Adjust plots and rephrase problem statement

main.tex

@@ -38,6 +38,7 @@ sorting=ynt
 \newtheorem{lemma}{Lemma}[section]
 \newtheorem{fact}{Fact}[section]
 \newtheorem{example}{Example}[section]
+\newtheorem{problem}{Problem Statement}
 
 \begin{document}
 
@@ -642,22 +643,33 @@ finding all pseudo-walls when $\beta_{-}\in\QQ$. In section [ref], a different
 algorithm will be presented making use of the later theorems in this article,
 with the goal of cutting down the run time.
 
-\subsection*{Problem statement}
-\label{subsect:problem-statement}
+\begin{problem}[sufficiently large `left' pseudo-walls]
+\label{problem:problem-statement-1}
 
 Fix a Chern character $v$ with positive rank, $\Delta(v) \geq 0$,
 and $\beta_{-}(v) \in \QQ$.
-The goal is to find all Chern characters $u=(r,c\ell,d\ell^2)$ which satisfy the
-conditions of lemma \ref{lem:pseudo_wall_numerical_tests} using
-$P=(\beta_{-},0)$, $\chern_1^{\beta_{-}}(v-u)\geq 0$, as well as the Bogomolov inequalities:
-$\Delta(u),\Delta(v-u) \geq 0$ and $\Delta(u)+\Delta(v-u) \leq \Delta(v)$.
-We want to restrict our attention to pseudo-walls left of $V_v$ (condition (a) of
-lemma), because this is the side of $V_v$ containing the chamber for Gieseker
-stable objects, and the picture on the other side should be symmetric.
-Condition (c) of the lemma is there to restrict to objects most likely to
-semistabilizers of actual sheaves. The Chern characters which destabilize
-`outwards' can be recovered as $v-u$ for each solution $u$ to the current
-problem.
+The goal is to find all pseudo-semistabilizers $u=(r,c\ell,d\ell^2)$
+which give circular pseudo-walls containing some fixed point
+$P\in\Theta_v^-$.
+With the added restriction that $u$ `destabilizes' $v$ moving `inwards', that is,
+$\nu(u)>\nu(v)$ inside the circular pseudo-wall
+(`outward' destabilizers can be recovered as $v-u$).
+
+This will give all pseudo-walls between the chamber corresponding to Gieseker
+stability and the stability condition corresponding to $P$.
+\end{problem}
+
+\begin{problem}[all `left' pseudo-walls]
+\label{problem:problem-statement-1}
+
+Fix a Chern character $v$ with positive rank, $\Delta(v) \geq 0$,
+and $\beta_{-}(v) \in \QQ$.
+The goal is to find all solutions $u=(r,c\ell,d\ell^2)$
+to problem \ref{problem:problem-statement-1} with the choice
+$P=(\beta_{-},0)$.
+
+This will give all circular pseudo-walls left of $V_v$.
+\end{problem}
 
 \subsection*{Algorithm}
 
@@ -1310,14 +1322,14 @@ def plot_d_bound(
 \centering
 \begin{subfigure}{.45\textwidth}
   \centering
-	\sageplot[width=\linewidth]{plot_d_bound(v_example, 0)}
+	\sageplot[width=\linewidth]{plot_d_bound(v_example, 0, ymin=-0.5)}
 	\caption{$q = 0$ (all bounds other than green coincide on line)}
   \label{fig:d_bounds_xmpl_min_q}
 \end{subfigure}%
 \hfill
 \begin{subfigure}{.45\textwidth}
   \centering
-	\sageplot[width=\linewidth]{plot_d_bound(v_example, 4, ymin=-3)}
+	\sageplot[width=\linewidth]{plot_d_bound(v_example, 4, ymin=-3, ymax=3)}
 	\caption{$q = \chern^{\beta}(F)$ (all bounds other than blue coincide on line)}
   \label{fig:d_bounds_xmpl_max_q}
 \end{subfigure}
@@ -1365,7 +1377,7 @@ Some of the details around the associated numerics are explored next.
 \centering
 \sageplot[
 	width=\linewidth
-]{plot_d_bound(v_example, 2, ymax=6, ymin=-2, aspect_ratio=1)}
+]{plot_d_bound(v_example, 2, ymax=4, ymin=-2, aspect_ratio=1)}
 \caption{
 	Bounds on $d:=\chern_2(E)$ in terms of $r:=\chern_0(E)$ for a fixed
 	value $\chern_1^{\beta}(F)/2$ of $q:=\chern_1^{\beta}(E)$.