From e71d0cf7fa289b09045a57f487258a4f870aad5d Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Sun, 11 Aug 2024 00:16:03 +0100
Subject: [PATCH] Implement Jun28 feedback (minus big lemma)
---
tex/setting-and-problems.tex | 34 +++++++++++++++++++++-------------
1 file changed, 21 insertions(+), 13 deletions(-)
diff --git a/tex/setting-and-problems.tex b/tex/setting-and-problems.tex
index b86f6cc..1580ea3 100644
--- a/tex/setting-and-problems.tex
+++ b/tex/setting-and-problems.tex
@@ -39,9 +39,11 @@ affect the results.
be considered but are left out for now as they do not have a great impact on
the finiteness of pseudo-walls.
In the case of a principally polarised abelian surface, the main example in
- this thesis, the Euler characteristic condition is vacuous and the extension
- group condition eliminates possibities with lower rank, and often none at all
- for small values of $\chern_0(v)$.
+ this thesis, the Euler characteristic condition is vacuous.
+ The extension group condition can be shown to only
+ be redundant among the other conditions for sufficiently large rank of $u$
+ (when $\chern_0(u) \geq \chern_0(v)/2$), so does not affect the finiteness
+ of pseudo-semistabilisers.
\end{remark}
Later, Chern characters will be written $(r,c\ell,d\ell^2)$ because operations
@@ -119,7 +121,7 @@ $d \in \frac{1}{\lcm(m,2)}\ZZ$.
\section{Characteristic Curves for Pseudo-semistabilisers}
These characteristic curves introduced in
-subsection \ref{subsec:charact-curves}
+Subsection \ref{subsec:charact-curves}
are convenient tools to think about the
numerical conditions that can be used to test for pseudo-semistabilisers, and
for solutions to the problems
@@ -159,10 +161,12 @@ are equivalent to the following more numerical conditions:
\item $\chern_1^{\beta(P)}(u) < \chern_1^{\beta(P)}(v)$
\item $\Delta(v-u) \geq 0$
\item $\chern_2^{P}(u)>0$
+ \footnote{Here $\chern_2^{P} = \chern_2^{\alpha, \beta}$ where $P=(\alpha, \beta)$.}
\end{enumerate}
\end{lemma}
-\begin{proof}[Proof for $\chern_0(v)>0$ case.]
+\begin{proof}
+First, consider the case where $\chern_0(v)>0$.
Let $u,v$ be Chern characters with
$\Delta(u),\Delta(v) \geq 0$, and $v$ has positive rank.
@@ -186,9 +190,9 @@ ${
(consequence 5).
Notice that $0 < \chern_1^{\beta(P)}(u)$
follows from consequences 1 and 2, this is why it is not included in consequence 5.
-If $u$ were to have 0 rank, it's tilt slope would be decreasing as $\beta$
+If $u$ were to have 0 rank, its tilt slope would be decreasing as $\beta$
increases, contradicting supposition b. So $u$ must have strictly non-zero rank,
-and we can consider it's characteristic curves (or that of $-u$ in case of
+and we can consider its characteristic curves (or that of $-u$ in case of
negative rank).
$\nu_Q(v)=0$, and hence $\nu_Q(u)=0$ too. This means that $\Theta_{\pm u}$ must
intersect $\Theta_v^-$ at $Q$. Considering the shapes of the hyperbolae alone,
@@ -250,14 +254,14 @@ right of $\Theta_v^-$ at $\alpha=\alpha(P)$, but crosses to the left side as
$\alpha \to +\infty$, intersection at some point $Q$ above $P$.
This implies that the characteristic curves for $u$ and $v$ are in the
configuration illustrated in Fig \ref{fig:correct-hyperbol-intersection}.
-We then have $\nu(u)=\nu(v)$ along a circle to the left of $V_u$ reaching it's
+We then have $\nu(u)=\nu(v)$ along a circle to the left of $V_u$ reaching its
apex at $Q$, and encircling $P$. This along with consequence 3 implies that $u$
is a pseudo-semistabiliser at the point on the circle with $\beta=\beta(P)$.
-Therefore, it's also a pseudo-semistabiliser further along the circle at $Q$
+Therefore, it is also a pseudo-semistabiliser further along the circle at $Q$
(supposition a).
Finally, consequence 4 along with $P$ being to the left of $V_u$ implies
$\nu_P(u) > 0$ giving supposition b.
-\end{proof}
+
\begin{sagesilent}
from rank_zero_case_curves import pseudo_semistab_char_curves_rank_zero
\end{sagesilent}
@@ -268,7 +272,8 @@ from rank_zero_case_curves import pseudo_semistab_char_curves_rank_zero
with rank 0 destabilising `downwards'}
\label{fig:hyperbol-intersection-rank-zero}
\end{figure}
-\begin{proof}[Proof for $\chern_0(v)=0$ case.]
+
+Now consider the case where $\chern_0(v)=0$ but $\chern_1(v)>0$.
Let $u,v$ be Chern characters with
$\Delta(u),\Delta(v) \geq 0$, and $\chern_0(v)=0$ but $\chern_1(v)>0$.
So $\Theta_v^-$ is the vertical line at $\beta = \beta_{-}(v)$, and
@@ -406,7 +411,7 @@ 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 specialisation 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).
@@ -449,6 +454,7 @@ problem using Lemma \ref{lem:pseudo_wall_numerical_tests}.
\item $0<\chern_1^{\beta(P)}(u)<\chern_1^{\beta(P)}(v)$
\label{item:chern1bound:lem:num_test_prob1}
\item $\chern_2^{P}(u)>0$
+ \footnote{Here $\chern_2^{P} = \chern_2^{\alpha, \beta}$ where $P=(\alpha, \beta)$.}
\label{item:radiuscond:lem:num_test_prob1}
\end{multicols}
\end{enumerate}
@@ -497,6 +503,8 @@ problem using Lemma \ref{lem:pseudo_wall_numerical_tests}.
\end{theorem}
\begin{proof}
- This is a specialization of the previous Lemma, using $P=(\beta_{-},0)$.
+ This is a specialisation of the previous Theorem
+ \ref{lem:num_test_prob1},
+ using $P=(\beta_{-},0)$.
\end{proof}
--
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