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Commit a500675b authored by Luke Naylor's avatar Luke Naylor
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Extend problem statement to rank 0

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...@@ -372,6 +372,7 @@ degenerate_characteristic_curves ...@@ -372,6 +372,7 @@ degenerate_characteristic_curves
\subsubsection{Rank Zero Case} \subsubsection{Rank Zero Case}
\label{subsubsect:rank-zero-case-charact-curves}
\begin{sagesilent} \begin{sagesilent}
from rank_zero_case import Theta_v_plot from rank_zero_case import Theta_v_plot
...@@ -634,7 +635,7 @@ are trying to solve for. ...@@ -634,7 +635,7 @@ 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 positive rank, and $\Delta(v) \geq 0$. Fix a Chern character $v$ with non-negative rank, 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
$P\in\Theta_v^-$. $P\in\Theta_v^-$.
...@@ -669,19 +670,25 @@ $v-u$ for each solution $u$ of the problem. ...@@ -669,19 +670,25 @@ $v-u$ 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 positive rank, $\Delta(v) \geq 0$, Fix a Chern character $v$ with non-negative rank, $\Delta(v) \geq 0$,
and $\beta_{-}(v) \in \QQ$. 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}
This is a specialization of problem (\ref{problem:problem-statement-1}) This is a specialization of problem (\ref{problem:problem-statement-1})
with the choice $P=(\beta_{-},0)$. 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). 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 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, solutions. As mentioned in the introduction (\ref{sec:intro}), this is known,
however this will also be proved again in passing in this article. however this will also be proved again in passing in this article.
This description still holds for the case of rank 0 case if we consider $V_v$ to
be infinitely far to the right
(see section \ref{subsubsect:rank-zero-case-charact-curves}).
Note also that the $\beta_-(v)$ condition always holds for $v$ rank 0.
\subsection{Numerical Formulations of the Problems} \subsection{Numerical Formulations of the Problems}
The problems introduced in this section are phrased in the context of stability The problems introduced in this section are phrased in the context of stability
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