Cylinders in del Pezzo surfaces with du Val singularities

TL;DR

This paper proves that del Pezzo surfaces with du Val singularities and degree 1, which contain a $-K_X$-polar cylinder, also contain an $H$-polar cylinder for any ample divisor $H$, expanding understanding of their geometric structures.

Contribution

It establishes that the existence of a $-K_X$-polar cylinder on such surfaces implies the existence of an $H$-polar cylinder for any ample divisor $H$, a new result in the study of del Pezzo surfaces.

Findings

Surfaces with $-K_X$-polar cylinders also have $H$-polar cylinders for any ample $H$.

The result applies specifically to degree 1 del Pezzo surfaces with du Val singularities.

The proof connects the existence of certain cylinders to the ample divisors on the surface.

Abstract

We consider del Pezzo surfaces $X$ with du Val singularities. Assume that $X$ has a $-K_X$-polar cylinder and $\deg X=1$. Let $H$ be an ample divisor. We'll prove that $X$ has a $H$-polar cylinder.