admit polar cylinders.

Provided new tools for studying the geometry of singular del Pezzo surfaces.

Abstract

Let $S$ be a del Pezzo surface with at worst Du Val singularities of degree $2$ such that $S$ admits an $(-K_S)$-polar cylinder. In this article, we construct an $H$-polar cylinder for any ample $\mathbb{Q}$-divisor $H$ on $S$.