Cylinder conjecture for low-degree del Pezzo surfaces

Let $S$ be a del Pezzo surface of anticanonical degree at most $2$ with at worst Du~Val singularities. The affine Fano variety $\mathbb{P}\setminus S$ is the complement of $S$ in the ambient projective variety, and a $(-K_S)$-polar cylinder is an open cylinder in $S$ whose boundary is supported by an anticanonical divisor. Cylinder conjecture for low-degree del Pezzo surfaces. The affine Fano variety $\mathbb{P}\setminus S$ contains an open $\mathbb{A}^1$-cylinder if and only if $S$ contains a $(-K_{S})$-polar cylinder. The preceding discussion establishes the analogous equivalence for cubic surfaces, while the claim for anticanonical degrees at most $2$ is presented as a natural expectation and remains unresolved in the supplied text.