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%% Write basic article template
\documentclass[12pt]{article}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{graphicx}
\usepackage{hyperref}
\usepackage{color}
\usepackage{subcaption}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\firsttilt}[1]{\mathcal{B}^{#1}}
\newcommand{\bddderived}{\mathcal{D}^{b}}
\newcommand{\centralcharge}{\mathcal{Z}}
\newcommand{\minorheading}[1]{{\noindent\normalfont\normalsize\bfseries #1}}
\newtheorem{theorem}{Theorem}[section]
Luke Naylor
committed
\newtheorem{corrolary}{Corrolary}[section]
\newtheorem{lemmadfn}{Lemma/Definition}[section]
\newtheorem{dfn}{Definition}[section]
\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}
[ref] shows that for any rational $\beta_0$,
the vertical line $\{\sigma_{\alpha,\beta_0} \colon \alpha \in \RR_{>0}\}$ only
intersects finitely many walls. A consequence of this is that if
$\beta_{-}$ is rational, then there can only be finitely many circular walls to the
On the other hand, when $\beta_{-}$ is not rational, [ref] showed that there are
infinitely many walls.
This dichotomy does not only hold for real walls, realised by actual objects in
$\bddderived(X)$, but also for pseudowalls. Here pseudowalls are defined as
`potential' walls, induced by hypothetical Chern characters of destabilizers
which satisfy certain numerical conditions which would be satisfied by any real
destabilizer, regardless of whether they are realised by actual semistabilizers
in $\bddderived(X)$.
Since real walls are a subset of pseudowalls, the irrational $\beta_{-}$ case
follows immediately from the corresponding case for real walls.
However, the rational $\beta_{-}$ case involves showing that the following
conditions only admit finitely many solutions (despite the fact that the same
conditions admit infinitely many solutions when $\beta_{-}$ is irrational).
For a destabilizing sequence
$E \hookrightarrow F \twoheadrightarrow G$ in $\mathcal{B}^\beta$
we have the following conditions.
There are some Bogomolov-Gieseker type inequalities:
$0 \leq \Delta(E), \Delta(G)$ and $\Delta(E) + \Delta(G) \leq \Delta(F)$.
We also have a condition relating to the tilt category $\firsttilt\beta$:
$0 \leq \chern^\beta_1(E) \leq \chern^\beta_1(F)$.
Finally, there is a condition ensuring that the radius of the circular wall is
strictly positive: $\chern^{\beta_{-}}_2(E) > 0$.
For any fixed $\chern_0(E)$, the inequality
$0 \leq \chern^{\beta}_1(E) \leq \chern^{\beta}_1(F)$,
allows us to bound $\chern_1(E)$. Then, the other inequalities allow us to
bound $\chern_2(E)$. The final part to showing the finiteness of pseudowalls
would be bounding $\chern_0(E)$. This has been hinted at in [ref] and done
explicitly by Benjamin Schmidt within a computer program which computes
pseudowalls. Here we discuss these bounds in more detail, along with the methods
used, followed by refinements on them which give explicit formulae for tighter
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):
alpha = stability.Tilt().alpha
beta = stability.Tilt().beta
coords_range = (beta, -4, 5), (alpha, 0, 4)
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="x-large", clip=True)
+ text(r"$ch_1^{\alpha, \beta}(v)=0$", [0.45, 1.5], rotation=90, fontsize="x-large", clip=True)
+ text(r"$ch_2^{\alpha, \beta}(v)=0$", [-2, 2], rotation=-45, fontsize="x-large", clip=True)
+ text(r"$\nu_{\alpha, \beta}(v)>0$", [-3, 1], rgbcolor="black", fontsize="x-large", clip=True)
+ text(r"$\nu_{\alpha, \beta}(v)<0$", [-1, 3], rgbcolor="black", fontsize="x-large", clip=True)
+ text(r"$\nu_{\alpha, \beta}(-v)>0$", [2, 3], rgbcolor="black", fontsize="x-large", clip=True)
+ text(r"$\nu_{\alpha, \beta}(-v)<0$", [4, 1], rgbcolor="black", fontsize="x-large", clip=True)
)
p.xmax(5)
p.xmin(-4)
p.ymax(4)
p.axes_labels([r"$\beta$", r"$\alpha$"])
return p
v1 = Chern_Char(3, 2, -2)
v2 = Chern_Char(3, 2, 2/3)
\end{sagesilent}
%\begin{figure}
% \centering
% \sageplot[width=\textwidth]{charact_curves(v1)}
% \caption{}
% \label{fig:charact_curves_vis}
%\end{figure}
\begin{subfigure}{.49\textwidth}
\centering
\sageplot[width=\textwidth]{charact_curves(v1)}
\caption{$\Delta(v)>0$}
\label{fig:charact_curves_vis_bgmvlPos}
\end{subfigure}%
\hfill
\begin{subfigure}{.49\textwidth}
\centering
\sageplot[width=\textwidth]{charact_curves(v2)}
\caption{$\Delta(v)=0$}
\label{fig:charact_curves_vis_bgmlv0}
\end{subfigure}
\caption{
Characteristic curves ($\chern_i^{\alpha,\beta}(v)=0$) of stability conditions
associated to Chern characters $v$ with $\Delta(v) \geq 0$ and positive rank.
}
\label{fig:charact_curves_vis}
\end{figure}
\begin{sagesilent}
v = Chern_Char(3, 2, -2)
u = Chern_Char(1, 0, 0)
def charact_curve_with_wall_plot(u,v):
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"$\alpha$"])
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