diff --git a/constants/74a.md b/constants/74a.md index f1e5dba..be83ed8 100644 --- a/constants/74a.md +++ b/constants/74a.md @@ -20,7 +20,7 @@ $$ $$ where “nef” means having nonnegative intersection with every effective curve on $X$. [HR2009-nef-alt] -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 $$ \frac{117}{370}\ \le\ C_{74}\ =\ \varepsilon_{10}\ \le\ \frac{1}{\sqrt{10}}. $$ @@ -38,10 +38,13 @@ $$ | ----- | --------- | -------- | | $\frac{4}{13}$ | [[Eck2008](#Eck2008)] | Lower bound for $10$ general points on $\mathbb{P}^2$ obtained by the asymptotic Dumnicki method. [Eck2008-lb-4-13] | | $\frac{177}{560}$ | [[HR2009](#HR2009)] | Obtained by proving that $(560/177)L-(E_1+\cdots+E_{10})$ is nef. [HR2009-lb-177-560] | -| $\frac{117}{370}$ | [[Eck2011](#Eck2011)] | Eckl’s improvement on the earlier $55/174$ bound. [Eck2011-lb-117-370] | +| $\frac{55}{174}$ | [[Eck2011](#Eck2011)] | Intermediate lower bound, later improved to $117/370$. | +| $\frac{117}{370}$ | [[Eck2011](#Eck2011)] | Eckl’s improvement on the earlier $55/174$ bound. Record among the tabulated lower bounds. [Eck2011-lb-117-370] | ## Additional comments and links +- **Historical 55/174 bound.** The $55/174$ row is the intermediate Eckl bound cited in the $117/370$ comment, omitted from the table previously. + - **Conjectural value.** For $r=10$, Nagata predicts the maximal value $$ \varepsilon_{10}\ =\ \frac{1}{\sqrt{10}},