Nagata's conjecture: the multipoint Seshadri constant of the projective plane at general points and full Kähler packings by equal balls

Nagata's conjecture (1959), presented as the central open problem in the source paper on Kähler packings of projective complex manifolds (Fleming, European Journal of Mathematics 7, 2021, Conjectures 1.1 and 1.2), asserts that for $k\ge 9$ points $P_1,\dots,P_k$ in general position in the complex projective plane $\mathbb{P}^2$ the multipoint Seshadri constant of the hyperplane bundle attains its universal volume upper bound: $\epsilon(\mathbb{P}^2,\mathcal{O}_{\mathbb{P}^2}(1);P_1,\dots,P_k)=1/\sqrt{k}$. Equivalently, every plane curve of degree $d$ passing through the points with multiplicities $m_i$ satisfies $\sqrt{k}\,d\ge\sum_i m_i$; and, by the equivalence between multipoint Seshadri constants and maximal Kähler packing radii established by Eckl for surfaces and by Fleming in arbitrary dimension, $\mathbb{P}^2$ admits Kähler packings by $k$ equal balls filling the surface up to an arbitrarily small volume deficit. Nagata proved the conjecture when $k$ is a perfect square ($k\ge 9$), and Biran proved the symplectic analogue for all $k\ge 9$; the algebraic and Kähler statement remains open for every non-square $k\ge 10$.

Background

Let $X$ be a smooth complex projective variety and $L$ an ample line bundle on it. Seshadri constants, introduced by Demailly in 1992, measure how much positivity $L$ retains near finitely many points. If $\pi:\tilde X\to X$ is the blow-up of $X$ at distinct points $P_1,\dots,P_k$ (a blow-up replaces each point by a copy of projective space, the exceptional divisor $E_i=\pi^{-1}(P_i)$), then the multipoint Seshadri constant is $\epsilon(X,L;P_1,\dots,P_k)=\sup\{\epsilon\ge 0:\ \pi^*L-\epsilon(E_1+\cdots+E_k)\ \text{is nef}\},$ where a $\mathbb{Q}$-divisor is called nef if it has non-negative intersection number with every irreducible curve. Equivalently, $\epsilon(X,L;P_1,\dots,P_k)=\inf_C\ \frac{L\cdot C}{\sum_{P_i\in C}\mathrm{mult}_{P_i}(C)},$ the infimum running over irreducible curves $C$ passing through at least one of the points, with $\mathrm{mult}_{P_i}(C)$ the multiplicity of $C$ at $P_i$. A volume comparison gives the universal upper bound $\epsilon(X,L;P_1,\dots,P_k)\le(L^{\dim X}/k)^{1/\dim X}$; for the hyperplane bundle $\mathcal{O}_{\mathbb{P}^2}(1)$ on the complex projective plane this reads $\epsilon\le 1/\sqrt{k}$.

The question of whether this bound is attained on $\mathbb{P}^2$ originates in Nagata's 1959 paper that produced counterexamples to Hilbert's fourteenth problem on finite generation of rings of invariants. Nagata conjectured that for $k\ge 9$ points of $\mathbb{P}^2$ in general position (configurations avoiding a countable union of proper closed algebraic conditions) every algebraic curve of degree $d$ having multiplicity at least $m_i$ at $P_i$ satisfies $\sqrt{k}\,d\ge m_1+\cdots+m_k$; in Seshadri language, $\epsilon(\mathbb{P}^2,\mathcal{O}_{\mathbb{P}^2}(1);P_1,\dots,P_k)=1/\sqrt{k}$. For few points the bound is not attained: for $k=2$ the line through the two points gives $\epsilon\le 1/2<1/\sqrt{2}$, and for $k=3$ the triangle of three lines through the pairs of points gives $\epsilon\le 1/2<1/\sqrt{3}$; this is why the conjecture is made only from $k\ge 9$ onward. Nagata himself proved the conjecture when $k$ is a perfect square ($k=9,16,25,\dots$) by degenerating general points to a square grid. The conjecture is implied by the Segre–Harbourne–Gimigliano–Hirschowitz (SHGH) conjecture, the standard (still open) conjectural dimension theory for linear systems of plane curves with assigned base multiplicities (so-called fat points schemes).

