Add 10-point multi-point Seshadri constant on $\mathbb{P}^2$ to README

Author: damek

README.md

@@ -104,6 +104,7 @@ We are arbitrarily numbering the constants as $C_{1}$, $C_{2}$, etc., mostly bas
 | [71](https://teorth.github.io/optimizationproblems/constants/71a.html) | Fourier Entropy-Influence constant | 6.278 | $\infty$ |
 | [72](https://teorth.github.io/optimizationproblems/constants/72a.html) | Polya-Vinogradov best constant (squarefree asymptotic) | 0 | $\frac{1}{4\pi}\approx 0.07958$ |
 | [73](https://teorth.github.io/optimizationproblems/constants/73a.html) | Flatness constant in dimension 3 | $2+\sqrt{2}$ | $<3.972$ |
+| [74](https://teorth.github.io/optimizationproblems/constants/74a.html) | 10-point multi-point Seshadri constant on $\mathbb{P}^2$ | $\frac{117}{370}$ | $\frac{1}{\sqrt{10}}$ |
 
 
 ## Recent progress

constants/74a.md

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+# 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. <a href="#HR2009-very-general">[HR2009-very-general]</a>
+
+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 $x_i$. <a href="#HR2009-def-plane">[HR2009-def-plane]</a>
+
+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 $X$. <a href="#HR2009-nef-alt">[HR2009-nef-alt]</a>
+
+The best established range in the references below is
+$$
+\frac{117}{370}\ \le\ C_{74}\ =\ \varepsilon_{10}\ \le\ \frac{1}{\sqrt{10}}.
+$$
+<a href="#Eck2011-lb-117-370">[Eck2011-lb-117-370]</a> <a href="#Gal2025-upper-submax">[Gal2025-upper-submax]</a>
+
+## Known upper bounds
+
+| Bound | Reference | Comments |
+| ----- | --------- | -------- |
+| $\frac{1}{\sqrt{10}}$ | [[Gal2025](#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 $r=10$. <a href="#Gal2025-upper-submax">[Gal2025-upper-submax]</a> |
+
+## Known lower bounds
+
+| Bound | Reference | Comments |
+| ----- | --------- | -------- |
+| $\frac{4}{13}$ | [[Eck2008](#Eck2008)] | Lower bound for $10$ general points on $\mathbb{P}^2$ obtained by the asymptotic Dumnicki method. <a href="#Eck2008-lb-4-13">[Eck2008-lb-4-13]</a> |
+| $\frac{177}{560}$ | [[HR2009](#HR2009)] | Obtained by proving that $(560/177)L-(E_1+\cdots+E_{10})$ is nef. <a href="#HR2009-lb-177-560">[HR2009-lb-177-560]</a> |
+| $\frac{117}{370}$ | [[Eck2011](#Eck2011)] | Eckl’s improvement on the earlier $55/174$ bound. <a href="#Eck2011-lb-117-370">[Eck2011-lb-117-370]</a> |
+
+## Additional comments and links
+
+- **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. <a href="#Gal2025-open-nonsquare">[Gal2025-open-nonsquare]</a>
+
+- **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. <a href="#Gal2025-upper-submax">[Gal2025-upper-submax]</a>
+
+- **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 $0$. <a href="#Gal2025-mori-ray">[Gal2025-mori-ray]</a>
+
+## References
+
+- <a id="HR2009"></a>**[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](https://doi.org/10.36045/bbms/1260369405). arXiv PDF: [math/0309064v3](https://arxiv.org/pdf/math/0309064v3.pdf). [Google Scholar](https://scholar.google.com/scholar?q=Harbourne+Roe+Computing+multi-point+Seshadri+constants+on+P2)
+	- <a id="HR2009-def-plane"></a>**[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.”
+	- <a id="HR2009-very-general"></a>**[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 $p_1,\ldots,p_n$.”
+	- <a id="HR2009-nef-alt"></a>**[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.”
+	- <a id="HR2009-lb-177-560"></a>**[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 $\varepsilon(10)\ge 177/560$.”
+
+- <a id="Eck2008"></a>**[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](https://doi.org/10.1007/s10711-008-9291-8). arXiv PDF: [0801.2926v2](https://arxiv.org/pdf/0801.2926v2.pdf). [Google Scholar](https://scholar.google.com/scholar?q=Eckl+An+asymptotic+version+of+Dumnicki%27s+algorithm+for+linear+systems+in+CP2)
+	- <a id="Eck2008-lb-4-13"></a>**[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 $P^2$.”
+
+- <a id="Eck2011"></a>**[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](https://doi.org/10.1016/j.jpaa.2010.06.016). arXiv PDF: [0907.4425v1](https://arxiv.org/pdf/0907.4425v1.pdf). [Google Scholar](https://scholar.google.com/scholar?q=Eckl+Ciliberto-Miranda+degenerations+of+CP2+blown+up+in+10+points)
+	- <a id="Eck2011-lb-117-370"></a>**[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 $\mathbb{CP}^2$.”
+
+- <a id="Gal2025"></a>**[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](https://doi.org/10.1007/s40687-025-00500-2). arXiv PDF: [2208.11041](https://arxiv.org/pdf/2208.11041.pdf). [Google Scholar](https://scholar.google.com/scholar?q=Galindo+Monserrat+Moreno-Avila+Moyano-Fernandez+On+the+valuative+Nagata+conjecture)
+	- <a id="Gal2025-open-nonsquare"></a>**[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.”
+	- <a id="Gal2025-upper-submax"></a>**[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}.”
+	- <a id="Gal2025-mori-ray"></a>**[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.