Start plots for characteristic curves

main.tex

@@ -98,6 +98,62 @@ bounds on $\chern_0(E)$ of potential destabilizers $E$ of $F$.
 \section{Characteristic Curves of Stability Conditions Associated to Chern
 Characters}
 
+\begin{dfn}[Pseudo-semistabilizers]
+	Given a Chern Character $v$, and a given stability condition $\sigma$, 
+	a pseudo-semistabilizing $u$ is a `potential' Chern character:
+	\[
+		u = \left(r, c\ell, d \frac{1}{2} \ell^2\right)
+	\]
+	which has the same slope as $v$: $\mu_{\sigma}(u) = \mu_{\sigma}(v)$.
+
+	Note $u$ does not need to be a Chern character of an actual sub-object of some
+	object in the stability condition's heart with Chern character $v$.
+\end{dfn}
+
+At this point, and in this document, we do not care about whether
+pseudo-semistabilizers are even Chern characters of actual elements of
+$\bddderived(X)$, some other sources may have this extra restriction too.
+
+Considering the stability conditions with two parameters $\alpha, \beta$ on
+Picard rank 1 surfaces.
+We can draw 2 characteristic curves for any given Chern character $v$ with
+$\Delta(v) \geq 0$ and positive rank.
+
+\begin{sagesilent}
+def charact_curves(v):
+    var("alpha beta")
+    coords_range = (beta, -10, 10), (alpha, 0, 8)
+    p = (
+      implicit_plot(stability.Tilt().degree(v), *coords_range )
+      + line([(mu(v),0),(mu(v),5)], linestyle = "dashed")
+      + text(r"$ch_2^{\alpha, \beta}(v)=0$",[3.5, 2], rotation=45, fontsize="large")
+      + text(r"$ch_1^{\alpha, \beta}(v)=0$", [0.45, 1.5], rotation=90, fontsize="large")
+      + text(r"$ch_2^{\alpha, \beta}(v)=0$", [-2, 2], rotation=-45, fontsize="large")
+      + text(r"$\nu_{\alpha, \beta}(v)>0$", [-3, 1], rgbcolor="black", fontsize="large")
+      + text(r"$\nu_{\alpha, \beta}(v)<0$", [-1, 3], rgbcolor="black", fontsize="large")
+      + text(r"$\nu_{\alpha, \beta}(-v)>0$", [2, 3], rgbcolor="black", fontsize="large")
+      + text(r"$\nu_{\alpha, \beta}(-v)<0$", [4, 1], rgbcolor="black", fontsize="large")
+    )
+    p.xmax(5)
+    p.xmin(-4)
+    p.ymax(4)
+    p.axes_labels([r"$\beta$", r"$\alpha$"])
+    p.tick_label_color("white")
+    return p
+
+v1 = Chern_Char(3, 2, -2)
+v2 = Chern_Char(3, 2, 2/3)
+\end{sagesilent}
+
+\begin{figure}
+\centering
+\resizebox{\textwidth}{!}{
+\sageplot{charact_curves(v1)}
+}
+\caption{capt 1}
+\label{fig:charact_curves_vis}
+\end{figure}
+
 \begin{sagesilent}
 v = Chern_Char(3, 2, -2)
 u = Chern_Char(1, 0, 0)