Nagata's conjecture for ten general points: emptiness of the homogeneous linear systems $\mathcal{L}_d(m^{10})$ below slope $\sqrt{10}$

For positive integers $d,m$, let $\mathcal{L}_d(m^{10})$ be the linear system of complex plane curves of degree $d$ passing with multiplicity at least $m$ through each of ten points in general position in the projective plane $\mathbb{P}^2$. Nagata's conjecture for ten general points predicts that $\mathcal{L}_d(m^{10})$ is empty whenever $d/m<\sqrt{10}\approx 3.16228$, or equivalently that the ten-point Seshadri constant of the plane equals $1/\sqrt{10}$. The conjecture goes back to Nagata's 1959 counterexample to Hilbert's fourteenth problem, and the ten-point case is the critical open base case of the conjecture for all $r\ge 10$ points. The best known uniform emptiness threshold is $117/37\approx 3.16216$, established in 2009 by Ciliberto–Dumitrescu–Miranda–Roé and independently by Eckl; closing the remaining gap to $\sqrt{10}$ (or exhibiting a counterexample) is the open problem.

Background

A linear system $\mathcal{L}_d(m_1,\dots,m_r)$ of plane curves is the projective space of degree-$d$ curves in $\mathbb{P}^2$ having multiplicity at least $m_i$ at $r$ fixed points $p_1,\dots,p_r$; imposing a point of multiplicity $m_i$ is expected to cut the dimension by $m_i(m_i+1)/2$, giving an expected dimension $\max\{-1,\,d(d+3)/2-\sum_i m_i(m_i+1)/2\}$, where the value $-1$ means the system is expected to be empty. The points are called general (or very general) when they avoid every proper algebraic condition, i.e. no special position is imposed; a linear system of plane curves through general points is called special if its dimension exceeds the expected one. Equivalently, one studies the blow-up $X_r$ of $\mathbb{P}^2$ at the points — the smooth surface obtained by replacing each $p_i$ with the set of tangent directions at $p_i$ — a rational surface whose Picard group (divisor classes modulo linear equivalence) is generated by the pull-back $H$ of a line and the exceptional divisors $E_i$ (the curves replacing the points), so that $\mathcal{L}_d(m_1,\dots,m_r)$ corresponds to the complete linear system $|dH-\sum_i m_iE_i|$. The Segre–Harbourne–Gimigliano–Hirschowitz (SHGH) conjecture predicts that speciality for general points is entirely explained by $(-1)$-curves (curves of self-intersection $-1$ on $X_r$) via the so-called $(-1)$-special systems; it is known for $r\le 9$ but open for ten and more points. The problem studied here originates in Nagata's 1959 negative solution of Hilbert's fourteenth problem: studying invariant rings led Nagata to prove that when $r$ is a perfect square at least $16$, the homogeneous system $\mathcal{L}_d(m^r)$ is empty whenever $d/m<\sqrt{r}$, and the analogous statement was conjectured for every $r\ge 10$. In modern language the conjecture for $r$ general points is equivalent to the statement that the multi-point Seshadri constant of the hyperplane bundle — defined as $\varepsilon(\mathcal{O}_{\mathbb{P}^2}(1);p_1,\dots,p_r)=\inf_C d/(\sum_i\operatorname{mult}_{p_i}C)$, the infimum over all curves $C$ of degree $d$ through the points — equals $1/\sqrt{r}$; the upper bound $\le 1/\sqrt{r}$ being easy, the conjecture amounts to the lower bound. For a finite set $Z$ of points with homogeneous ideal $I(Z)$, the $m$-th symbolic power $I(mZ)$ is the ideal of plane curves vanishing to order at least $m$ at every point of $Z$, its initial degree $\alpha(I(mZ))$ is the smallest degree of a curve it contains, and the Waldschmidt constant is the limit $\widehat\alpha(I(Z))=\lim_{m\to\infty}\alpha(I(mZ))/m$. For $r=10$ the conjectured critical ratio is $\sqrt{10}\approx 3.162278$, and the known uniform emptiness thresholds have crept up through Harbourne–Roé's $177/56\approx 3.1607$, Ciliberto–Miranda's $550/174\approx 3.1609$, Dumnicki's $313/99\approx 3.16162$, to $117/37\approx 3.162162$ (Ciliberto–Dumitrescu–Miranda–Roé, and independently Eckl), still strictly below $\sqrt{10}$. For bounded multiplicities the conjecture is verified computationally (Dumnicki's cutting-diagram method handles the homogeneous case up to $m=42$), but the uniform statement over all pairs $(d,m)$ has resisted all methods: Cremona-transform techniques (Cremona transformations are birational self-maps of the plane, such as the standard quadratic map, which act on linear systems by changing degrees and multiplicities) and degeneration techniques tend to produce only rational thresholds, and the conjectural limit slope $\sqrt{10}$ is irrational. The ten-point case is the base case of the general conjecture: Ciliberto–Harbourne–Miranda–Roé showed that Nagata's conjecture for $n\le 89$ points, combined with a stronger statement at $n=10$, implies the conjecture for all $n\ge 90$.

Problem Statement

Prove or refute Nagata's conjecture for ten general points in the projective plane: for all pairs of positive integers $(d,m)$ with $d/m<\sqrt{10}$, the homogeneous linear system $\mathcal{L}_d(m^{10})$ — the system of plane curves of degree $d$ having multiplicity at least $m$ at each of ten points in general position — is empty. Equivalently, prove or refute that the multi-point Seshadri constant of the hyperplane bundle at ten general points equals $1/\sqrt{10}$, i.e. $\varepsilon(\mathcal{O}_{\mathbb{P}^2}(1);p_1,\dots,p_{10})=1/\sqrt{10}$, or again equivalently that the initial degree of the $m$-th symbolic power of the ideal of ten general points satisfies $\alpha(I(mZ))>m\sqrt{10}$ for every $m\ge 1$ (here $Z=\{p_1,\dots,p_{10}\}$ and $I(mZ)$ is the ideal of curves vanishing to order at least $m$ at each $p_i$). The single scientific objective is the exact determination of the critical emptiness ratio for ten general points; currently the best established uniform threshold is $117/37\approx 3.162162$, and the gap to $\sqrt{10}\approx 3.162278$ is precisely what remains. No narrowing of the multiplicities, degrees, point count, or answer form is intended: the statement must hold for all $(d,m)$, and a single explicit pair $(d,m)$ with $d/m<\sqrt{10}$ for which $\mathcal{L}_d(m^{10})$ is nonempty for general points would refute the conjecture.

References

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