From 543f5d72e25e6f154937cce2c49aaacbe823792d Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Fri, 30 Jun 2023 15:23:18 +0100
Subject: [PATCH] Trim out old outline bullet points
---
main.tex | 23 +----------------------
1 file changed, 1 insertion(+), 22 deletions(-)
diff --git a/main.tex b/main.tex
index 5e4b908..aac27e6 100644
--- a/main.tex
+++ b/main.tex
@@ -626,17 +626,6 @@ with the goal of cutting down the run time.
\subsection*{Problem statement}
-Goals:
-\begin{itemize}
- \item link repo
- \item Calc max destab rank
- \item Decrease mu(E) starting from mu(F) taking on all poss frac vals
- \item iterate over something else
- \item Stop when conditions fail
- \item method works same way for both rational beta_{-} but also for walls
- larger than certain amount
-\end{itemize}
-
Fix a Chern character $v$ with positive rank, $\Delta(v) \geq 0$,
and $\beta_{-}(v) \in \QQ$.
The goal is to find all Chern characters $u=(r,c\ell,d\ell^2)$ which satisfy the
@@ -674,7 +663,6 @@ the Bogomolov inequalities and consequence 3 of lemma
\ref{lem:pseudo_wall_numerical_tests}
($\chern_2^{\beta_{-}}(u)>0$).
-
\subsubsection*{Finding $d$ for fixed $r$ and $c$}
$\Delta(u) \geq 0$ induces an upper bound $\frac{c^2}{2r}$ on $d$, and the
@@ -682,17 +670,8 @@ $\chern_2^{\beta_{-}}(u)>0$ condition induces a lower bound on $d$.
The values in the range can be tested individually, to check that
the rest of the conditions are satisfied.
-
-
\subsection*{Limitations}
-Goals:
-\begin{itemize}
- \item large rank forces mu to beta_{-}, so many vals of mu(E) checked
- needlessly
- \item noticeably slow (show benchmark)
-\end{itemize}
-
The main downside of this algorithm is that many $r$,$c$ pairs which are tested
end up not yielding any solutions for the problem.
In fact, solutions $u$ to our problem with high rank must have $\mu(u)$ close to
@@ -712,7 +691,7 @@ Here are some benchmarks to illustrate the performance benefits of the
alternative algorithm which will later be described in this article [ref].
\begin{center}
-\begin{tabular}{ |r|l|l| }
+\begin{tabular}{ |r|l|l| }
\hline
Choice of $v$ on $\mathbb{P}^2$
& $(3, 2\ell, -2)$
--
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