The valuative Nagata conjecture: minimality of very general divisorial valuations of the projective plane

The valuative Nagata conjecture replaces the $r$ very general points of Nagata's classical conjecture on linear systems in the projective plane by a very general plane valuation $\nu$ whose inverse normalized volume $[\mathrm{vol}^N(\nu)]^{-1}$ plays the role of a (real) number of points: it predicts that every very general valuation with $[\mathrm{vol}^N(\nu)]^{-1}\ge 9$ is minimal, i.e. that its Seshadri-type constant $\hat\mu^N(\nu)$, the asymptotic maximal vanishing order of plane curves along $\nu$, attains the universal lower bound $\sqrt{[\mathrm{vol}^N(\nu)]^{-1}}$. Stated for quasimonomial valuations by Dumnicki–Harbourne–Küronya–Roé–Szemberg (2017) and extended to all divisorial and irrational valuations by Galindo–Monserrat–Moyano-Fernández (2018), the conjecture implies the classical Nagata conjecture and follows from the Greuel–Lossen–Shustin conjecture. Galindo–Monserrat–Moreno-Ávila–Moyano-Fernández [1] prove seven equivalent formulations for divisorial valuations — in terms of a new valuative Seshadri constant $\varepsilon(D,\nu_r)$, Newton–Okounkov bodies, Zariski chambers, and submaximal curves (Conjecture C) — and pose the problem of finding effective, ideally combinatorial, conditions guaranteeing minimality. The problem asks for a proof or refutation of the conjecture in its divisorial form. The conjecture is open.

Background

Nagata's 1959 counterexample to Hilbert's fourteenth problem gave rise to the Nagata conjecture, one of the central open problems on linear systems in the complex projective plane $\mathbb{P}^2$: for $r\ge 10$ very general points $x_1,\dots,x_r$ (points avoiding countably many proper Zariski-closed subsets of the configuration space), every plane curve $C$ of degree $d$ passing through them with multiplicities $\mathrm{mult}_{x_i}(C)\ge m_i>0$ satisfies $d>\frac{1}{\sqrt{r}}\sum_{i=1}^r m_i$. Equivalently, the multi-point Seshadri constant — the supremum $\epsilon(\mathbb{P}^2,L;x_1,\dots,x_r)$ of $t\ge 0$ such that the pullback of a line $L$ minus $t\sum_i E_i$ is nef (has nonnegative intersection with every curve) on the blow-up at the points — attains its universal upper bound $1/\sqrt{r}$. Nagata proved the case where $r$ is a square; the general case has been open for more than sixty years.

The valuative approach replaces the finite set of points by a single valuation. Blowing up a chain of $r$ infinitely near points (each point lying on the exceptional divisor of the previous blow-up) produces an exceptional divisor $E_r$, and the order of vanishing along $E_r$ defines a divisorial plane valuation $\nu_r$ of the local ring $R=\mathcal{O}_{\mathbb{P}^2,p}$. Its volume $\mathrm{vol}(\nu_r)=\lim_{\alpha\to\infty}\mathrm{length}(R/\mathcal{P}_\alpha)/(\alpha^2/2)$, where $\mathcal{P}_\alpha=\{f:\nu_r(f)\ge\alpha\}$, measures the asymptotic growth of valuation ideals; it satisfies $\mathrm{vol}(\nu_r)^{-1}=\bar\beta_{g+1}(\nu_r)$, the last member of the sequence of maximal contact values $\bar\beta_0,\bar\beta_1,\dots,\bar\beta_{g+1}$ attached to $\nu_r$, whose length is determined by the dual graph of the blow-up (the graph recording which earlier exceptional divisors each infinitely near point lies on), and in fact $\bar\beta_{g+1}=\sum_i\nu_r(\mathfrak m_i)^2$. The normalized valuation $\nu_r^N=\nu_r/\nu_r(\mathfrak m)$ ($\mathfrak m$ the maximal ideal) has normalized volume $\mathrm{vol}^N(\nu_r)=\mathrm{vol}(\nu_r^N)$, and $[\mathrm{vol}^N(\nu_r)]^{-1}$ plays the role of the number of points. The Seshadri-type constant $\hat\mu(\nu)$, introduced by Boucksom–Küronya–Maclean–Szemberg in the study of vanishing sequences and Okounkov bodies, is the asymptotic maximal vanishing order of plane curves along $\nu$: $\hat\mu(\nu)=\lim_d\max\{\nu(f):\deg f\le d\}/d$. It always satisfies $\hat\mu^N(\nu)\ge\sqrt{[\mathrm{vol}^N(\nu)]^{-1}}$, and $\nu$ is called minimal when equality holds — the valuation-theoretic analogue of a Seshadri constant being maximal. A valuation is very general if, within the family $\mathcal{V}_\Gamma$ of valuations sharing its dual graph $\Gamma$ (parametrized by a Zariski-open subset of an affine space), the relevant property holds on a countable intersection of nonempty Zariski-open sets — the analogue of very general points.

