Iron out work up to dfn/lemma of e_{v,q}

tex/bounds-on-semistabilisers.tex

@@ -141,7 +141,7 @@ $\chern_1^{\sage{extravagant.betaminus}}(F) = \sage{extravagant.twisted.ch[1]}$.
 Using the above Theorem \ref{thm:loose-bound-on-r}, we get that the ranks of
 tilt semistabilisers for $v$ are bounded above by $\sage{extravagant.loose_bound}$.
 However, when computing all tilt semistabilisers for $v$ on $\PP^2$, the maximum
-rank that appears turns out to be $\sage{extravagant.actual_rmax}$.
+rank that appears turns out to be $\sage{round(extravagant.actual_rmax, 1)}$.
 \end{example}
 
 
@@ -811,49 +811,54 @@ from plots_and_expressions import q_sol, bgmlv_v, psi
 	line bundle $L$ with $c_1(L)$ generating $\neronseveri(X)$ and
 	$m\coloneqq\ell^2$.
 	Let $v$ be a fixed Chern character on this surface and
-	$R\coloneqq\chern_0(v) \leq \frac{1}{2}\lcm(m,2n^2){\chern_1^{\beta_{-}(v)}(v)}^2$.
-	Then the ranks of the pseudo-semistabilisers for $v$,
+	$R\coloneqq\chern_0(v)$.
+	Then the ranks of the pseudo-semistabilisers $u$ of $v$,
 	which are solutions to problem \ref{problem:problem-statement-2},
-	are bounded above by the following expression.
-
-	\begin{equation*}
-		\sage{main_theorem1.corollary_r_bound}
-	\end{equation*}
+	are bounded above as follows.
 
-	\noindent
-	If $R > \frac{1}{2}\lcm(m,2n^2){\chern_1^{\beta_{-}(v)}(v)}^2$,
-	then these ranks of pseudo-semistabilisers can instead bounded above by the
-	following.
-	\begin{equation*}
-		\frac{\Delta(v)\lcm(m,2n^2)}{2m}
-	\end{equation*}
+	\begin{align*}
+		r &\leq \sage{main_theorem1.corollary_r_bound}
+			&\text{if } R < \frac{\Delta(v)\lcm(m,2n^2)}{2m}
+		\\
+		r &\leq \frac{\Delta(v)\lcm(m,2n^2)}{2m}
+			&\text{if } R \geq \frac{\Delta(v)\lcm(m,2n^2)}{2m}
+	\end{align*}
 \end{corollary}
 
