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

The statement

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 tabulated lower bounds increase as $4/13 < 177/560 < 55/174 < 117/370$ . 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]

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.