Irrational Seshadri Constants from the Nagata Conjecture

Definitions and mathematical statement

Let $r\geq 9$ be an integer, and let \[ \pi:X_r\longrightarrow \mathbb P^2_k \] be the blowup at $r$ distinct general points. Write $H$ for the pullback of the class of a line and $E_1,\ldots,E_r$ for the exceptional divisors. The supplied record does not specify the ground field $k$, its characteristic, or the precise meaning of generality of the centers; these data require confirmation.

For an ample line bundle $L$ on $X_r$ and a geometric point $x\in X_r$, define the Seshadri constant by \[ \varepsilon(L;x)=\inf_{\substack{C\subset X_r\ \text{an integral curve}\\x\in C}} \frac{c_1(L)\cdot C}{\operatorname{mult}_x C}. \] Here $\operatorname{mult}_x C$ is the multiplicity of $C$ at $x$. The phrase "at a very general point" means that the asserted property holds outside a countable union of proper Zariski closed subsets, in a ground-field setting where this interpretation is meaningful.

One standard non-strict numerical formulation of the Nagata hypothesis is \[ \operatorname{Nag}_{\mathrm{nef}}(X_r):\qquad \sqrt r\,H-\sum_{i=1}^r E_i\ \text{is nef}. \] A real divisor class is nef if its intersection with every integral curve is nonnegative. Equivalently, this hypothesis requires \[ \sqrt r\,\deg B\ \geq\ \sum_{i=1}^r\operatorname{mult}_{p_i}B \] for every integral plane curve $B$, where $p_1,\ldots,p_r$ are the blowup centers.

Question, conditional on this interpretation of the Nagata hypothesis. For every integer $r\geq 9$, does \[ \operatorname{Nag}_{\mathrm{nef}}(X_r) \quad\Longrightarrow\quad \exists\,L\in\operatorname{Pic}(X_r)\ \text{ample such that} \varepsilon(L;x)\in\mathbb R\setminus\mathbb Q \ \text{for very general }x\in X_r \] hold, assuming only the Nagata conjecture for this same $X_r$? The line bundle is chosen before the very general point. No Nagata hypothesis for a different number of blowup centers, or additional conjectural hypothesis, is supplied.

The source does not state the precise version of the Nagata conjecture it intends. Consequently, the displayed implication is a formulation template requiring confirmation of that hypothesis and the ground-field conventions. In particular, a strict-inequality version cannot be substituted without checking its scope and the endpoint $r=9$.

Short English statement

For each $r\geq 9$, the question asks whether the Nagata conjecture alone for the plane blown up at $r$ general points guarantees an ample line bundle whose Seshadri constant is irrational at a very general point. The ground field, the meaning of general centers, and the precise Nagata hypothesis remain unspecified. The supplied record labels the question open; this draft does not independently verify that status.

Sources

• Local project registry: supplied UnsolvedMath source record, permanent ID $30003978$, source problem code catalogue entry, title "Irrational Seshadri Constants from the Nagata Conjecture." The record belongs to the Oberwolfach Reports Open Problems collection. No individual proposer is supplied.
• Supplied primary report citation: Mini-Workshop: Asymptotic Invariants of Homogeneous Ideals (2019), Oberwolfach Reports, pp. 2739--2767, DOI 10.4171/owr/2018/45. https://doi.org/10.4171/owr/2018/45. The record separately gives proposed year 2018 and the tag owr-year:2018; these supplied dates have not been reconciled.
• Supplied literature source: "Rationality of Seshadri constants." https://doi.org/10.1016/j.jpaa.2020.106345.
• Supplied literature source: "Irrational Seshadri constants." https://www.cmi.ac.in/~krishna/irrational-seshadri-constants.pdf.
• Supplied literature assessment dated exactly 2026-08-21: the record labels the question open and reports that later work provides conditional results rather than a proof from Nagata alone. This is supplied dated curation derived from the OWR material, not an independently verified assessment. The supplied background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry assigns statement status corrected_verified, explaining that notation, typography, and sentence structure were repaired while preserving mathematical content. This is supplied editorial metadata, not independent mathematical verification by this draft.