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Record Eckl's 55/174 bound on C74 #182

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Record Eckl's 55/174 bound on C74 #182

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68dd8c7
Record Eckl's intermediate 55/174 lower bound.
Chessing234 Sep 8, 2026
ae90869
Mark 117/370 as the tabulated lower-bound record.
Chessing234 Sep 8, 2026
7bb5068
Explain why 55/174 sits between Harbourne–Roé and 117/370.
Chessing234 Sep 8, 2026
8ab6d08
Record the numerical order of the C74 lower bounds.
Chessing234 Sep 8, 2026
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7 changes: 5 additions & 2 deletions constants/74a.md
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Expand Up @@ -20,7 +20,7 @@ $$
$$
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
The tabulated lower bounds increase as $4/13 < 177/560 < 55/174 < 117/370$. The best established range in the references below is
$$
\frac{117}{370}\ \le\ C_{74}\ =\ \varepsilon_{10}\ \le\ \frac{1}{\sqrt{10}}.
$$
Expand All @@ -38,10 +38,13 @@ $$
| ----- | --------- | -------- |
| $\frac{4}{13}$ | [[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> |
| $\frac{55}{174}$ | [[Eck2011](#Eck2011)] | Intermediate lower bound, later improved to $117/370$. |
| $\frac{117}{370}$ | [[Eck2011](#Eck2011)] | Eckl’s improvement on the earlier $55/174$ bound. Record among the tabulated lower bounds. <a href="#Eck2011-lb-117-370">[Eck2011-lb-117-370]</a> |

## Additional comments and links

- **Historical 55/174 bound.** The $55/174$ row is the intermediate Eckl bound cited in the $117/370$ comment, omitted from the table previously.

- **Conjectural value.** For $r=10$, Nagata predicts the maximal value
$$
\varepsilon_{10}\ =\ \frac{1}{\sqrt{10}},
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