 \begin{proof}
 The ranks of the pseudo-semistabilisers for $v$ are bounded above by the
-maximum over $q\in [0, \chern_1^{\beta}(F)]$ of the expression in Theorem
+maximum over $q\in [0, \chern_1^{\beta_{-}}(v)]$ of the expression in Theorem
 \ref{thm:rmax_with_uniform_eps}.
-Noticing that the expression is a maximum of two quadratic functions in $q$:
+Noticing that the expression is a maximum of two quadratic functions in $q$
+($\beta_0=\beta_{-}(v)$ in this context):
 \begin{equation*}
 	f_1(q)\coloneqq\sage{main_theorem1.r_upper_bound1} \qquad
 	f_2(q)\coloneqq\sage{main_theorem1.r_upper_bound2}
 \end{equation*}
-These have their minimums at $q=0$ and $q=\chern_1^{\beta}(v)$ respectively.
-It suffices to find their intersection in
-$q\in [0, \chern_1^{\beta}(v)]$, if it exists,
-and evaluating on of the $f_i$ there.
-The intersection exists, provided that
-$f_1(\chern_1^{\beta}(v)) \geq f_2(\chern_1^{\beta}(v))=R$,
-or equivalently,
-$R \leq \frac{1}{2}\lcm(m,2n^2){\chern_1^{\beta}(F)}^2$.
-If this is not the case, then
-$\min(f_1, f_2)$ reaches its maximum over the domain
-$[0, \chern_1^{\beta_{-}}(v)]$
-at $q=\chern_1^{\beta_{-}}(v)$, and so is
-bounded by $f_1(\chern_1^{\beta_{-}(v)})$.
-
-In the case where
-$R \leq \frac{1}{2}\lcm(m,2n^2){\chern_1^{\beta}(F)}^2$,
+These have their minimums at $q=0$ and $q=\chern_1^{\beta_{-}}(v)$ respectively,
+with values 0 and $R>0$ respectively.
+So provided that
+$f_2\left(\chern^{\beta_{-}}_1(v)\right) < f_1\left(\chern^{\beta_{-}}_1(v)\right)$,
+the maximum is achieved at their intersection.
+Otherwise, the maximum is achieved at
+$\chern^{\beta_{-}}_1(v)$.
+So we can say that
+
+\begin{align*}
+	r &\leq
+		f_{1}(q_{\mathrm{max}})
+		&\text{if } f_2\left(\chern^{\beta_{-}}_1(v)\right) <
+		f_1\left(\chern^{\beta_{-}}_1(v)\right)
+ \\ &&
+		\text{where $q_{\mathrm{max}}$ is the $q$-value where the $f_i$ intersect}
+	\\
+	r &\leq f_1\left(\chern^{\beta_{-}}(v)\right)
+		&\text{if } f_2\left(\chern^{\beta_{-}}_1(v)\right) \geq
+		f_1\left(\chern^{\beta_{-}}_1(v)\right)
+\end{align*}
+
+\noindent
+In the first case,
 solving for $f_1(q)=f_2(q)$ yields
 \begin{equation*}
 	q=\sage{q_sol.expand()}
@@ -862,8 +867,10 @@ And evaluating $f_1$ at this $q$-value gives:
 \begin{equation*}
 	\sage{main_theorem1.corollary_intermediate}
 \end{equation*}
-Finally, noting that $\Delta(v)=(\chern_1^{\beta_{-}(v)}(v))^2\ell^2$, we get the bound as
-stated in the corollary.
+
+\noindent
+Finally, noting that $\Delta(v)=\left(\chern_1^{\beta_{-}(v)}(v)\right)^2\ell^2$,
+we get the bounds as stated in the statement of the Corollary.
 
 \end{proof}
 
@@ -876,7 +883,8 @@ giving $n=\sage{recurring.n}$.
 
 Using the above corollary \ref{cor:direct_rmax_with_uniform_eps}, we get that
 the ranks of tilt semistabilisers for $v$ are bounded above by
-$\sage{recurring.corrolary_bound} \approx  \sage{float(recurring.corrolary_bound)}$,
+$\sage{recurring.corrolary_bound} \approx
+\sage{round(float(recurring.corrolary_bound), 1)}$,
 which is much closer to real maximum 25 than the original bound 144.
 \end{example}
 
@@ -889,7 +897,8 @@ giving $n=\sage{extravagant.n}$.
 
 Using the above corollary \ref{cor:direct_rmax_with_uniform_eps}, we get that
 the ranks of tilt semistabilisers for $v$ are bounded above by
-$\sage{extravagant.corrolary_bound} \approx  \sage{float(extravagant.corrolary_bound)}$,
+$\sage{extravagant.corrolary_bound} \approx
+\sage{round(float(extravagant.corrolary_bound), 1)}$,
 which is much closer to real maximum $\sage{extravagant.actual_rmax}$ than the
 original bound 215296.
 \end{example}
@@ -913,28 +922,7 @@ $q$, then iterate through values of $r$ within the bounds (dependent on $q$),
 which would then determine $c$, and then find the corresponding possible values
 for $d$.
 
-
-Firstly, we only need to consider $r$-values for which $c\coloneqq\chern_1(E)$ is
-integral:
-
-\begin{sagesilent}
-from plots_and_expressions import c_in_terms_of_q	
-\end{sagesilent}
-\begin{equation*}
-	c =
-	\sage{c_in_terms_of_q.subs([q_value_expr,beta_value_expr])}
-	\in \ZZ
-\end{equation*}
-
-\noindent
-That is, $r \equiv -\aa^{-1}\bb$ mod $n$ ($\aa$ is coprime to
-$n$, and so invertible mod $n$).
-
-
-\noindent
-Let $\aa^{'}$ be an integer representative of $\aa^{-1}$ in $\ZZ/n\ZZ$.
-
-Next, we seek to find a larger $\epsilon$ to use in place of $\epsilon_F$ in the
+Next, we seek to find a larger $\epsilon$ to use in place of $\epsilon_v$ in the
 proof of Theorem \ref{thm:rmax_with_uniform_eps}:
 
 \begin{lemmadfn}[