Polar Cylinders and Affine Complements of Del Pezzo Surfaces

Definitions and mathematical statement

Let $k$ denote the ground field, which is not specified in the supplied record. Under the usual convention, a cylinder in a $k$-variety $X$ is a nonempty Zariski-open subset $U\subseteq X$ admitting a $k$-isomorphism \[ U\cong Z\times_k{\mathbb{A}}^1_k \] for a nonempty affine $k$-variety $Z$. Call $X$ cylindrical if it contains such a subset, and non-cylindrical otherwise. These definitions concern the existence of an open cylinder, rather than a product decomposition of all of $X$.

For a projective surface $S$ with $K_S$ ${\mathbb{Q}}$-Cartier, the conventional meaning of a $(-K_S)$-polar cylinder is a cylinder of the form \[ S\setminus\operatorname{Supp}(D), \qquad D\geq 0,\qquad D\sim_{{\mathbb{Q}}}-K_S, \] where $D$ is an effective ${\mathbb{Q}}$-Cartier ${\mathbb{Q}}$-divisor and $\sim_{{\mathbb{Q}}}$ denotes ${\mathbb{Q}}$-linear equivalence. The source does not specify its divisor convention; this conventional interpretation requires confirmation.

The supplied question concerns a del Pezzo surface $S$ embedded in its ambient weighted projective space $P$, with affine complement $P\setminus S$. Its background assessment refers specifically to del Pezzo hypersurfaces. For each pair $(S,P)$ in the intended class, the proposed equivalence is \[ \begin{aligned} &S\text{ contains no }(-K_S)\text{-polar cylinder}\\ &\hspace{2em}\Longleftrightarrow P\setminus S\text{ is non-cylindrical}. \end{aligned} \] The supplied record labels the forward implication as known. Thus its remaining proposed implication is \[ P\setminus S\text{ is non-cylindrical} \quad\Longrightarrow\quad S\text{ contains no }(-K_S)\text{-polar cylinder}. \] These are source-reported assertions and status qualifications, not independently verified results here.

The intended class of pairs $(S,P)$ is unresolved: the record supplies no ground-field assumptions, singularity hypotheses, weights, hypersurface degree, or precise embedding conditions. In particular, it does not determine whether "del Pezzo surface" means a smooth projective surface with ample anticanonical divisor or a specified class of singular surfaces. Consequently, the display preserves the extracted mathematical target but does not assert the equivalence for every field, every singular del Pezzo surface, or every weighted-projective embedding.

Short English statement

For the intended del Pezzo hypersurfaces, the question asks whether having no anticanonically polar cylinder is equivalent to the ambient affine complement containing no open cylinder. The supplied record reports the forward implication as proved and the converse as conjectural. The ground field, allowed singularities, embedding hypotheses, and precise polar-divisor convention require clarification before this becomes a fully specified universally quantified statement.

Sources

• Supplied local problem registry record, permanent UnsolvedMath ID $30003862$, exact source problem code catalogue entry, title: Polar Cylinders and Affine Complements of Del Pezzo Surfaces. The record labels the problem partially_solved and the statement corrected_verified; these are supplied registry labels, not independent verification. No individual proposer is supplied.
• Supplied report citation: Subgroups of Cremona Groups (2019), Oberwolfach Reports, DOI 10.4171/owr/2018/28, pp. 1685--1743. https://doi.org/10.4171/owr/2018/28. The registry separately gives proposed year $2018$; the relationship between these year labels is not explained.
• Supplied literature source: Park, Automorphism groups of complements. https://ems.press/content/serial-article-files/46750. No fuller bibliographic details or theorem locator are supplied.
• Literature assessment checked at 2026-08-21: the supplied curation states that, for del Pezzo hypersurfaces, absence of a polar cylinder implies non-cylindricity of the complement, while the converse remains conjectural. This is supplied dated curation derived from the Oberwolfach Reports record, not independently verified here. The record expressly requests independent verification of time-sensitive status and cited claims.