The problem also has an exact reformulation as a packing problem, which is the viewpoint of the source paper. A Kähler packing of a projective manifold $X$ with ample line bundle $L$ by $k$ balls of radii $r_1,\dots,r_k$ centered at prescribed points $P_i$ is a holomorphic embedding $\coprod_{q=1}^k B(r_q)\hookrightarrow X$ of disjoint flat complex balls, with the center of the $q$-th ball mapped to $P_q$, such that the pullback of some Kähler form representing $c_1(L)$ restricts to the standard Kähler form on each ball; the supremal equal radius is the packing constant $\gamma_k$. McDuff and Polterovich (1994) showed that symplectic packings of $\mathbb{CP}^2$ — the same notion with the holomorphicity requirement dropped — are governed by symplectic blow-ups, and Biran (1997) used this to prove the symplectic analogue of Nagata's conjecture: for $k\ge 9$ equal symplectic balls the volume bound is the only obstruction, so $\mathbb{CP}^2$ admits a full symplectic packing by $k$ equal balls for every $k\ge 9$. Eckl (2017, for surfaces) and Fleming (2021, in arbitrary dimension and for arbitrary point configurations) proved the Kähler counterpart of this correspondence: the square of the maximal Kähler packing constant equals the multipoint Seshadri constant, $\gamma_k(X,L;P_1,\dots,P_k)^2=\epsilon(X,L;P_1,\dots,P_k)$. Consequently Nagata's conjecture is equivalent to the statement that for every $k\ge 9$ the projective plane admits Kähler packings by $k$ equal balls whose total volume is arbitrarily close to the volume of $\mathbb{P}^2$ — a full Kähler packing by $k$ equal balls. The flexibility exploited in the symplectic case has no known analogue for holomorphic embeddings, so Biran's theorem does not settle the algebraic problem.

Problem Statement

Prove or refute Nagata's conjecture, stated as Conjecture 1.2 (equivalently Conjecture 1.1) of the source paper:

Conjecture (Nagata). Let $P_1,\dots,P_k$ be points of $\mathbb{P}^2$ in general position. Then for $k\ge 9$ the multipoint Seshadri constant of the hyperplane bundle satisfies $\epsilon\big(\mathbb{P}^2,\mathcal{O}_{\mathbb{P}^2}(1);P_1,\dots,P_k\big)=\frac{1}{\sqrt{k}}.$

Equivalently: for $k\ge 9$, every plane curve of degree $d$ with multiplicity at least $m_i$ at $P_i$ satisfies $\sqrt{k}\,d\ge\sum_{i=1}^k m_i$ (strict inequality holds automatically when $k$ is not a perfect square, since $\sqrt{k}$ is then irrational). Equivalently again: for every $k\ge 9$ and every $\epsilon<1/\sqrt{k}$, the divisor class $\pi^*H-\epsilon(E_1+\cdots+E_k)$ on the blow-up of $\mathbb{P}^2$ at the points, where $H$ denotes the class of a line and $E_1,\dots,E_k$ the exceptional divisors, is ample, i.e. it is represented by a Kähler form; and, by the equivalence $\gamma_k=\sqrt{\epsilon}$ between maximal Kähler packing radii and multipoint Seshadri constants, $\mathbb{P}^2$ admits a full Kähler packing by $k$ equal balls up to an arbitrarily small volume deficit.

Nagata proved the conjecture for $k$ a perfect square with $k\ge 9$ (so $k=9$ is a theorem), and Biran proved the corresponding full symplectic packing statement for all $k\ge 9$. The open core of the problem is therefore: prove, or refute, the conjecture for every non-square $k\ge 10$. A refutation consists of exhibiting some non-square $k\ge 10$ for which the multipoint Seshadri constant at general points is strictly smaller than $1/\sqrt{k}$.

References

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