The pluripotential versions of the Nagata conjecture in $\mathbb{C}^n$: Nivoche's conjectures (P1)–(P3) on general point configurations and their equivalent algebraic forms

Nivoche (2021) introduced three pluripotential conjectures in $\mathbb{C}^n$ - on the collision limit of multipoled pluricomplex Green functions as a configuration of $r$ general points collapses to a point, on the equality of two plurisubharmonic singularity invariants, and on the existence of entire plurisubharmonic functions with prescribed Lelong numbers and minimal logarithmic growth - and proved them equivalent to the weak Nagata/Iarrobino conjectures for general points in projective space, which assert that the singular degree satisfies $\Omega(S)=r^{1/n}$ and the dual invariant $\omega(S)=r^{1-1/n}$. The equivalence itself is settled mathematics; what remains open is the truth of the conjectures. In dimension two, for non-square $r\geq 10$, they are exactly the classical Nagata conjecture on plane curves, open since 1959; in higher dimensions only cardinalities of the form $r=s^n$ (Évain), perfect squares in the plane (Nagata), and nearly-square plane cases (Ciliberto–Miranda) are settled. The problem is to prove or refute these conjectures in their natural generality: every dimension $n\geq 2$ and every integer $r$ except possibly finitely many exceptional cardinalities.

Background

The Nagata conjecture, posed by Masayoshi Nagata in 1959 in his work on Hilbert's fourteenth problem, concerns algebraic curves through point configurations in general position. A property is said to hold for r general points of the complex projective plane P^2 if it holds for every configuration in some Zariski-open subset of the r-fold product (P^2)^r. The conjecture predicts that for $r\geq 10$ general points, every algebraic curve of degree $d$ passing through all of them with multiplicity at least $l$ satisfies $d>\sqrt{r}\,l$. Nagata proved it when $r$ is a perfect square; for every non-square $r\geq 10$ it has remained open since 1959, and it is known to fail for $r\leq 9$. The conjecture is equivalent to the statement that the Seshadri constants - the standard invariants measuring how far a line bundle is from being ample near a point configuration - of general point blow-ups of $\mathbb{P}^2$ take their maximal expected values, and it governs optimal symplectic ball packings of the plane.

The conjecture has an asymptotic formulation through two invariants of a finite point set $S$. Let $\Omega(S,l)$ be the smallest degree of a polynomial vanishing to order at least $l$ at every point of $S$; the limit $\Omega(S)=\lim_{l\to\infty}\Omega(S,l)/l$ exists and is called the singular degree of $S$ (it is the Waldschmidt constant, the standard asymptotic least-degree invariant of the ideal of $S$). Dually, let $\omega(S)=\sup_{P}\sum_{p\in S}\mathrm{ord}(P,p)/\deg P$, the supremum over nonzero polynomials $P$, where $\mathrm{ord}(P,p)$ is the vanishing order of $P$ at $p$. A classical interpolation bound of Waldschmidt gives $\Omega(S)\leq|S|^{1/n}$ for every set of $|S|$ points in $\mathbb{C}^n$, and dually $\omega(S)\geq|S|^{(n-1)/n}$; finding the reverse inequalities for general configurations is the difficult open direction. In $\mathbb{P}^2$ it is exactly Nagata's conjecture. In $\mathbb{P}^n$ it is Iarrobino's conjecture (1997), motivated by his inverse-systems study of fat points: hypersurfaces of degree $d$ through $r$ general points with multiplicity $l$ should satisfy $d\geq(rl^{n})^{1/n}$. This is known for numbers of points of the form $r=s^n$, proved by Évain (2005) by degeneration to grid configurations, and in the plane for square and nearly-square $r$ (Nagata; Ciliberto–Miranda 2006); for all other cardinalities it is open.

Stéphanie Nivoche (2021) translated this circle of problems into pluripotential theory - the several-complex-variables theory of plurisubharmonic (psh) functions, i.e. upper semicontinuous functions that are subharmonic under composition with every holomorphic map, and of the complex Monge–Ampère measure $(dd^{c}u)^{n}$ of Bedford–Taylor. On a bounded hyperconvex domain $D\subset\mathbb{C}^{n}$ (a domain admitting a continuous negative psh exhaustion function), the pluricomplex Green function $g_{D}(S,\cdot)$ with logarithmic poles on a finite set $S\subset D$ is the supremum of the nonpositive psh functions that lie below $\log|z-p|+O(1)$ near each $p\in S$; the strength of such a singularity is measured by the Lelong number $\nu(u,z)$, the residual Monge–Ampère density of $u$ at $z$. Building on convergence results of Rashkovskii–Thomas for Green functions with colliding poles, Nivoche studied the collapse of a configuration $z_{0}+tS$ toward $z_{0}$ as $t\to 0$: the Green functions $g_{D}(z_{0}+tS,\cdot)$ converge, locally uniformly outside $z_{0}$, to a canonical psh limit $g_{\infty}$, and she proved $\nu(g_{\infty},z_{0})=\Omega(S)$ together with the Monge–Ampère mass bounds $\Omega(S)^{n}\leq(dd^{c}g_{\infty})^{n}(\{z_{0}\})\leq|S|$. The Nagata-type question becomes whether all inequalities in this chain are equalities for general points.

