From f31a146377aefb2368b25bff38cdbe211b963acb Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Tue, 2 Jul 2024 19:18:33 +0100
Subject: [PATCH] Complete statement of main lemma about fixing q value
---
tex/bounds-on-semistabilisers.tex | 58 ++++++++++++++++++++++++++++++-
1 file changed, 57 insertions(+), 1 deletion(-)
diff --git a/tex/bounds-on-semistabilisers.tex b/tex/bounds-on-semistabilisers.tex
index 7e40dec..d3156e5 100644
--- a/tex/bounds-on-semistabilisers.tex
+++ b/tex/bounds-on-semistabilisers.tex
@@ -198,12 +198,68 @@ from plots_and_expressions import c_in_terms_of_q
\qquad 0 \leq q \coloneqq \chern_1^{\beta}(u) \leq \chern_1^{\beta}(v)
\end{equation}
-Furthermore, if $\beta$ is rational, $\chern_1(u) \in \ZZ$ so we only need to consider
+Furthermore, $\chern_1(u) \in \ZZ$, so if $\beta$ is rational we only need to consider
$q \in \frac{1}{n} \ZZ \cap [0, \chern_1^{\beta}(F)]$,
where $n$ is the denominator of $\beta$.
For the next subsections, we consider $q$ to be fixed with one of these values,
and we shall be varying $\chern_0(E) = r$ to see when certain inequalities fail.
+\begin{lemma}
+ Consider the Problem \ref{problem:problem-statement-1} (or \ref{problem:problem-statement-2}),
+ and $\beta_{0}\coloneqq \beta(P)$ (or $\beta_{-}(v)$ resp.).
+
+ \noindent
+ If $u$ is a solution to the Problem then $u$ satisfies:
+ \begin{align}
+ q\coloneqq \chern^{\beta_0}_1(u) &\in \left( 0, \chern_1^{\beta_0}(v) \right)
+ \label{lem:eqn:cond-for-fixed-q}
+ \\
+ \chern_0(u) &> \frac{q}{\mu(v) - \beta_0}
+ \nonumber
+ \end{align}
+
+ \noindent
+ Conversely, any $u = (r,c\ell,d\ell^2)$
+ with $r,c\in \ZZ$ and $d\in \frac{1}{\lcm(m,2)}\ZZ$
+ satisfying the above Equations \ref{lem:eqn:cond-for-fixed-q}
+ is a solution to the Problem if and only if the following are satisfied:
+ \begin{multicols}{3}
+ \begin{itemize}
+ \item $\Delta(u) \geq 0$
+ \item $\Delta(v-u) \geq 0$
+ \item $\chern^P_2(u) \geq 0$
+ \end{itemize}
+ \end{multicols}
+
+ \noindent
+ Furthermore, suppose $\beta_0$ is rational, and written $\beta_0=\frac{a_v}{n}$ for
+ some coprime integers $a_v$, $n$ with $n>0$.
+ Then any solution $u$ satisfies:
+ \begin{align*}
+ \chern^{\beta_0}_1(u)
+ &= \frac{b_q}{n}
+ &\text{for some }
+ b_q \in \left\{ 1, 2, \ldots, n\chern_1^{\beta_0}(v) - 1 \right\}
+ \\
+ a_v r &\equiv -b_q \pmod{n}
+ \end{align*}
+ And any $u = (r,c\ell,d\ell^2)$
+ with $r,c\in \ZZ$ and $d\in \frac{1}{\lcm(m,2)}\ZZ$
+ satisfying these equations is a solution to the Problem if and only if, again,
+ the following are satisfied:
+ \begin{multicols}{3}
+ \begin{itemize}
+ \item $\Delta(u) \geq 0$
+ \item $\Delta(v-u) \geq 0$
+ \item $\chern^P_2(u) \geq 0$
+ \end{itemize}
+ \end{multicols}
+
+\end{lemma}
+
+\begin{proof}
+ proof
+\end{proof}
\subsection{Bounds on \texorpdfstring{`$d$'}{d}-values for Solutions of Problems}
--
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