Complete proof for Schmidt Bound

figures/schmidt-arg-diag.pdf_tex

@@ -57,7 +57,7 @@
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     \put(0,0){\includegraphics[width=\unitlength,page=2]{schmidt-arg-diag.pdf}}%
-    \put(0.09550205,0.78995793){\color[rgb]{0,0,0.70980392}\rotatebox{-90.02923919}{\makebox(0,0)[lt]{\lineheight{1.25}\smash{\begin{tabular}[t]{l}\textbf{$\chern_0(u) \leq 0$}\end{tabular}}}}}%
+    \put(0.09550205,0.70529121){\color[rgb]{0,0,0.70980392}\rotatebox{-90.02923919}{\makebox(0,0)[lt]{\lineheight{1.25}\smash{\begin{tabular}[t]{l}\textbf{$\chern_0(u) \leq 0$}\end{tabular}}}}}%
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   \end{picture}%

figures/schmidt-arg-diag.svg

@@ -284,7 +284,7 @@
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tex/bounds-on-semistabilisers.tex

@@ -40,12 +40,12 @@ The Bogomolov form applied to the twisted Chern character is the same as the
 untwisted one.
 
 \noindent
-\begin{minipage}{0.59\linewidth}
+\begin{minipage}{0.57\linewidth}
 	So $0 \leq \Delta(u)$ (condition 2 from Corollary \ref{cor:num_test_prob2})
 	yields:
 	\begin{equation}
 		\label{eqn-bgmlv-on-E}
-		2\chern^\beta_0(u) \chern^\beta_2(u) \leq \chern^\beta_1(u)^2
+		2\chern_0(u) \chern^{\beta_{-}}_2(u) \leq \chern^{\beta_{-}}_1(u)^2
 	\end{equation}
 
 	\noindent
@@ -58,9 +58,10 @@ untwisted one.
 	\end{equation}
 
 	\noindent
-	The restrictions on $\chern^{\beta_-}_0(u)$ and $\chern^{\beta_-}_2(v)$
-	is best seen with the following graph:
-	% TODO: hyperbola restriction graph (shaded)
+	The induced restrictions on possible pairs $\chern^{\beta_-}_0(u)$ and
+	$\chern^{\beta_-}_2(u)$,
+	as well as conditions 1 and 6 from Corollary \ref{cor:num_test_prob2}
+	are illustrated here on the right, with the invalid regions shaded.
 \end{minipage}
 \hfill
 \begin{minipage}{0.39\linewidth}
@@ -71,40 +72,40 @@ untwisted one.
 	\subimport{../figures/}{schmidt-arg-diag.pdf_tex}
 	}
 	\end{center}
+	\vspace{3pt}
 \end{minipage}
 
-This is where the rationality of $\beta_{-}$ comes in. If
-$\beta_{-} = \frac{a_v}{n}$ for some $a_v,n \in \ZZ$.
-Then $\chern^{\beta_-}_2(E) \in \frac{1}{\lcm(m,2n^2)}\ZZ$.
-In particular, since $\chern_2^{\beta_-}(E) > 0$ (by using 	$P=(\beta_-,0)$ in
-lemma \ref{lem:pseudo_wall_numerical_tests}) we must also have
-$\chern^{\beta_-}_2(E) \geq \frac{1}{\lcm(m,2n^2)}$, which then in turn gives a
-bound for the rank of $E$:
+Currently, the unshaded region in the diagram above, corresponding to possible
+values for $\chern_0(u)$ and $\chern^{\beta_{-}}_2(u)$ that satisfy the
+currently considered restrictions, is unbounded.
+This is where the rationality of $\beta_{-}$ comes in.
+If $\beta_{-} = \frac{a_v}{n}$ for some $a_v,n \in \ZZ$,
+then $\chern^{\beta_-}_2(u) \in \frac{1}{\lcm(m,2n^2)}\ZZ$.
+In particular, since $\chern_2^{\beta_-}(u) > 0$ we must also have
+$\chern^{\beta_-}_2(u) \geq \frac{1}{\lcm(m,2n^2)}$, which then in turn gives a
+bound for the rank of $u$:
 
 \begin{align}
-	\chern_0(E) &= \chern^{\beta_-}_0(E) \nonumber \\
-	&\leq \frac{\lcm(m,2n^2) \chern^{\beta_-}_1(E)^2}{2} \nonumber \\
-	&= \frac{mn^2 \chern^{\beta_-}_1(F)^2}{\gcd(m,2n^2)}
+	\chern_0(u)
+	&\leq \frac{\chern^{\beta_-}_1(u)^2}{2\chern^{\beta_{-}}_2(u)} \nonumber \\
+	&\leq \frac{\lcm(m,2n^2) \chern^{\beta_-}_1(u)^2}{2} \nonumber \\
+	&= \frac{mn^2 \chern^{\beta_-}_1(u)^2}{\gcd(m,2n^2)}
 	\label{proof:first-bound-on-r}
 \end{align}
-
-In fact Equation \ref{eqn-tilt-cat-cond} can be tightened slightly:
-we cannot have equality $\chern^{\beta_{-}}_1(E) = \chern^{\beta_{-}}_1(F)$
-otherwise we would have $\chern^{\beta_{-}}_1(G)=0$ for the quotient $G$.
-This would imply $\mu(G)=\beta_{-}$, but since $\Theta_G$ is bounded above in the
-upper-half plane by the assymptotes crossing the $\beta$-axis at $\pm45^\circ$ at
-$\beta=\beta_{-}(v)$. So $\Theta_G$ cannot intersect $\Theta_v$ at any point
-with $\alpha > 0$, so there is no point with $\nu(E)=\nu(F)=\nu(G)=0$, which would
-have to hold at the top of the pseudo-wall if it were to exist.
-Therefore we must have a strict inequality
-$\chern^{\beta_{-}}_1(E) < \chern^{\beta_{-}}_1(F)$,
-and since these are elements of $\frac{1}{n}\ZZ$, we can also conclude:
+\noindent
+Which we can then immediately bound using Equation \ref{eqn-tilt-cat-cond}.
+Alternatively, given that
+$\chern_1^{\beta_{-}}(u)$, $\chern_1^{\beta_{-}}(v)\in\frac{1}{n}\ZZ$,
+we can tighten this bound on $\chern_1^{\beta_{-}}(u)$ given by that Equation to:
+\[
+	n\chern^{\beta_{-}}_1(u) \leq n\chern^{\beta_{-}}_1(v) - 1
+\]
+allowing us to bound the expression in Equation \ref{proof:first-bound-on-r} to
+the following:
 \[
-	n\chern^{\beta_{-}}_1(E) \leq n\chern^{\beta_{-}}_1(F) - 1
+	\chern_0(u)
+	&\leq \frac{m\left(n \chern^{\beta_-}_1(v) - 1\right)^2}{\gcd(m,2n^2)}
 \]
-which then tightens the upper bound found for $\chern_0(E)$
-in Equation \ref{proof:first-bound-on-r}
-to the bound in the statement of the Lemma.
 
 \end{proof}