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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
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\newtheorem{corrolary}{Corrolary}[section]
\newtheorem{lemmadfn}{Lemma/Definition}[section]
\newtheorem{dfn}{Definition}[section]
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\newtheorem{lemma}{Lemma}[section]
\newtheorem{fact}{Fact}[section]
\newtheorem{example}{Example}[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
def beta_minus(v):
beta = stability.Tilt().beta
solutions = solve(
stability.Tilt(alpha=0).degree(v)==0,
beta)
return min(map(lambda s: s.rhs(), solutions))
class Object(object):
pass
\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 tilt 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.
These are given by the equations $\chern_i^{\alpha,\beta}(v)=0$ for $i=1,2$, and
are illustrated in Fig \ref{fig:charact_curves_vis}
(dotted line for $i=1$, solid for $i=2$).
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\begin{dfn}[Characteristic Curves $V_v$ and $\Theta_v$]
Given a Chern character $v$, with positive rank and $\Delta(v) \geq 0$, we
define two characteristic curves on the $(\alpha, \beta)$-plane:
\begin{align*}
V_v &\colon \chern_1^{\alpha, \beta}(v) = 0 \\
\Theta_v &\colon \chern_2^{\alpha, \beta}(v) = 0
\end{align*}
\end{dfn}
\begin{fact}[Geometry of Characteristic Curves]
The following facts can be deduced from the formulae for $\chern_i^{\alpha, \beta}(v)$
as well as the restrictions on $v$:
\begin{itemize}
\item $V_v$ is a vertical line at $\beta=\mu(v)$
\item $\Theta_v$ is a hyperbola with assymptotes angled at $\pm 45^\circ$
crossing where $V_v$ meets the $\beta$-axis: $(\mu(v),0)$
\item $\Theta_v$ is oriented with left-right branches (as opposed to up-down).
The left branch shall be labelled $\Theta_v^-$ and the right $\Theta_v^+$.
\item The gap along the $\beta$-axis between either branch of $\Theta_v$
and $V_v$ is $\sqrt{\Delta(v)}/\chern_0(v)$.
\item When $\Delta(v)=0$, $\Theta_v$ degenerates into a pair of lines, but the
labels $\Theta_v^\pm$ will still be used for convenience.
\end{itemize}
\end{fact}
\minorheading{Relevance of the vertical line $V_v$}
By definition of the first tilt $\firsttilt\beta$, objects of Chern character
$v$ can only be in $\firsttilt\beta$ on the left of $V_v$, where
$\chern_1^{\alpha,\beta}(v)>0$, and objects of Chern character $-v$ can only be
in $\firsttilt\beta$ on the right, where $\chern_1^{\alpha,\beta}(-v)>0$. In
fact, if there is a Gieseker semistable coherent sheaf $E$ of Chern character
$v$, then $E \in \firsttilt\beta$ if and only if $\beta<\mu(E)$ (left of the
$V_v$), and $E[1] \in \firsttilt\beta$ if and only if $\beta\geq\mu(E)$.
Because of this, when using these characteristic curves, only positive ranks are
considered, as negative rank objects are implicitly considered on the right hand
side of $V_v$.
\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"$\Theta_v^+$",[3.5, 2], rotation=45, **text_args)
+ text(r"$V_v$", [0.43, 1.5], rotation=90, **text_args)
+ text(r"$\Theta_v^-$", [-2.2, 2], rotation=-45, **text_args)
+ text(r"$\nu_{\alpha, \beta}(v)>0$", [-3, 1], **black_text_args)
+ text(r"$\nu_{\alpha, \beta}(v)<0$", [-1, 3], **black_text_args)
+ text(r"$\nu_{\alpha, \beta}(-v)>0$", [2, 3], **black_text_args)
+ text(r"$\nu_{\alpha, \beta}(-v)<0$", [4, 1], **black_text_args)
)
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
\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{
$\Delta(v)=0$: hyperbola collapses
}
\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}
\minorheading{Relevance of the hyperbola $\Theta_v$}
Since $\chern_2^{\alpha, \beta}$ is the numerator of the tilt slope
$\nu_{\alpha, \beta}$. The curve $\Theta_v$, where this is 0, firstly divides the
$(\alpha$-$\beta)$-half-plane into regions where the signs of tilt slopes of
objects of Chern character $v$ (or $-v$) are fixed. Secondly, it gives more of a
fixed target for some $u=(r,c\ell,d\frac{1}{2}\ell^2)$ to be a
pseudo-semistabilizer of $v$, in the following sense: If $(\alpha,\beta)$, is on
$\Theta_v$, then for any $u$, $u$ is a pseudo-semistabilizer of $v$ iff
$\nu_{\alpha,\beta}(u)=0$, and hence $\chern_2^{\alpha, \beta}(u)=0$. In fact,
this allows us to use the characteristic curves of some $v$ and $u$ (with
$\Delta(v), \Delta(u)\geq 0$ and positive ranks) to determine the location of
the pseudo-wall where $u$ pseudo-semistabilizes $v$. This is done by finding the
intersection of $\Theta_v$ and $\Theta_u$.
