Bounded Negativity for Rational Surfaces

Definitions and mathematical statement

Let $k$ be a field. A smooth projective surface $X$ over $k$ is $k$-rational if it is birational over $k$ to ${\mathbb{P}}^2_k$; it is geometrically rational if $X_{\overline{k}}$ is birational over $\overline{k}$ to ${\mathbb{P}}^2_{\overline{k}}$. The supplied record says only "rational" and does not specify which interpretation is intended over a nonalgebraically closed field.

For a curve $C\subset X$, write $C^2=C\cdot C$ for its self-intersection number in the intersection theory of the smooth projective surface $X$. A reduced irreducible curve means an integral closed subscheme of dimension one over $k$; geometric irreducibility is not imposed by the supplied wording.

Supplied question, with the meaning of rationality unresolved. For every field $k$ and every smooth projective rational surface $X$ over $k$, does there exist a real constant $B_X$, depending on $X$ but independent of $C$, such that \[ C^2\geq B_X \qquad\text{for every reduced irreducible curve }C\subset X? \] This is the literal unrestricted-field formulation in the clean statement. No characteristic assumption or algebraic-closedness hypothesis is supplied; such a hypothesis must not be inserted without source clarification. The rationality ambiguity above must also be resolved before this becomes a single precise conjecture.

Additional formulation in the original statement. The original statement instead asks for a constant $B_X^{\mathrm{red}}$ such that \[ C^2\geq B_X^{\mathrm{red}} \qquad\text{for every reduced curve }C\subset X, \] and asserts that it suffices to consider reduced irreducible curves. Here a reduced curve may have finitely many distinct irreducible components, so that $C=C_1+\cdots+C_r$ and \[ C^2=\sum_{i=1}^{r}C_i^2+2\sum_{1\leq i<j\leq r}C_i\cdot C_j. \] The claimed equivalence is retained as a source assertion requiring review: the displayed identity alone does not convert an irreducible-curve lower bound into a uniform lower bound independent of the number of components. The intended target, including any hypotheses needed for that equivalence, remains unresolved.

Short English statement

The supplied question asks whether each smooth projective rational surface has a uniform lower bound for the self-intersections of its reduced irreducible curves. The field assumptions and meaning of rationality need clarification. The original version also includes reducible reduced curves and claims an equivalence that requires review. The supplied open label and literature assessment are dated curation, not independently verified conclusions.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30004603$, title Bounded Negativity for Rational Surfaces, Oberwolfach Reports Open Problems collection, exact source problem code catalogue entry. The record supplies proposed year 2021 and no named proposer. Its status is supplied as open; its statement status is supplied as corrected_verified, with a claimed manual semantic, typographic, and LaTeX review. These are registry annotations, not independent verification in this draft.
• Supplied report citation: Logarithmic Vector Fields and Freeness of Divisors and Arrangements: New perspectives and applications (2022), Oberwolfach Reports, pp. 225--300, DOI 10.4171/owr/2021/5. https://doi.org/10.4171/owr/2021/5. The background describes the report as documenting a mathematical workshop and its open-problem discussions. The literature list labels this same reference "Oberwolfach Report 5/2021." The supplied citation does not identify an exact page or passage for this problem.
• Supplied literature source: Urzua, Proyecciones (2022). https://www.scielo.cl/pdf/proy/v41n2/0716-0917-proy-41-02-517.pdf. No article title or more detailed attribution is supplied.
• Supplied literature source: Urzua, International Mathematics Research Notices. https://doi.org/10.1093/imrn/rnu236. No article title or publication year is supplied in this entry.
• Supplied literature assessment, checked on 2026-08-22: the record labels the status open and states that extremal $H$-constant questions for reduced plane curves remain unresolved, with bounds and examples but no definitive minimum or general classification. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment. Its relationship to the bounded-negativity formulation above is not established by the supplied record. The background explicitly requests independent verification of time-sensitive status and cited claims.