diff --git a/main.tex b/main.tex index 1dfedd51b49d0a89d061aca7b817bd057a0e2c3f..18b33dee24be0a3f209470d7b186b89d57b4ea7a 100644 --- a/main.tex +++ b/main.tex @@ -29,6 +29,16 @@ \begin{document} +\begin{sagesilent} +# Requires extra package: +#! sage -pip install "pseudowalls==0.0.3" --extra-index-url https://gitlab.com/api/v4/projects/43962374/packages/pypi/simple + +from pseudowalls import * + +Δ = lambda v: v.Q_tilt() +mu = stability.Mumford().slope +\end{sagesilent} + \title{Explicit Formulae for Bounds on the Ranks of Tilt Destabilizers and Practical Methods for Finding Pseudowalls} @@ -85,6 +95,41 @@ bounds on $\chern_0(E)$ of potential destabilizers $E$ of $F$. \section{Characteristic Curves of Stability Conditions Associated to Chern Characters} +\begin{sagesilent} +v = Chern_Char(3, 2, -2) +u = Chern_Char(1, 0, 0) + +alpha = stability.Tilt().alpha +beta = stability.Tilt().beta + +coords_range = (beta, -5, 5), (alpha, 0, 5) + +charact_curve_plot = ( + implicit_plot(stability.Tilt().degree(u), *coords_range , rgbcolor = "red") + + implicit_plot(stability.Tilt().degree(v), *coords_range ) + + line([(mu(v),0),(mu(v),5)], linestyle = "dashed", legend_label = + r"$(3,2\ell,-4\ell^2/2)$") + + line([(mu(u),0),(mu(u),5)], rgbcolor = "red", linestyle = + "dashed", legend_label = r"$(1,0,0)$") + + implicit_plot(stability.Tilt().wall_eqn(u,v)/alpha, + *coords_range , rgbcolor = "black") +) +charact_curve_plot.xmax(1) +charact_curve_plot.xmin(-2) +charact_curve_plot.ymax(1.5) +charact_curve_plot.axes_labels([r"$\beta$", r"$\alpha$"]) +\end{sagesilent} + +\begin{figure} + \centering + \sageplot[width=\linewidth]{charact_curve_plot} + \caption{} + \label{fig:characteristic-curves-example} +\end{figure} + +Talk about figure \ref{fig:characteristic-curves-example}. + + \section{Loose Bounds on $\chern_0(E)$ for Semistabilizers Along Fixed $\beta\in\QQ$}