From 7d3975481cb541086e134a36e4b23abc7aa5c345 Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Mon, 17 Apr 2023 13:27:03 +0100
Subject: [PATCH] Start refinements
---
main.tex | 47 ++++++++++++++++++++++++++++++++++++++++++++++-
1 file changed, 46 insertions(+), 1 deletion(-)
diff --git a/main.tex b/main.tex
index 8aed269..40e3314 100644
--- a/main.tex
+++ b/main.tex
@@ -135,11 +135,56 @@ for the rank of $E$:
\section{Refinement}
-
+To get tighter bounds on the rank of destabilizers $E$ of some $F$ with some
+fixed Chern character, we will need to consider each of the values which
+$\chern_1^{\beta_{-}}(E)$ can take.
+Doing this will allow us to eliminate possible values of $\chern_0(E)$ for which
+each $\chern_1^{\beta_{-}}(E)$ leads to the failure of at least one of the inequalities.
+As opposed to only eliminating possible values of $\chern_0(E)$ for which all
+corresponding $\chern_1^{\beta_{-}}(E)$ fail one of the inequalities (which is what
+was implicitly happening before).
+
+First, let's fix a Chern character for $F$,
+$\chern(F) = (R,C,D)$, and consider the possible Chern characters
+$\chern(E) = (r,c,d)$ of some semistabilizer $E$.
+
\begin{sagesilent}
from pseudowalls import *
+
+ v = Chern_Char(*var("R C D", domain="real"))
+ u = Chern_Char(*var("r c d", domain="real"))
+
+ ts = stability.Tilt
+
+ beta_min = min(map(lambda sol: sol.rhs(),
+ solve(ts(alpha=0).degree(v), ts().beta)
+ ))
+\end{sagesilent}
+
+Here, we have $\beta_{-} = \sage{beta_min}$.
+Recall [ref] that $\chern_1^{\beta_{-}}$ has fixed bounds in terms of
+$\chern(F)$, and so we can write:
+
+\begin{equation}
+ \chern_1(E) = r\beta_{-} + m
+ \qquad 0 \leq m \leq \chern_1^{\beta_{-}}(F)
+\end{equation}
+
+Furthermore, $\chern_1 \in \ZZ$ so we only need to consider
+$m \in \frac{1}{n} \ZZ \cap [0, \chern_1^{\beta_{-}}(F)]$.
+For the next subsections, we consider $m$ to be fixed with one of these values,
+and we shall be varying $\chern_0(E) = r$ to see when certain inequalities fail.
+
+\begin{sagesilent}
+ c_lower_bound = -(ts(beta=beta_min).rank(u)/ts().alpha).expand() + c
+
+ var("m", domain="real")
+ c_val = c_lower_bound + m
\end{sagesilent}
+
+
+
\section{Conclusion}
\end{document}
--
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