From 00baee18b8126da0f94b3b348c2d0aa906d97207 Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Tue, 2 Jul 2024 18:27:55 +0100
Subject: [PATCH] Complete proof for Schmidt Bound
---
figures/schmidt-arg-diag.pdf | Bin 4116 -> 4116 bytes
figures/schmidt-arg-diag.pdf_tex | 2 +-
figures/schmidt-arg-diag.svg | 2 +-
tex/bounds-on-semistabilisers.tex | 63 +++++++++++++++---------------
4 files changed, 34 insertions(+), 33 deletions(-)
diff --git a/figures/schmidt-arg-diag.pdf b/figures/schmidt-arg-diag.pdf
index bc33ced3f77b5eb80beacc01db2c9bad727c0f8b..1b67aac2046843e094625420ef73385fa7845d4f 100644
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diff --git a/figures/schmidt-arg-diag.pdf_tex b/figures/schmidt-arg-diag.pdf_tex
index 7f3ed3c..7e33843 100644
--- a/figures/schmidt-arg-diag.pdf_tex
+++ b/figures/schmidt-arg-diag.pdf_tex
@@ -57,7 +57,7 @@
\put(0.72051844,0.18401749){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\lineheight{1.25}\smash{\begin{tabular}[t]{l}\textbf{$\chern_0(u)$}\end{tabular}}}}%
\put(-0.00341324,0.92240317){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\lineheight{1.25}\smash{\begin{tabular}[t]{l}\textbf{$\chern_2^{\beta_{-}}(u)$}\end{tabular}}}}%
\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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\put(0.35268151,0.97647054){\color[rgb]{0.6627451,0.11372549,0}\rotatebox{-44.62594593}{\makebox(0,0)[lt]{\lineheight{1.25}\smash{\begin{tabular}[t]{l}\textbf{$2\chern_0(u)\chern_2^{\beta_{-}}(u) \geq \chern_1^{\beta_{-}}(v)^2$}\end{tabular}}}}}%
\end{picture}%
diff --git a/figures/schmidt-arg-diag.svg b/figures/schmidt-arg-diag.svg
index 4ad411a..7d80f17 100644
--- a/figures/schmidt-arg-diag.svg
+++ b/figures/schmidt-arg-diag.svg
@@ -284,7 +284,7 @@
</g>
<text
xml:space="preserve"
- transform="matrix(-1.3502221e-4,0.2645833,-0.2645833,-1.3502221e-4,46.737196,16.645156)"
+ transform="matrix(-1.3502221e-4,0.2645833,-0.2645833,-1.3502221e-4,46.737196,25.111828)"
id="text7"
style="font-weight:500;font-size:26.6667px;font-family:'FiraMono Nerd Font Mono';-inkscape-font-specification:'FiraMono Nerd Font Mono Medium';text-align:start;white-space:pre;shape-inside:url(#rect7);display:inline;fill:#0000b5;fill-opacity:1;stroke:none;stroke-width:4.71307;stroke-linecap:round;stroke-linejoin:round;stroke-miterlimit:3.2;stroke-dasharray:4.71307, 14.1392;stroke-dashoffset:0;stroke-opacity:1;paint-order:stroke fill markers"><tspan
x="16.546875"
diff --git a/tex/bounds-on-semistabilisers.tex b/tex/bounds-on-semistabilisers.tex
index 13a6ce2..7e40dec 100644
--- a/tex/bounds-on-semistabilisers.tex
+++ b/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}
--
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