Tweak up to the main corollary with global bound for problem 2 ranks

tex/bounds-on-semistabilisers.tex

@@ -547,7 +547,7 @@ bounds do not share the same assymptote as the lower bound
 	d &\leq
 	\sage{problem1.bgmlv3_d_upperbound_terms.linear}
 	+ \sage{problem1.bgmlv3_d_upperbound_terms.const}
-	+ \sage{problem1.bgmlv3_d_upperbound_terms.hyperbolic}
+	 \sage{problem1.bgmlv3_d_upperbound_terms.hyperbolic}
 	&\text{when }r>R
 	\label{eqn:prob1:bgmlv3}
 \end{align}
@@ -678,33 +678,33 @@ following Lemma \ref{lem:prob1:convenient_r_bound}.
 \ref{problem:problem-statement-2}}
 \label{subsec:bounds-on-semistab-rank-prob-2}
 
-Now, the inequalities from the above subsubsection
+Now, the inequalities from the above Subsubsection
 \ref{subsubsect:all-bounds-on-d-prob2} will be used to find, for
 each given $q=\chern^{\beta}_1(E)$, how large $r$ needs to be in order to leave
-no possible solutions for $d$. At that point, there are no solutions
-$u=(r,c\ell,d\ell^2)$ to problem \ref{problem:problem-statement-2}.
-The strategy here is similar to what was shown in Theorem
-\ref{thm:loose-bound-on-r}.
+no possible solutions for $d$ in the context of Problem
+\ref{problem:problem-statement-2}.
+At that point, there are no solutions
+$u=(r,c\ell,d\ell^2)$ to the Problem.
+
+In the context of Problem \ref{problem:problem-statement-2}, $\beta_{-}(v)$ is
+assumed to be rational.
+Considering Corollary \ref{cor:rational-beta:fixed-q-semistabs-criterion},
+for any solution $u$, we have $q\coloneqq\chern^{\beta_{-}(v)}(u) = \frac{b_q}{n}$
+where $\beta_{-}(v) = \frac{a_v}{n}$ in lowest terms and $b_q$ is an integer
+between 1 and $n\chern_1^{\beta_0}(v) - 1$ (inclusive),
+and $a_v r \equiv -b_q \pmod{n}$.
+The Corollary then gives a lower bound for $r$, and states that any $u$ of the
+form from Equation \ref{eqn:u-coords} satisfying these conditions so far, is a
+solution to Problem \ref{problem:problem-statement-2} if and only if it
+satisfies the conditions
+$\chern^{\beta_{-}(v)}(u)>0$, $\Delta(u) \geq 0$, and $\Delta(v-u) \geq 0$.
 
 
 \renewcommand{\aa}{{a_v}}
 \newcommand{\bb}{{b_q}}
-Suppose $\beta = \frac{\aa}{n}$ for some coprime $n \in \NN,\aa \in \ZZ$.
-Then fix a value of $q$:
-\begin{equation}
-	q\coloneqq \chern_1^{\beta}(E)
-	  =\frac{\bb}{n}
-	\in
-	\frac{1}{n} \ZZ
-	\cap [0, \chern_1^{\beta}(F)]
-\end{equation}
-as noted at the beginning of this section \ref{sec:refinement} so that we are
-considering $u$ which satisfy \ref{item:chern1bound:lem:num_test_prob2}
-in corollary \ref{cor:num_test_prob2}.
-
-Substituting the current values of $q$ and $\beta$ into the condition for the
-radius of the pseudo-wall being positive
-(eqn \ref{eqn:radiuscond_d_bound_betamin}) we get:
+Substituting more specialised values of $q$ and $\beta_0=\beta_{-}(v)$ into the condition
+$\chern^{\beta_0}(u) > 0$
+(Equation \ref{eqn:radiuscond_d_bound_betamin}) we get:
 
 \begin{sagesilent}
 from plots_and_expressions import \
@@ -723,16 +723,25 @@ beta_value_expr
 	\frac{1}{2n^2}\ZZ
 \end{equation}
 
+\noindent
+This fact will be leveraged to give tighter lower bounds in a similar way to the
+proof of Theorem
+\ref{thm:loose-bound-on-r}.
+
 
 \begin{sagesilent}
 from plots_and_expressions import main_theorem1
 \end{sagesilent}
 \begin{theorem}[Bound on $r$ \#1]
 \label{thm:rmax_with_uniform_eps}
-	Let $v = (R,C\ell,D\ell^2)$ be a fixed Chern character. Then the ranks of the
-	pseudo-semistabilisers for $v$,
-	which are solutions to problem \ref{problem:problem-statement-2},
-	with $\chern_1^\beta = q$
+	Let $X$ be a smooth projective surface with Picard rank 1 and choice of ample
+	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 with positive rank
+	(or rank 0 and $c_1(v)>0$), and $\Delta(v)\geq 0$.
+	Then the ranks of the pseudo-semistabilisers $u$ for $v$,
+	which are solutions to Problem \ref{problem:problem-statement-2},
+	with $\chern_1^{\beta_{-}(v)}(u) = q$
 	are bounded above by the following expression.
 
