From 937f8b936fc605e80bf064611c960533448f6213 Mon Sep 17 00:00:00 2001 From: Luke Naylor <l.naylor@sms.ed.ac.uk> Date: Sun, 23 Jul 2023 19:34:15 +0100 Subject: [PATCH] Complete proof and misc in problems sect --- main.tex | 31 ++++++++++++++++++------------- 1 file changed, 18 insertions(+), 13 deletions(-) diff --git a/main.tex b/main.tex index abe5699..a351601 100644 --- a/main.tex +++ b/main.tex @@ -654,14 +654,13 @@ $v-u$ for each solution $u$ of the problem. Fix a Chern character $v$ with positive rank, $\Delta(v) \geq 0$, and $\beta_{-}(v) \in \QQ$. -The goal is to find all solutions $u$ -to problem \ref{problem:problem-statement-1} with the choice -$P=(\beta_{-},0)$. +The goal is to find all pseudo-semistabilizers $u$ which give circular +pseudo-walls on the left side of $V_v$. \end{problem} This is a specialization of problem (\ref{problem:problem-statement-1}) -which will give all circular pseudo-walls left of $V_v$. -This is because all circular walls left of $V_v$ intersect $\Theta_v^-$. +with the choice $P=(\beta_{-},0)$. +This is because all circular walls left of $V_v$ intersect $\Theta_v^-$ (once). The $\beta_{-}(v) \in \QQ$ condition is to ensure that there are finitely many solutions. As mentioned in the introduction (\ref{sec:intro}), this is known, however this will also be proved again in passing in this article. @@ -680,17 +679,22 @@ problem with the help of lemma \ref{lem:pseudo_wall_numerical_tests}. Are precisely given by integers $r,c,d$ satisfying the following conditions: \begin{enumerate} \item $r > 0$ - \item $\Delta(u) \geq 0$ - \item $\Delta(v-u) \geq 0$ + \item $\Delta(u) \geq 0$ + \item $\Delta(v-u) \geq 0$ \item $\beta(P)<\mu(u)=\frac{c}{r}<\mu(v)$ - \item $\chern_1^{\beta(P)}(v-u)\geq0$ + \item $\chern_1^{\beta(P)}(u)\leq\chern_1^{\beta(P)}(v)$ \item $\chern_2^{P}(u)>0$ \end{enumerate} \end{lemma} \begin{proof} - % TODO complete - Use main lemma \ref{lem:pseudo_wall_numerical_tests} TODO + Consider the context of $v$ being a Chern character with positive rank and + $\Delta \geq 0$, and $u$ being a Chern character with $\Delta(u) \geq 0$. + Lemma \ref{lem:pseudo_wall_numerical_tests} gives that the remaining + conditions for $u$ being a solution to problem + \ref{problem:problem-statement-1} are precisely equivalent to the + remaining conditions in this lemma. + \end{proof} \begin{corrolary}[Numerical Tests for All `left' Pseudo-walls] @@ -701,16 +705,17 @@ problem with the help of lemma \ref{lem:pseudo_wall_numerical_tests}. Are precisely given by integers $r,c,d$ satisfying the following conditions: \begin{enumerate} \item $r > 0$ - \item $\Delta(u) \geq 0$ - \item $\Delta(v-u) \geq 0$ + \item $\Delta(u) \geq 0$ + \item $\Delta(v-u) \geq 0$ \item $\beta(P)<\mu(u)=\frac{c}{r}<\mu(v)$ - \item $\chern_1^{\beta_{-}}(v-u)\geq0$ + \item $\chern_1^{\beta(P)}(u)\leq\chern_1^{\beta(P)}(v)$ \item $\chern_2^{\beta_{-}}(u)>0$ \end{enumerate} \end{corrolary} \begin{proof} This is a specialization of the previous lemma, using $P=(\beta_{-},0)$. + \end{proof} -- GitLab