10-point multi-point Seshadri constant on $\mathbb{P}^2$
Description of constant
Let $x_1,\dots,x_{10}$ be very general points of $\mathbb{P}^2$, and let $\pi:X\to \mathbb{P}^2$ be the blow-up of $\mathbb{P}^2$ at these points. Let $L$ denote the pullback to $X$ of the class of a line in $\mathbb{P}^2$, and let $E_1,\dots,E_{10}$ denote the corresponding exceptional divisors. We define \(C_{74}\ :=\ \varepsilon_{10}\ :=\ \varepsilon\!\left(\mathbb{P}^2,\mathcal O_{\mathbb{P}^2}(1);x_1,\dots,x_{10}\right),\) the multi-point Seshadri constant of $\mathcal O_{\mathbb{P}^2}(1)$ at $10$ very general points. [HR2009-very-general]
Specializing the general multipoint definition to the plane, one gets \(\varepsilon_{10}\ =\ \inf_C \frac{\deg C}{\sum_{i=1}^{10}\operatorname{mult}_{x_i} C},\) where the infimum runs over all plane curves $C\subset \mathbb{P}^2$ passing through at least one of the points, and $\operatorname{mult}_{x_i} C$ denotes the multiplicity of $C$ at $x_i$. [HR2009-def-plane]
Equivalently, \(\varepsilon_{10}\ =\ \sup\left\{t>0 : \frac{1}{t}L-(E_1+\cdots+E_{10}) \text{ is nef on } X\right\},\) where “nef” means having nonnegative intersection with every effective curve on $X$. [HR2009-nef-alt]
The best established range in the references below is \(\frac{117}{370}\ \le\ C_{74}\ =\ \varepsilon_{10}\ \le\ \frac{1}{\sqrt{10}}.\) [Eck2011-lb-117-370] [Gal2025-upper-submax]
Known upper bounds
| Bound | Reference | Comments |
|---|---|---|
| $\frac{1}{\sqrt{10}}$ | [Gal2025] | This is the general bound $\epsilon(S,D,x_1,\dots,x_r)\le \sqrt{D^2/r}$ applied to $S=\mathbb{P}^2$, $D=\mathcal O_{\mathbb{P}^2}(1)$, and $r=10$. [Gal2025-upper-submax] |
Known lower bounds
| Bound | Reference | Comments |
|---|---|---|
| $\frac{4}{13}$ | [Eck2008] | Lower bound for $10$ general points on $\mathbb{P}^2$ obtained by the asymptotic Dumnicki method. [Eck2008-lb-4-13] |
| $\frac{177}{560}$ | [HR2009] | Obtained by proving that $(560/177)L-(E_1+\cdots+E_{10})$ is nef. [HR2009-lb-177-560] |
| $\frac{117}{370}$ | [Eck2011] | Eckl’s improvement on the earlier $55/174$ bound. [Eck2011-lb-117-370] |
Additional comments and links
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Conjectural value. For $r=10$, Nagata predicts the maximal value \(\varepsilon_{10}\ =\ \frac{1}{\sqrt{10}},\) and this remains open because $10$ is not a square. [Gal2025-open-nonsquare]
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Submaximal-curve formulation. In the plane case, the equality $\varepsilon_{10}=1/\sqrt{10}$ is equivalent to the non-existence of submaximal curves for $10$ very general points. [Gal2025-upper-submax]
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Mori-cone formulation. Nagata’s conjecture for $r=10$ is equivalently the statement that the ray generated by \(\sqrt{10}\,\pi^*H-(E_1+\cdots+E_{10})\) is wonderful, where $H$ is the class of a line on $\mathbb{P}^2$ and “wonderful” means an irrational nef ray with self-intersection $0$. [Gal2025-mori-ray]
References
- [HR2009] Harbourne, Brian; Roé, Joaquim. Computing multi-point Seshadri constants on $P^2$. Bulletin of the Belgian Mathematical Society - Simon Stevin 16 (2009), no. 5, 887–906. DOI: 10.36045/bbms/1260369405. arXiv PDF: math/0309064v3. Google Scholar
- [HR2009-def-plane] loc: arXiv v3 PDF p.1, Section 1 (“Introduction”). quote: “Given a positive integer $n$, the codimension 1 multipoint Seshadri constant for points $p_1,\ldots,p_n$ of $P^N$ is the real number \(\varepsilon(N,p_1,\ldots,p_n)=\sqrt[N-1]{\inf\left\{\frac{\deg(Z)}{\sum_{i=1}^n \operatorname{mult}_{p_i} Z}\right\}},\) where the infimum is taken with respect to all hypersurfaces $Z$, through at least one of the points.”
