From 1d94f415029289d486e502192add0b9e8b028205 Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Tue, 2 May 2023 17:47:48 +0100
Subject: [PATCH] Add alternative expression for hyperbolic term in third bound
for d
---
main.tex | 12 +++++++-----
1 file changed, 7 insertions(+), 5 deletions(-)
diff --git a/main.tex b/main.tex
index ab36005..e66912c 100644
--- a/main.tex
+++ b/main.tex
@@ -413,12 +413,14 @@ This can be rearranged to express a bound on $d$ as follows:
v.twist(beta_min).ch[2]
+ beta_min*q
).expand()
+ assert bgmlv3_d_upperbound_exp_term == (
+ R*v.twist(beta_min).ch[2]
+ + (C - q)^2/2
+ + R*beta_min*q
+ - D*R
+ )/(r-R)
\end{sagesilent}
-Hyperbolic term:
-\begin{equation}
- \sage{bgmlv3_d_upperbound_exp_term}
-\end{equation}
\noindent
Viewing equation \ref{eqn-bgmlv3_d_upperbound} as an upper bound for $d$ give:
@@ -429,7 +431,7 @@ The linear term in $r$ is
$\sage{bgmlv3_d_upperbound_linear_term}$.
Finally, there's an hyperbolic term in $r$ which tends to 0 as $r \to \infty$,
and can be written:
-$?$.
+$\frac{R\chern^{\beta}_2(F) + (C-q)^2/2 + R\beta q - DR}{r-R}$.
In the case $\beta = \beta_{-}$ (or $\beta_{+}$) we have
$\chern^{\beta}_2(F) = 0$,
so some of these expressions simplify, and in particular, the constant and
--
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