She then formulated three pluripotential conjectures in $\mathbb{C}^{n}$ and proved them equivalent to the algebraic ones. To state the second and third, associate to an entire psh function $u$ its logarithmic growth $\gamma_{u}=\limsup_{|z|\to\infty}u(z)/\log\|z\|$ and the ratio $\omega(S,u)=\sum_{p\in S}\nu(u,p)/\gamma_{u}$, and define $\omega_{psh}(S)$ and $\omega_{psh}^{+}(S)$ as the suprema of $\omega(S,u)$ over all entire psh functions and over those locally bounded off $S$, respectively. The conjectures, each stated for a general set $S$ of $r$ points with $r$ allowed only finitely many exceptional values, are: (P1) the collision limit of the pluricomplex Green functions is the single-pole Green function scaled by $r^{1/n}$; (P2) $\omega_{psh}(S)=\omega_{psh}^{+}(S)$; (P3) for every $\epsilon>0$ there is an entire continuous psh function, locally bounded off $S$, with Lelong number at least one at each point of $S$ and growth at most $(1+\epsilon)r^{1/n}$. Her Theorem 1.3 proves that each of (P1), (P2), (P3) is equivalent to the algebraic conjectures (A1) $\Omega(S)=r^{1/n}$ and (A2) $\omega(S)=r^{1-1/n}$ for general sets in $\mathbb{P}^{n}$; in dimension two, for non-square $r$, (A1) is the original Nagata conjecture.

Problem Statement

Prove or refute the following conjectures, in the natural generality of the source - every dimension $n\geq 2$, every integer $r$ except possibly a finite number of exceptional cardinalities (in the plane the known exceptions are exactly $r\leq 9$), and general sets $S$ of $r$ points.

(P1) In $\mathbb{C}^{n}$, the pluricomplex Green functions of the collapsing configurations converge to a scaled single-pole Green function:

$$g_{B(O,1)}(tS,\cdot)\longrightarrow r^{1/n}\,g_{B(O,1)}(O,\cdot)\quad\text{locally uniformly on }B(O,1)\setminus\{O\}\text{ as }t\to 0 .$$

(P2) The two plurisubharmonic singularity invariants coincide: $\omega_{psh}(S)=\omega_{psh}^{+}(S)$.

(P3) For every $\epsilon>0$ there exists an entire continuous plurisubharmonic function $v$ in $\mathbb{C}^{n}$, locally bounded on $\mathbb{C}^{n}\setminus S$, with Lelong number $\nu(v,p)\geq 1$ for every $p\in S$ and logarithmic growth $\gamma_{v}\leq(1+\epsilon)\,r^{1/n}$.

Equivalently:

(A1) In $\mathbb{P}^{n}$, $\Omega(S)=r^{1/n}$ for every general set $S$ of $r$ points.

(A2) In $\mathbb{P}^{n}$, $\omega(S)=r^{1-1/n}$ for every general set $S$ of $r$ points.

The mutual equivalence of all five formulations is already proven (Nivoche 2021, Theorem 1.3), so a resolution of any single formulation - analytic or algebraic - resolves the entire problem; the open content is the truth of the conjectures, not their equivalence. A positive answer must hold for all dimensions $n\geq 2$ and all but finitely many $r$. A refutation must exhibit failure for general configurations for infinitely many $r$ in some fixed dimension $n\geq 2$ (a single failing cardinality refutes only the unhedged version of Iarrobino's conjecture, not the finitely-many-exceptions formulation above). For $n=2$ and non-square $r\geq 10$ the statement is exactly the classical Nagata conjecture; the cases $r=s^{n}$ (Évain), perfect squares in $\mathbb{P}^{2}$ (Nagata), and square or nearly-square $r$ in $\mathbb{P}^{2}$ (Ciliberto–Miranda) are known.

References

1. Stéphanie Nivoche, "Transcendental versions in C^n of the Nagata conjecture", Annales de l'Institut Fourier 71 (2021), 27–52. DOI: 10.5802/aif.3402; arXiv:1906.08518; https://doi.org/10.5802/aif.3402
2. Masayoshi 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. A. Iarrobino, "Inverse system of a symbolic power III: thin algebras and fat points", Compositio Mathematica 108 (1997), 319–356. DOI: 10.1023/a:1000155612073; https://doi.org/10.1023/a:1000155612073
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7. Ciro Ciliberto and Rick Miranda, "Nagata's conjecture for a square or nearly-square number of points", Ricerche di Matematica 55 (2006), no. 1, 71–78. DOI: 10.1007/s11587-006-0005-y; https://doi.org/10.1007/s11587-006-0005-y
8. Ciro Ciliberto, Rick Miranda and Joaquim Roé, "Quadric cones on the boundary of the Mori cone for very general blowups of the plane", arXiv:2310.10507 (2023); https://arxiv.org/abs/2310.10507
9. Shijie Bao, Qi'an Guan, Zhitong Mi and Zheng Yuan, "On the multipoled global Zhou weights and semi-continuity for Zhou numbers", arXiv:2311.06459 (2023); https://arxiv.org/abs/2311.06459
10. Carlos Galindo, Francisco Monserrat, Carlos-Jesús Moreno-Ávila and Julio José Moyano-Fernández, "On the valuative Nagata conjecture", Research in the Mathematical Sciences 12 (2025), no. 1. arXiv:2208.11041; DOI: 10.1007/s40687-025-00500-2; https://arxiv.org/abs/2208.11041