- [HR2009-very-general] loc: arXiv v3 PDF p.1, Section 1 (“Introduction”). quote: “We also take $\varepsilon(N,n)$ to be defined as $\sup{\varepsilon(N,p_1,\ldots,p_n)}$, where the supremum is taken with respect to all choices of $n$ distinct points $p_i$ of $P^N$. It is not hard to see that $\varepsilon(N,n)=\varepsilon(N,p_1,\ldots,p_n)$ for very general points $p_1,\ldots,p_n$.”
- [HR2009-nef-alt] loc: arXiv v3 PDF p.1, Section 1 (“Introduction”). quote: “The divisor class group $\mathrm{Cl}(X)$ has $\mathbb{Z}$-basis given by the classes $L,E_1,\ldots,E_n$, where $L$ is the pullback of the class of a line and $E_i$ is the class of $\pi^{-1}(p_i)$. This terminology provides an alternate description of Seshadri constants: $\varepsilon(n)$ is the supremum of all real numbers $t$ such that $F=(1/t)L-(E_1+\cdots+E_n)$ is nef.”
- [HR2009-lb-177-560] loc: arXiv v3 PDF p.6, Section 2 (discussion after Table 1). quote: “Thus $F=(560/177)L-(E_1+\cdots+E_{10})$ is nef, $F\cdot C(177,56,0)=0$, and we have $\varepsilon(10)\ge 177/560$.”
- [Eck2008] Eckl, Thomas. An asymptotic version of Dumnicki’s algorithm for linear systems in $\mathbb{CP}^2$. Geometriae Dedicata 137 (2008), 149–162. DOI: 10.1007/s10711-008-9291-8. arXiv PDF: 0801.2926v2. Google Scholar
- [Eck2011] Eckl, Thomas. Ciliberto-Miranda degenerations of $\mathbb{CP}^2$ blown up in 10 points. Journal of Pure and Applied Algebra 215 (2011), no. 4, 672–696. DOI: 10.1016/j.jpaa.2010.06.016. arXiv PDF: 0907.4425v1. Google Scholar
- [Gal2025] Galindo, Carlos; Monserrat, Francisco; Moreno-Ávila, Carlos-Jesús; Moyano-Fernández, Julio-José. On the valuative Nagata conjecture. Research in the Mathematical Sciences 12 (2025), Article 18. DOI: 10.1007/s40687-025-00500-2. arXiv PDF: 2208.11041. Google Scholar
- [Gal2025-open-nonsquare] loc: arXiv PDF p.2, Section 1 (“Introduction”). quote: “Nagata proved this result when $r$ is a square and it is an open problem in the remaining cases.”
- [Gal2025-upper-submax] loc: arXiv PDF p.2, Section 1 (“Introduction”). quote: “It holds that $\epsilon(S,D,x_1,x_2 \ldots, x_r) \leq \sqrt{D^2/r}$ and, when the bound is attained, one says that $\epsilon(S,D,x_1,x_2 \ldots, x_r)$ is maximal. Otherwise, there exists a submaximal curve, that is a curve $C$ on $S$ going through at least a point $x_i$ such that $\epsilon(S,D,x_1,x_2 \ldots, x_r)= \frac{D\cdot C}{\sum_{i=1}^{r} \text{mult}{x_i} C}$ \cite[Proposition 1.1]{BauSze}. Setting $S= \mathbb{P}^2$ and $D=L$, the Nagata conjecture is equivalent to the non-existence of submaximal curves (for very general points ${x_i}{i=1}^r$, $r\geq 10$) and it can be generalized to the Nagata-Biran-Szemberg conjecture, which can be stated as follows: $\epsilon(S,D,x_1,x_2 \ldots, x_r)$ is maximal for $D$ ample, $r$ large enough and ${x_i}_{i=1}^r$ very general points in an arbitrary smooth projective surface $S$ \cite[Section 2]{StrSze} and \cite[Section 5.1]{Laz1}.”
- [Gal2025-mori-ray] loc: arXiv PDF p.2, Section 1 (“Introduction”). quote: “The Nagata conjecture can be equivalently stated by claiming that, when ${x_i}{i=1}^r$ are very general points and $r\geq 10$, the ray generated by the class of the $\mathbb{R}$-divisor $ \sqrt{r} \pi^* L - \sum{i=1}^r E_i $ is wonderful, where $\pi^*$ means pull-back and the $E_i$’s denote the exceptional divisors created by $\pi$. Recall that a wonderful ray is an irrational nef ray with vanishing self-intersection.”
Contribution notes
Prepared with assistance from ChatGPT 5.2 Pro.