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.

The conjecture, formulated in this broader form in 2015, asks whether an open $\mathbb{A}^1$ -cylinder in the affine complement is equivalent to a $(-K_S)$ -polar cylinder on a low-degree del Pezzo surface; the cubic case is known.

Known results

Degree $2$ : a $(-K_S)$ -polar cylinder exists unless all singularities are of type $A_1$ (Cheltsov, Park, Won, 2013).
Degree $1$ : the listed exceptional configurations have singularities only of types $A_1$ , $A_2$ , $A_3$ , or $D_4$ (Cheltsov, Park, Won, 2013).
A tiger criterion characterizes existence of $(-K_S)$ -polar cylinders (Cheltsov, Park, Won, 2018).
These results concern cylinders on $S$ , not the affine-complement equivalence.

December 2024--December 2025 related cylinder results

A December 2024 paper claims the ample-cylinder conjecture for Du Val del Pezzo surfaces of degree at least $2$ , including degree $2$ . A December 2025 paper claims the analogous result for degree- $1$ surfaces with Picard rank $2$ . These are claims about a neighboring surface formulation and do not explicitly prove the affine-Fano statement.

Current status (as of August 2026): The affine-complement equivalence remains unsettled; degree- $2$ and some degree- $1$ surface-cylinder statements are claimed, but their implication here is unverified.

Cylinder conjecture for low-degree del Pezzo surfaces

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Progress summary

Solutions 0