Complete statement of main lemma about fixing q value

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}