Polar Cylinders and Complement Automorphisms

Definitions and mathematical statement

The supplied record does not specify a ground field, the ambient space $P$, or the embedding $S\subset P$. It also leaves the allowed singularities of the del Pezzo surface unspecified. Consequently, the assertion below records the intended equivalence with these parameters unresolved.

Under the usual smooth interpretation, a del Pezzo surface over a field $k$ is a smooth projective surface $S$ with ample anticanonical divisor $-K_S$, where $K_S$ denotes a canonical divisor. A conventional meaning of a $(-K_S)$-polar cylinder is an open subset \[ U=S\setminus\operatorname{Supp}(D)\cong Z\times_k\mathbb A^1_k, \qquad D\geq 0,\qquad D\sim_{\mathbb Q}-K_S, \] where $D$ is a rational divisor and $Z$ is an affine curve over $k$. Here $\sim_{\mathbb Q}$ denotes rational linear equivalence. These conventions, including the permitted curves $Z$ and any singular version of the definition, require confirmation against the source.

For a specified embedding $S\subset P$, write \[ \operatorname{Aut}_k(P,S)=\{g\in\operatorname{Aut}_k(P):g(S)=S\}. \] Restriction gives a homomorphism \[ \rho:\operatorname{Aut}_k(P,S)\longrightarrow\operatorname{Aut}_k(P\setminus S). \] The intended question, for every del Pezzo surface and ambient embedding in the source's intended class, is whether \[ \left(S\text{ contains no }(-K_S)\text{-polar cylinder}\right) \quad\Longleftrightarrow\quad \rho\left(\operatorname{Aut}_k(P,S)\right)=\operatorname{Aut}_k(P\setminus S). \] Thus the right-hand side means that every automorphism of the complement extends to an automorphism of $P$ preserving $S$. The source writes this as $\operatorname{Aut}(P\setminus S)=\operatorname{Aut}(P,S)$; interpreting it as an identification of groups through restriction additionally requires injectivity of $\rho$. The missing ambient, field, embedding, and surface hypotheses prevent a fully specified universally quantified conjecture.

Short English statement

The proposed equivalence asks whether a del Pezzo surface has no anticanonically polar cylinder exactly when every automorphism of its complement in the intended ambient space extends to an ambient automorphism preserving the surface. The ambient space, embedding, ground field, and precise surface and cylinder conventions remain unspecified.

Sources

• Supplied local registry record: permanent UnsolvedMath ID $30003863$, source problem code catalogue entry, title Polar Cylinders and Complement Automorphisms, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2018.
• Supplied report citation: Subgroups of Cremona Groups (2019), Oberwolfach Reports, pp. 1685--1743, DOI 10.4171/owr/2018/28. https://doi.org/10.4171/owr/2018/28. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference: Park, Automorphism groups of complements. https://ems.press/content/serial-article-files/46750. No further bibliographic details are supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, identifies the proposed automorphism-group equality as Conjecture 5, and reports that known results cover some degrees and rule out connected groups in others, while the general assertion remains unresolved. This is supplied dated curation, not an independently verified literature assessment; the degrees, hypotheses, and precise meaning of the connected-group claim are not given.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review claiming self-containment and preservation of the target. These are supplied review claims; the referenced sources have not been independently inspected for this draft.