Nagata's conjecture for linear systems of plane curves at non-square numbers of general points

For $n\ge 10$ points in general position in the complex projective plane, Nagata's conjecture (1959) predicts that every non-empty linear system $\mathcal{L}(d;m_1,\dots,m_n)$ of plane curves of degree $d>0$ with multiplicities $m_i$ at the points satisfies $\sqrt{n}\,d>m_1+\cdots+m_n$. Equivalently, the Nagata ray $\sqrt{n}H-E_1-\cdots-E_n$ is nef on the blow-up $X_n$ of $\mathbb{P}^2$ at the points, and the Seshadri constant of $\mathbb{P}^2$ at $n$ general points is at least $1/\sqrt{n}$. The conjecture is proved for $n\le 9$ (classical) and for square $n$ (Nagata's theorem), but it remains open for every non-square $n\ge 10$, including $n=10$ even in the homogeneous form. The source record, Ciliberto–Harbourne–Miranda–Roé's "Variations on Nagata's Conjecture", embeds this question in the study of the boundary of the cone of curves of $X_n$: good rays (rational non-effective boundary rays) were constructed there for all $n\ge 10$, and the wonderful-ray existence question it posed (irrational nef rays of self-intersection zero) was resolved affirmatively for all $n\ge 10$ in 2022 - but the Nagata ray itself, the conjecture's core, is still not known to be nef. The problem asks for a proof or refutation of Nagata's conjecture in full generality.

Background

A classical interpolation problem asks which linear systems of plane curves exist through points in general position. Fix $n$ points $x_1,\dots,x_n$ in general position in the complex projective plane $\mathbb{P}^2$ (that is, avoiding all algebraic coincidences: the configuration lies outside a countable union of proper closed subsets of $(\mathbb{P}^2)^n$), and non-negative integers $m_1,\dots,m_n$. The linear system $\mathcal{L}(d;m_1,\dots,m_n)$ is the vector space of plane curves of degree $d$ having multiplicity at least $m_i$ at $x_i$; it may be empty. The standard model is the surface $X_n$ obtained by blowing up $\mathbb{P}^2$ at the points (replacing each point by the line of directions through it); the system corresponds to the divisor class $dH-\sum_i m_iE_i$, where $H$ is the pull-back of a line and the $E_i$ are the exceptional curves, with self-intersection $d^2-\sum_i m_i^2$. Blowing up introduces infinitely many curves of self-intersection $-1$ (lines through pairs of points, conics through five points, and so on). A naive parameter count predicts $\dim\mathcal{L}=d(d+3)/2-\sum_i m_i(m_i+1)/2$; systems exceeding the prediction are called special, and the Segre–Harbourne–Gimigliano–Hirschowitz (SHGH) conjecture - open for over half a century - asserts that every special system with general base points is $(-1)$-special, that is, forced by a curve of self-intersection $-1$ contained in it. SHGH is known for $n\le 9$ (going back to Castelnuovo).

In 1959, while constructing his counterexamples to Hilbert's fourteenth problem on finite generation of invariant rings, Nagata proved by a degeneration argument that for $n=k^2\ge 9$ general points every non-empty system satisfies $d\ge(m_1+\cdots+m_n)/\sqrt n$, and he conjectured the same bound for all $n\ge 10$. For non-square $n$ the inequality is automatically strict, since $\sqrt n$ is irrational. This is Nagata's conjecture (N). It is equivalent to the statement that the Nagata ray $\nu_n=\sqrt n\,H-E_1-\cdots-E_n$ is nef (meets every effective curve non-negatively) on $X_n$, and to the lower bound $\varepsilon\ge 1/\sqrt n$ for the Seshadri constant of $\mathcal{O}_{\mathbb{P}^2}(1)$ at the $n$ points, where $\varepsilon=\inf_C d_C/\sum_i\operatorname{mult}_{x_i}C$ is the invariant measuring local positivity; for square $n$ the exact value $\varepsilon=1/\sqrt n$ follows from Nagata's theorem.

The source record of this problem, "Variations on Nagata's Conjecture" by Ciliberto, Harbourne, Miranda and Roé, studies (N) through the geometry of the Mori cone $\overline{\mathrm{NE}}(X_n)$, the closed cone of effective curve classes. By Riemann–Roch, the nonnegative cone $\mathcal{Q}_n=\{\xi:\xi\cdot H\ge 0,\ \xi^2\ge 0\}$ (a quadric cone in the real Néron–Severi space) is contained in $\overline{\mathrm{NE}}(X_n)$, so all effectivity phenomena beyond $\mathcal{Q}_n$ concentrate on its boundary $\partial\mathcal{Q}_n$. The authors call a rational non-effective ray of $\partial\mathcal{Q}_n$ a good ray and an irrational nef ray of $\partial\mathcal{Q}_n$ a wonderful ray; (N) holds exactly when the Nagata ray - irrational of self-intersection zero when $n$ is not a square - is wonderful. Good rays themselves yield counterexamples to Hilbert's fourteenth problem by Nagata's original construction, and the paper produced good rays for every $n\ge 10$. De Fernex showed that for $n\ge 10$ the Mori cone is not even countably generated, and formulated conjectures, refined in the source paper into the $\Delta$-conjecture and strong $\Delta$-conjecture phrased via the de Fernex point $D_n=(\sqrt{n-1};1^n)$, which imply (N) but remain unproven, as do the Strong Nagata conjecture (SN) and SHGH above them. The paper also proved (Proposition 6.1) that (N) for all $n\le 89$, combined with (SN) at $n=10$, implies (N) for every $n\ge 90$, reducing the infinite problem to a finite verification range - conditional on an open strengthening at $n=10$.

