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Commit 8ba0438a authored by Luke Naylor's avatar Luke Naylor
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Extend numerical formulations to rank 0

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...@@ -506,7 +506,8 @@ semistabilizing sequence. ...@@ -506,7 +506,8 @@ semistabilizing sequence.
\begin{lemma}[Numerical tests for left-wall pseudo-semistabilizers] \begin{lemma}[Numerical tests for left-wall pseudo-semistabilizers]
\label{lem:pseudo_wall_numerical_tests} \label{lem:pseudo_wall_numerical_tests}
Let $v$ and $u$ be Chern characters with $\Delta(v), Let $v$ and $u$ be Chern characters with $\Delta(v),
\Delta(u)\geq 0$, and $v$ has non-negative rank. Let $P$ be a point on $\Theta_v^-$. \Delta(u)\geq 0$, and $v$ has non-negative rank (and $\chern_1(v)>0$ if rank 0).
Let $P$ be a point on $\Theta_v^-$.
\noindent \noindent
The following conditions: The following conditions:
...@@ -635,7 +636,8 @@ are trying to solve for. ...@@ -635,7 +636,8 @@ are trying to solve for.
\begin{problem}[sufficiently large `left' pseudo-walls] \begin{problem}[sufficiently large `left' pseudo-walls]
\label{problem:problem-statement-1} \label{problem:problem-statement-1}
Fix a Chern character $v$ with non-negative rank, and $\Delta(v) \geq 0$. Fix a Chern character $v$ with non-negative rank (and $\chern_1(v)>0$ if rank 0),
and $\Delta(v) \geq 0$.
The goal is to find all pseudo-semistabilizers $u$ The goal is to find all pseudo-semistabilizers $u$
which give circular pseudo-walls containing some fixed point which give circular pseudo-walls containing some fixed point
$P\in\Theta_v^-$. $P\in\Theta_v^-$.
...@@ -670,8 +672,8 @@ $v-u$ for each solution $u$ of the problem. ...@@ -670,8 +672,8 @@ $v-u$ for each solution $u$ of the problem.
\begin{problem}[all `left' pseudo-walls] \begin{problem}[all `left' pseudo-walls]
\label{problem:problem-statement-2} \label{problem:problem-statement-2}
Fix a Chern character $v$ with non-negative rank, $\Delta(v) \geq 0$, Fix a Chern character $v$ with non-negative rank (and $\chern_1(v)>0$ if rank 0),
and $\beta_{-}(v) \in \QQ$. $\Delta(v) \geq 0$, and $\beta_{-}(v) \in \QQ$.
The goal is to find all pseudo-semistabilizers $u$ which give circular The goal is to find all pseudo-semistabilizers $u$ which give circular
pseudo-walls on the left side of $V_v$. pseudo-walls on the left side of $V_v$.
\end{problem} \end{problem}
...@@ -697,7 +699,9 @@ problem with the help of lemma \ref{lem:pseudo_wall_numerical_tests}. ...@@ -697,7 +699,9 @@ problem with the help of lemma \ref{lem:pseudo_wall_numerical_tests}.
\begin{lemma}[Numerical Tests for Sufficiently Large `left' Pseudo-walls] \begin{lemma}[Numerical Tests for Sufficiently Large `left' Pseudo-walls]
\label{lem:num_test_prob1} \label{lem:num_test_prob1}
Given a Chern character $v$ with positive rank and $\Delta(v) \geq 0$, Given a Chern character $v$ with non-negative rank
(with $\chern_1(v)>0$ if rank 0)
and $\Delta(v) \geq 0$,
and a choice of point $P$ on $\Theta_v^-$. and a choice of point $P$ on $\Theta_v^-$.
Solutions $u=(r,c\ell,\frac{e}{\lcm(m,2)}\ell^2)$ Solutions $u=(r,c\ell,\frac{e}{\lcm(m,2)}\ell^2)$
to problem \ref{problem:problem-statement-1}. to problem \ref{problem:problem-statement-1}.
...@@ -718,7 +722,9 @@ problem with the help of lemma \ref{lem:pseudo_wall_numerical_tests}. ...@@ -718,7 +722,9 @@ problem with the help of lemma \ref{lem:pseudo_wall_numerical_tests}.
\end{lemma} \end{lemma}
\begin{proof} \begin{proof}
Consider the context of $v$ being a Chern character with positive rank and Consider the context of $v$ being a Chern character with non-negative rank
(and $\chern_1(v)>0$ if rank 0)
and
$\Delta \geq 0$, and $u$ being a Chern character with $\Delta(u) \geq 0$. $\Delta \geq 0$, and $u$ being a Chern character with $\Delta(u) \geq 0$.
Lemma \ref{lem:pseudo_wall_numerical_tests} gives that the remaining Lemma \ref{lem:pseudo_wall_numerical_tests} gives that the remaining
conditions for $u$ being a solution to problem conditions for $u$ being a solution to problem
......
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