Dumnicki–Harbourne–Küronya–Roé–Szemberg (2017) conjectured, for quasimonomial (monomial-type) valuations, that every very general valuation with $[\mathrm{vol}^N(\nu)]^{-1}\ge 9$ is minimal, and proved that this valuative Nagata conjecture implies the classical Nagata conjecture (via degenerating general points to a curvilinear configuration, a "collision de front", plus semicontinuity) and that it follows from the Greuel–Lossen–Shustin conjecture on dimensions of linear systems with prescribed cluster multiplicities, a statement in the same circle as the Segre–Harbourne–Gimigliano–Hirschowitz conjecture on speciality of fat-point linear systems. They computed $\hat\mu$ explicitly for inverse volumes up to $7+1/9$ and sporadic values beyond, and conjectured the sharp threshold $8+1/36$. Galindo–Monserrat–Moyano-Fernández (2018) extended the conjecture to all divisorial and irrational plane valuations, proved the Greuel–Lossen–Shustin conjecture implies it, constructed infinitely many families of very general minimal valuations, and proved an asymptotic result $\lim_{t\to\infty}\hat\mu(\nu_t^N)/\sqrt{t}=1$ supporting the conjecture.

The paper [1] develops the conjecture for divisorial valuations. It introduces a valuative Seshadri constant $\varepsilon(D,\nu_r)$ — the supremum of $t\ge 0$ such that $D^*-t\sum_i\nu_r(\mathfrak m_i)E_i^*$ is nef on the blow-up surface $\tilde S$ — and proves that minimality of $\nu_r$ with respect to an ample divisor $D$ is equivalent to a web of conditions involving: the Newton–Okounkov body of $D$ (the convex body encoding the asymptotics of the graded linear series of $D$ along a flag) being the expected triangle for every exceptional curve valuation extending $\nu_r$ (an exceptional curve valuation is the rank-two valuation defined by a flag $\tilde S\supset E_r\supset\{q\}$); nefness of an explicit $\mathbb{R}$-divisor $P_\mu$ with $P_\mu^2=0$; the identity $\hat\mu=\varepsilon\cdot[\mathrm{vol}]^{-1}$; the segment $\{[L^*-tE_r]\}$ crossing only one Zariski chamber (the big cone of a surface decomposes into Zariski chambers, regions on which the stable base locus of divisor classes is constant); and the absence of submaximal curves (curves computing $\varepsilon$ strictly below its expected value). For $S=\mathbb{P}^2$ and $D=L$ these give seven equivalent forms of the valuative Nagata conjecture, stated as Conjecture C. The paper also proves that non-minimality forces at most $\rho(S)$ submaximal curves ($\rho(S)$ the Picard number, the rank of the Néron–Severi group) and at most $2+\rho(S)-n$ Zariski-chamber crossings, where $n$ is the number of submaximal curves computing $\varepsilon(D,\nu_r)$, and it closes by posing the problem of finding effective — ideally purely combinatorial, in terms of the dual graph and contact values — conditions guaranteeing minimality.

Problem Statement

Prove or disprove the valuative Nagata conjecture in its divisorial form (Conjecture C of [1]). Let $\nu_r$ be a very general divisorial valuation of $\mathbb{P}^2_{\mathbb{C}}$, let $L$ be a general projective line, and let $\mathrm{vol}^N(\nu_r)$ be its normalized volume. Prove, or refute, the statement:

$$\text{if }[\mathrm{vol}^N(\nu_r)]^{-1}\ge 9\text{, then }\nu_r\text{ is minimal with respect to }L,$$

that is, $\hat\mu(\nu_r)=\sqrt{1/\mathrm{vol}(\nu_r)}$, equivalently $\hat\mu^N(\nu_r)=\sqrt{[\mathrm{vol}^N(\nu_r)]^{-1}}$ — the universal lower bound is attained. Equivalently, establish any one of the seven equivalent properties of Conjecture C, for instance: $\varepsilon(L,\nu_r)=\sqrt{1/[\mathrm{vol}(\nu_r)]^{-1}}$ with no submaximal curve computing it; or the Newton–Okounkov body of $L$ with respect to any exceptional curve valuation with first component $\nu_r$ is the expected triangle $\mathfrak{T}_L(\nu)$; or the segment $\{[L^*-tE_r]\mid 0\le t\le\hat\mu_L(\nu_r)\}$ crosses only one Zariski chamber.

The conjecture is refuted by a proof that non-minimality holds on a nonempty Zariski-open subfamily of the family $\mathcal{V}_\Gamma$ of valuations with some dual graph $\Gamma$, inside the region $[\mathrm{vol}^N]^{-1}\ge 9$; by very generality, such a subfamily contradicts the conjecture. The stronger original form of the conjecture, stated for all divisorial and irrational plane valuations, may be addressed instead, since it implies the divisorial form. The authors' broader posed problem — effective conditions on a very general divisorial valuation of an arbitrary smooth surface $S$ and an ample divisor $D$ guaranteeing minimality with respect to $D$ — is the natural generalization.

References

1. Carlos Galindo, Francisco Monserrat, Carlos-Jesús Moreno-Ávila, Julio José Moyano-Fernández. On the valuative Nagata conjecture. Research in the Mathematical Sciences 12(1), 2025. DOI: 10.1007/s40687-025-00500-2; arXiv:2208.11041. https://arxiv.org/abs/2208.11041
2. Marcin Dumnicki, Brian Harbourne, Alex Küronya, Joaquim Roé, Tomasz Szemberg. Very general monomial valuations of $\mathbb{P}^2$ and a Nagata-type conjecture. Communications in Analysis and Geometry 25(1), 125–161, 2017. DOI: 10.4310/cag.2017.v25.n1.a4; arXiv:1312.5549. https://arxiv.org/abs/1312.5549
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