From 99dbc164f3c1bbe4e711af23250c978a088c2e39 Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Thu, 29 Jun 2023 17:02:52 +0100
Subject: [PATCH] Complete explanation for Schmidt's algorithm

---
 main.tex | 27 +++++++++++++++++++++------
 1 file changed, 21 insertions(+), 6 deletions(-)

diff --git a/main.tex b/main.tex
index e456ead..32b5823 100644
--- a/main.tex
+++ b/main.tex
@@ -589,7 +589,7 @@ finding all pseudo-walls when $\beta_{-}\in\QQ$. In section [ref], a different
 algorithm will be presented making use of the later theorems in this article,
 with the goal of cutting down the run time.
 
-\subsection{Strategy}
+\subsection*{Problem statement}
 
 Goals:
 \begin{itemize}
@@ -600,6 +600,7 @@ Goals:
 	\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 $\beta_{-}(v) \in \QQ$.
@@ -615,6 +616,9 @@ semistabilizers of actual sheaves. The Chern characters which destabilize
 `outwards' can be recovered as $v-u$ for each solution $u$ to the current
 problem.
 
+\subsection*{Algorithm}
+
+\subsubsection*{Finding possible $r$ and $c$}
 To do this, first calculate the upper bound $r_{max}$ on the ranks of tilt
 semistabilizers, as given by theorem \ref{thm:loose-bound-on-r}.
 
@@ -625,16 +629,27 @@ are no large than $r_{max}$ (giving a finite sequence). This can be done with Fa
 which there exist formulae to generate.
 
 These $\mu(u)$ values determine pairs $r,c$ up to multiples, we can then take
-all multiples which satisy $0<r\geq r_{max}$.
+all multiples which satisy $0<r\leq r_{max}$.
 
 We now have a finite sequence of pairs $r,c$ for which there might be a solution
-to our problem. In particular, any $(r,c\ell,d\ell^2)$ satisfies consequence 2
-of lemma \ref{lem:pseudo_wall_numerical_tests}, and the positive rank condition.
+$(r,c\ell,d\ell^2)$ to our problem. In particular, any $(r,c\ell,d\ell^2)$
+satisfies consequence 2 of lemma \ref{lem:pseudo_wall_numerical_tests}, and the
+positive rank condition. What remains is to find the $d$ values which satisfy
+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
+$\chern_2^{\beta_{-}}(u)>0$ 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.
 
 
-\end{itemize}
 
-\subsection{Limitations}
+\subsection*{Limitations}
 
 Goals:
 \begin{itemize}
-- 
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