Bounded Negativity for Root of Unity Blowups

Definitions and mathematical statement

Let $m\geq 1$ be an integer. The source does not specify a ground field. Conditional on fixing a field $k$ containing an element $\varepsilon\in k^{\times}$ of multiplicative order exactly $m$, define \[ Z(m)=\left\{(1:\varepsilon^{\alpha}:\varepsilon^{\beta})\in{\mathbb{P}}^{2}_{k}:1\leq\alpha,\beta\leq m\right\}, \qquad X(m)=\operatorname{Bl}_{Z(m)}({\mathbb{P}}^{2}_{k}). \] Here $Z(m)$ is given its reduced structure and consists of $m^{2}$ distinct points; the blowup is at these points. The existence of $\varepsilon$ requires that the characteristic of $k$, if positive, does not divide $m$.

To make the question precise, adopt the usual bounded-negativity convention that curves are reduced, irreducible curves on the smooth projective surface $X(m)$, and let $C^{2}$ denote the self-intersection number of such a curve. Define \[ b(X(m))=\sup\left(\{0\}\cup\{-C^{2}:C\subset X(m)\text{ is a reduced, irreducible curve}\}\right) \in{\mathbb{Z}}_{\geq0}\cup\{+\infty\}. \] Negativity is bounded precisely when there exists $B\in{\mathbb{Z}}_{\geq0}$ such that \[ \forall\, C\subset X(m)\text{ reduced and irreducible},\qquad C^{2}\geq-B. \] When finite, $b(X(m))$ is the least such $B$.

Question. For the root-of-unity blowups $X(m)$, is $b(X(m))$ finite for each positive integer $m$, and, whenever it is finite, what is its exact value? The bound may depend on $m$; the source does not explicitly ask for one bound independent of $m$.

Unresolved qualifications. The ground field, its characteristic, and whether curves are considered after passage to an algebraic closure are not specified in the supplied statement. The formulation above is conditional on a chosen field and uses curves over that field; it does not assert that the intended question ranges over all such fields. The reduced, irreducible curve convention and the normalization of $b(X(m))$ also require confirmation from the source.

Short English statement

Blow up the $m^{2}$ projective points whose last two coordinates are powers of a primitive $m$th root of unity. For each $m$, ask whether the self-intersection numbers of reduced, irreducible curves on the resulting surface have a lower bound, and determine the optimal bound. The ground field and the precise curve convention remain unspecified in the supplied source.

Sources

• Local UnsolvedMath registry: supplied record with permanent ID $30004334$, title Bounded Negativity for Root of Unity Blowups, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied.
• Supplied report citation: Mini-Workshop: Seshadri Constants (2020), Oberwolfach Reports, DOI 10.4171/owr/2019/53, pp. 3271--3313. https://doi.org/10.4171/owr/2019/53. The record separately gives proposed year $2019$; both year references are retained without reconciliation.
• Supplied literature reference: Farnik et al., negative curves on special rational surfaces. https://www.cmi.ac.in/~krishna/negative-curves-special-rational-surfaces.pdf. Full bibliographic details are not supplied.
• Supplied literature reference: OWR report on Seshadri constants. https://ems.press/content/serial-article-files/46833.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and reports that the cited work still poses bounded negativity and the value of its bound for $X(m)$ as open; it further states that results for the nine-point Fermat case do not cover all $m$. This is supplied dated curation, not an independently verified literature assessment. The record explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified; its accompanying qualification says that only formula delimiters, notation, grammar, and typography were repaired, with mathematical content and scope preserved. This label does not constitute independent mathematical verification of this draft.