Automorphisms of Rational Surfaces with Antipluricanonical Curves

Definitions and mathematical statement

Let $X$ be a smooth projective complex rational surface, so $X$ is birational to ${\mathbb{P}}^2_{\mathbb{C}}$. Let $\operatorname{Pic}(X)$ be its group of line bundles, and define \[ \operatorname{Aut}^*(X)=\operatorname{im}\left(\operatorname{Aut}(X)\to\operatorname{Aut}(\operatorname{Pic}(X))\right) \] using a consistent pullback-action convention. Assume this image is infinite. Infinitude of the automorphism group alone is not substituted for infinitude of its Picard action.

The source asks whether there is a birational morphism $\pi:X\to Y$ such that $Y$ has an antipluricanonical curve and an infinite subgroup of $\operatorname{Aut}^*(Y)$ lifts to $X$. Under the usual smooth-surface convention, the first condition means \[ \exists m\geq1\quad\exists 0\ne D\geq0,\qquad D\sim-mK_Y. \] No irreducibility, reducedness, or smoothness of the divisor is requested. The condition is on the target $Y$, not necessarily on $X$. The source does not explicitly state whether $Y$ is smooth projective or permits singular targets; the latter would require a specified canonical-divisor and pluricanonical-sheaf convention.

To make the usual lifting requirement precise, set \[ L_\pi=\{a\in\operatorname{Aut}(Y):\exists\widetilde a\in\operatorname{Aut}(X),\quad \pi\circ\widetilde a=a\circ\pi\}. \] The intended condition can be expressed as infinitude of the image of $L_\pi$ in $\operatorname{Aut}^*(Y)$. This gives actual liftable automorphisms whose Picard actions form an infinite subgroup; it does not require a section from all of $\operatorname{Aut}^*(Y)$ or lifting every automorphism of $Y$. If the source intends an action of a specified abstract subgroup with chosen representatives, that stronger convention must be recovered.

Thus for every indicated $X$ with infinite Picard-action image, the desired morphism and effective multiple of $-K_Y$ may depend on $X$, while the lifted Picard-action image must still be infinite. The supplied literature triage checked 2026-08-21 reports no authoritative complete resolution of the extracted question, without a separate theorem or construction.

Short English statement

Can a rational surface with infinitely many Picard-group automorphisms be contracted to a surface carrying an effective anticanonical multiple while retaining an infinite liftable Picard-action subgroup? The target and lifting conventions need confirmation.

Sources

• Local UnsolvedMath registry, permanent ID 30000652, catalogue entry, Automorphisms of Rational Surfaces with Antipluricanonical Curves.
• Affine Algebraic Geometry (2007), Oberwolfach Reports, pp. 5--82, DOI https://doi.org/10.4171/owr/2007/01.
• Supplied source-verification claim and literature triage checked 2026-08-21 cite only the report; no separate contraction or lifting theorem is supplied.