Problem Statement

Let $n$ be a positive integer, $x_1,\dots,x_n$ points in general position in the complex projective plane $\mathbb{P}^2$, and $X_n$ the blow-up of $\mathbb{P}^2$ at these points, with hyperplane class $H$ and exceptional curves $E_1,\dots,E_n$. Prove or disprove Nagata's conjecture:

(N) For every $n\ge 9$, every choice of $n$ points $x_1,\dots,x_n$ in general position, and every non-empty linear system $\mathcal{L}(d;m_1,\dots,m_n)$ of plane curves of degree $d\ge 1$ with multiplicity at least $m_i$ at each $x_i$,

$$\sqrt{n}\,d\ \ge\ m_1+\cdots+m_n,$$

with strict inequality whenever $n\ge 10$.

(The convention $d\ge 1$ is the standard one in this literature - the source paper's own follow-up work introduces its systems with $d>0$ - and excludes only the trivial system of constants, the single degree-zero exception to strictness. For non-square $n$ with $d\ge 1$ the inequality is automatically strict because $\sqrt{n}$ is irrational, so strictness is substantive only at square $n\ge 16$, where it is part of Nagata's theorem.)

The open cases are exactly the non-square $n\ge 10$: the case $n\le 9$ is classical and the square case is Nagata's theorem. Equivalent formulations of the open content are: (i) the Nagata ray $\nu_n=\sqrt{n}\,H-E_1-\cdots-E_n$ is nef on $X_n$ for every $n\ge 10$; (ii) in the terminology of the source paper, the Nagata ray is a wonderful ray for every non-square $n\ge 10$; (iii) the Seshadri constant of $\mathbb{P}^2$ at $n$ general points is at least $1/\sqrt{n}$ for every $n\ge 10$. A refutation consists of a single non-square $n\ge 10$ and an effective system with $n\,d^2<(m_1+\cdots+m_n)^2$; a proof must cover all non-square $n\ge 10$ and arbitrary multiplicity vectors, not only the homogeneous systems $\mathcal{L}(d;m^n)$. The reduction of the source paper - (N) for $n\le 89$ plus Strong Nagata at $n=10$ implies (N) for all $n\ge 90$ - is context, not a restriction of the target.

References

1. C. Ciliberto, B. Harbourne, R. Miranda, J. Roé, "Variations on Nagata's Conjecture", Clay Mathematics Proceedings 18 (2013), 185–203; arXiv:1202.0475, https://arxiv.org/abs/1202.0475
2. M. Nagata, "On the 14-th Problem of Hilbert", American Journal of Mathematics 81 (1959), 766–772, DOI 10.2307/2372927, https://doi.org/10.2307/2372927
3. C. Ciliberto, R. Miranda, J. Roé, "Irrational nef rays at the boundary of the Mori cone for very general blowups of the plane", Michigan Mathematical Journal 76 (2026), DOI 10.1307/mmj/20236356; arXiv:2201.08634, https://arxiv.org/abs/2201.08634
4. C. Ciliberto, R. Miranda, J. Roé, "Quadric cones on the boundary of the Mori cone for very general blowups of the plane", Collectanea Mathematica 76 (2025), 601–611, DOI 10.1007/s13348-024-00447-7; arXiv:2310.10507, https://arxiv.org/abs/2310.10507
5. C. Ciliberto, R. Miranda, J. Roé, "Boundedness results for planar linear systems assuming the Segre–Harbourne–Gimigliano–Hirschowitz Conjecture", arXiv:2508.01462, https://arxiv.org/abs/2508.01462
6. T. de Fernex, "On the Mori cone of blow-ups of the plane", arXiv:1001.5243, https://arxiv.org/abs/1001.5243
7. A. Laface, L. Ugaglia, M. Vilches, "Equivalent conjectures on blowing-ups of P^2", arXiv:2405.07716, https://arxiv.org/abs/2405.07716
8. J. Roé, P. Supino, "Nagata type statements", arXiv:1707.00583, https://arxiv.org/abs/1707.00583
9. C. Ciliberto, O. Dumitrescu, R. Miranda, J. Roé, "Emptiness of homogeneous linear systems with ten general base points", in Classification of Algebraic Varieties, C. Faber, G. van der Geer, E. Looijenga (eds.), EMS Series of Congress Reports vol. 3 (2011), 189–195, DOI 10.4171/007-1/8; arXiv:0910.1171, https://arxiv.org/abs/0910.1171
10. C. Ciliberto, R. Miranda, "Homogeneous interpolation on ten points", Journal of Algebraic Geometry 20 (2011), 685–726, DOI 10.1090/s1056-3911-2011-00545-6; arXiv:0812.0032, https://arxiv.org/abs/0812.0032
11. M. Dumnicki, A. Küronya, C. Maclean, T. Szemberg, "Seshadri constants via Okounkov functions and the Segre-Harbourne-Gimigliano-Hirschowitz Conjecture", arXiv:1304.0249, https://arxiv.org/abs/1304.0249