From 68dd8c76ad7ad5d2e9d5e4e30bbfd4a7eab31cfa Mon Sep 17 00:00:00 2001 From: Taksh Date: Tue, 8 Sep 2026 15:50:37 +0530 Subject: [PATCH 1/4] Record Eckl's intermediate 55/174 lower bound. --- constants/74a.md | 1 + 1 file changed, 1 insertion(+) diff --git a/constants/74a.md b/constants/74a.md index f1e5dba8..8b936da8 100644 --- a/constants/74a.md +++ b/constants/74a.md @@ -38,6 +38,7 @@ $$ | ----- | --------- | -------- | | $\frac{4}{13}$ | [[Eck2008](#Eck2008)] | Lower bound for $10$ general points on $\mathbb{P}^2$ obtained by the asymptotic Dumnicki method. [Eck2008-lb-4-13] | | $\frac{177}{560}$ | [[HR2009](#HR2009)] | Obtained by proving that $(560/177)L-(E_1+\cdots+E_{10})$ is nef. [HR2009-lb-177-560] | +| $\frac{55}{174}$ | [[Eck2011](#Eck2011)] | Intermediate lower bound, later improved to $117/370$. | | $\frac{117}{370}$ | [[Eck2011](#Eck2011)] | Eckl’s improvement on the earlier $55/174$ bound. [Eck2011-lb-117-370] | ## Additional comments and links From ae908695406b55144ec1eb58d5b5191b302309bc Mon Sep 17 00:00:00 2001 From: Taksh Date: Tue, 8 Sep 2026 15:50:37 +0530 Subject: [PATCH 2/4] Mark 117/370 as the tabulated lower-bound record. --- constants/74a.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/constants/74a.md b/constants/74a.md index 8b936da8..cc2fbd75 100644 --- a/constants/74a.md +++ b/constants/74a.md @@ -39,7 +39,7 @@ $$ | $\frac{4}{13}$ | [[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] | | $\frac{55}{174}$ | [[Eck2011](#Eck2011)] | Intermediate lower bound, later improved to $117/370$. | -| $\frac{117}{370}$ | [[Eck2011](#Eck2011)] | Eckl’s improvement on the earlier $55/174$ bound. [Eck2011-lb-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 From 7bb5068a535de18244d293decd6eba97e6e9981d Mon Sep 17 00:00:00 2001 From: Taksh Date: Tue, 8 Sep 2026 15:50:38 +0530 Subject: [PATCH 3/4] =?UTF-8?q?Explain=20why=2055/174=20sits=20between=20H?= =?UTF-8?q?arbourne=E2=80=93Ro=C3=A9=20and=20117/370.?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- constants/74a.md | 2 ++ 1 file changed, 2 insertions(+) diff --git a/constants/74a.md b/constants/74a.md index cc2fbd75..5c4af5a2 100644 --- a/constants/74a.md +++ b/constants/74a.md @@ -43,6 +43,8 @@ $$ ## Additional comments and links +- **Historical 55/174 bound.** The $55/174$ row is the intermediate Eckl bound cited in the $117/370$ comment, omitted from the table previously. + - **Conjectural value.** For $r=10$, Nagata predicts the maximal value $$ \varepsilon_{10}\ =\ \frac{1}{\sqrt{10}}, From 8ab6d0869a8a486c53b879a9088bf7758a8b5e2c Mon Sep 17 00:00:00 2001 From: Taksh Date: Tue, 8 Sep 2026 15:50:38 +0530 Subject: [PATCH 4/4] Record the numerical order of the C74 lower bounds. --- constants/74a.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/constants/74a.md b/constants/74a.md index 5c4af5a2..be83ed82 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 $$ \frac{117}{370}\ \le\ C_{74}\ =\ \varepsilon_{10}\ \le\ \frac{1}{\sqrt{10}}. $$