From 7db8090407d8b2a363e7729eb8512aa64bc74155 Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Wed, 10 May 2023 01:28:46 +0100
Subject: [PATCH] Add condition for positive radius
---
main.tex | 21 ++++++++++++++++++++-
1 file changed, 20 insertions(+), 1 deletion(-)
diff --git a/main.tex b/main.tex
index e9850dc..ba53da4 100644
--- a/main.tex
+++ b/main.tex
@@ -167,7 +167,7 @@ $\chern(F)$, and so we can write:
\begin{sagesilent}
ts = stability.Tilt
-beta = var("beta", domain="real")
+var("beta", domain="real")
c_lower_bound = -(
ts(beta=beta).rank(u)
@@ -571,6 +571,25 @@ vertical wall (TODO as discussed in ref).
+& \sage{bgmlv3_d_upperbound_exp_term_alt.subs(chbv == 0)}
\end{align}
+Furthermore, we get an extra bound for $d$ resulting from the condition that the
+radius of the circular wall must be positive. As discussed in (TODO ref), this
+is equivalent to $\chern^{\beta_{-}}_2(E) > 0$, which yields:
+
+\begin{sagesilent}
+ positive_radius_condition = (
+ (
+ (0 > - u.twist(beta).ch[2])
+ + d # rearrange for d
+ )
+ .subs(solve(q == u.twist(beta).ch[1], c)[0]) # express c in term of q
+ .expand()
+ )
+\end{sagesilent}
+
+\begin{equation*}
+ \sage{positive_radius_condition}
+\end{equation*}
+
\section{Conclusion}
--
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