10-point multi-point Seshadri constant on $\mathbb{P}^2$ ($C_{74}$)

• The $55/174$ bound is Ciliberto–Miranda's, not Eckl's. It was cited in the $117/370$ comment but missing from the table. [Eckl 2011, §0] attributes it explicitly: "a few months ago, Ciliberto and Miranda [CM08] presented a new method to improve these bounds, and obtained $55/174$" — their [CM08] is [CM2011] here — and later compares "$117/370 \approx 0.31621\ldots$: compared to $55/174 \approx 0.31609\ldots$ this is one decimal closer to $0.31622\ldots$".

• Conjectural value. For $r=10$, Nagata predicts the maximal value

and this remains open because $10$ is not a square. [Gal2025-open-nonsquare]

• 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]

• Mori-cone formulation. Nagata’s conjecture for $r=10$ is equivalently the statement that the ray generated by

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 (from the repository page)

• [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

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

• [Eck2008-lb-4-13] loc: arXiv v2 PDF p.1, Abstract. quote: “With this method we prove the lower bound $4/13$ for 10 general points on $P^2$.”

• [CM2011] Ciliberto, Ciro; Miranda, Rick. Homogeneous interpolation on ten points. Journal of Algebraic Geometry 20 (2011), 685–726. DOI: 10.1090/S1056-3911-2011-00545-6. arXiv: 0812.0032 (29 Nov 2008).

• [CM2011-lb-55-174] loc: arXiv:0812.0032, Abstract. quote: "we prove that for all pairs $(d,m)$ with $d/m \geq 174/55$, the linear system of plane curves of degree $d$ with ten general base points of multiplicity $m$ has the expected dimension."

• [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

• [Eck2011-lb-117-370] loc: arXiv v1 PDF p.1, Abstract. quote: “We obtain the lower bound $117/370$ for the 10-point Seshadri constant on $\mathbb{CP}^2$.”

• [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 v1 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 v1 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 $\epsilon(S,D,x_1,x_2 \ldots, x_r)= \frac{D\cdot C}{\sum_{i=1}^{r} \text{mult}_{x_i} C}$ 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 $S$ Section 2 [StrSze] and Section 5.1.”

• [Gal2025-mori-ray] loc: arXiv v1 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.”

Repository table (commit 2c1968c, 26 September 2026): best known lower bound $\frac{117}{370}$; best known upper bound $\frac{1}{\sqrt{10}}$. Values marked with an asterisk have minimal verification.

Known upper bounds

Known lower bounds