 	\begin{align*}
@@ -742,18 +751,13 @@ from plots_and_expressions import main_theorem1
 			\sage{main_theorem1.r_upper_bound2}
 		\right)
 	\end{align*}
-
-	Taking the maximum of this expression over
-	$q \in [0, \chern_1^{\beta}(F)]\cap \frac{1}{n}\ZZ$
-	would give an upper bound for the ranks of all solutions to problem
-	\ref{problem:problem-statement-2}.
+	\noindent
+	where $R\coloneqq \chern_0(v)$.
 \end{theorem}
 
 \begin{proof}
-
-\noindent
 Both $d$ and the lower bound in
-(eqn \ref{eqn:positive_rad_condition_in_terms_of_q_beta})
+(Equation \ref{eqn:positive_rad_condition_in_terms_of_q_beta})
 are elements of $\frac{1}{\lcm(m,2n^2)}\ZZ$.
 So, if any of the two upper bounds on $d$ come to within
 $\frac{1}{\lcm(m,2n^2)}$ of this lower bound, then there are no solutions for
@@ -803,8 +807,11 @@ from plots_and_expressions import q_sol, bgmlv_v, psi
 
 \begin{corollary}[Bound on $r$ \#2]
 \label{cor:direct_rmax_with_uniform_eps}
-	Let $v$ be a fixed Chern character and
-	$R\coloneqq\chern_0(v) \leq \frac{1}{2}\lcm(m,2n^2){\chern_1^{\beta}(F)}^2$.
+	Let $X$ be a smooth projective surface with Picard rank 1 and choice of ample
+	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 $v$,
 	which are solutions to problem \ref{problem:problem-statement-2},
 	are bounded above by the following expression.
@@ -812,6 +819,14 @@ from plots_and_expressions import q_sol, bgmlv_v, psi
 	\begin{equation*}
 		\sage{main_theorem1.corollary_r_bound}
 	\end{equation*}
+
+	\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*}
 \end{corollary}
 
 \begin{proof}
@@ -823,15 +838,23 @@ Noticing that the expression is a maximum of two quadratic functions in $q$:
 	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 $q=\chern_1^{\beta}(F)$ respectively.
+These have their minimums at $q=0$ and $q=\chern_1^{\beta}(v)$ respectively.
 It suffices to find their intersection in
-$q\in [0, \chern_1^{\beta}(F)]$, if it exists,
+$q\in [0, \chern_1^{\beta}(v)]$, if it exists,
 and evaluating on of the $f_i$ there.
 The intersection exists, provided that
-$f_1(\chern_1^{\beta}(F)) \geq f_2(\chern_1^{\beta}(F))=R$,
+$f_1(\chern_1^{\beta}(v)) \geq f_2(\chern_1^{\beta}(v))=R$,
 or equivalently,
 $R \leq \frac{1}{2}\lcm(m,2n^2){\chern_1^{\beta}(F)}^2$.
-Solving for $f_1(q)=f_2(q)$ yields
+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
+$R \leq \frac{1}{2}\lcm(m,2n^2){\chern_1^{\beta}(F)}^2$,
+solving for $f_1(q)=f_2(q)$ yields
 \begin{equation*}
 	q=\sage{q_sol.expand()}
 \end{equation*}
@@ -839,7 +862,7 @@ And evaluating $f_1$ at this $q$-value gives:
 \begin{equation*}
 	\sage{main_theorem1.corollary_intermediate}
 \end{equation*}
-Finally, noting that $\Delta(v)=\psi^2\ell^2$, we get the bound as
+Finally, noting that $\Delta(v)=(\chern_1^{\beta_{-}(v)}(v))^2\ell^2$, we get the bound as
 stated in the corollary.
 
 \end{proof}

tex/setting-and-problems.tex

@@ -389,9 +389,11 @@ Fix a Chern character $v$ with non-negative rank (and $\chern_1(v)>0$ if rank 0)
 $\Delta(v) \geq 0$, and $\beta_{-}(v) \in \QQ$.
 The goal is to find all pseudo-semistabilisers $u$ which give circular
 pseudo-walls on the left side of $V_v$.
+Where $u$ destabilises $v$ going `down' $\Theta_v^{-}$ (in the same sense as in
+Problem \ref{problem:problem-statement-1}.
 \end{problem}
 
-This is a specialization of problem (\ref{problem:problem-statement-1})
+This is a specialization of Problem \ref{problem:problem-statement-1}
 with the choice $P=(\beta_{-},0)$, the point where $\Theta_v^-$ meets the
 $\beta$-axis.
 This is because all circular walls left of $V_v$ intersect $\Theta_v^-$ (once).
@@ -403,7 +405,7 @@ where an algorithm is produced to find all solutions.
 
 This description still holds for the case of rank 0 case if we consider $V_v$ to
 be infinitely far to the right
-(see section \ref{subsubsect:rank-zero-case-charact-curves}).
+(see Section \ref{subsubsect:rank-zero-case-charact-curves}).
 Note also that the condition on $\beta_-(v)$ always holds for $v$ rank 0.
 
 \subsection{Numerical Formulations of the Problems}