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\subsection{Bertram's nested wall theorem}
Although Bertram's nested wall theorem can be proved more directly, it's also
important for the content of this document to understand the connection with
these characteristic curves.
Emanuele Macri noticed in (TODO ref) that any circular wall of $v$ reaches a critical
point on $\Theta_v$ (TODO ref). This is a consequence of
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$\frac{\delta}{\delta\beta} \chern_2^{\alpha,\beta} = -\chern_1^{\alpha,\beta}$.
This fact, along with the hindsight knowledge that non-vertical walls are
circles with centers on the $\beta$-axis, gives an alternative view to see that
the circular walls must be nested and non-intersecting.
\subsection{Characteristic curves for pseudo-semistabilizers}
\begin{lemma}[Numerical tests for left-wall pseudo-semistabilizers]
\label{lem:pseudo_wall_numerical_tests}
Let $v$ and $u$ be Chern characters with positive ranks and $\Delta(v),
\Delta(u)\geq 0$. Let $P$ be a point on $\Theta_v^-$.
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\noindent
Suppose that the following are satisfied:
\bgroup
\renewcommand{\labelenumi}{\alph{enumi}.}
\begin{enumerate}
\item $u$ gives rise to a pseudo-wall for $v$, left of the vertical line $V_v$
\item The pseudo-wall contains $P$ in it's interior
($P$ can be chosen to be the base of the left branch to target all left-walls)
\item $u$ destabilizes $v$ going `inwards', that is,
$\nu_{\alpha,\beta}(\pm u)<\nu_{\alpha,\beta}(v)$ outside the pseudo-wall, and
$\nu_{\alpha,\beta}(\pm u)>\nu_{\alpha,\beta}(v)$ inside.
Where we use $+u$ or $-u$ depending on whether $(\beta,\alpha)$ is on the left
or right (resp.) of $V_u$.
\end{enumerate}
\egroup
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\noindent
Then we have the following:
\item The pseudo-wall is left of $V_u$
(if this is a real wall then $v$ is being semistabilized by an object with
Chern character $u$, not $-u$)
\item $\beta(P)<\mu(u)<\mu(v)$, i.e., $V_u$ is strictly between $P$ and $V_v$.
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Furthermore, only the last two of these consequences are sufficient to recover
all of the suppositions above.
\end{lemma}
\begin{proof}
Let $u,v$ be Chern characters with positive ranks and
$\Delta(u),\Delta(v) \geq 0$.
For the forwards implication, assume that the suppositions of the lemma are
satisfied. The pseudo-wall intersects $\Theta_v^-$, at some point $Q$ further up
the hyperbola branch than $P$ (to satisfy supposition b). At $Q$, we have
$\nu_Q(v)=0$, and hence $\nu_Q(u)=0$ too. This means that $\Theta_u$ must
intersect $\Theta_v^-$ at $Q$. Considering the shapes of the hyperbolae alone,
there are 3 distinct ways that they can intersect, as illustrated in Fig
\ref{fig:hyperbol-intersection}. These cases are distinguished by whether it is
the left, or the right branch of $\Theta_u$ involved, as well as the positions
of the base. However, considering supposition b, only case 3 (green in
figure) is valid. This is because we need $\nu_{P}(u)>0$ (or $\nu_{P}(-u)>0$ in
case 1 involving $\Theta_u^+$).
Recalling how the sign of $\nu_{\alpha,\beta}(\pm u)$ changes (illustrated in
Fig \ref{fig:charact_curves_vis}), we can eliminate cases 1 and 2.
In passing, note that this implies consequence 3.
\begin{sagesilent}
def hyperbola_intersection_plot():
var("alpha beta", domain="real")
coords_range = (beta, -3, -1/2), (alpha, 0, 2.5)
delta1 = -sqrt(2)+1/100
delta2 = 1/2
pbeta=-1.5
text_args = {"fontsize":"large", "clip":True}
black_text_args = {"rgbcolor":"black", **text_args}
implicit_plot( beta^2 - alpha^2 == 2,
*coords_range , rgbcolor = "black", legend_label=r"a")
+ implicit_plot( (beta+4)^2 - (alpha)^2 == 2,
*coords_range , rgbcolor = "red")
+ implicit_plot( (beta+delta1)^2 - alpha^2 == (delta1-2)^2-2,
*coords_range , rgbcolor = "blue")
+ implicit_plot( (beta+delta2)^2 - alpha^2 == (delta2-2)^2-2,
*coords_range , rgbcolor = "green")
+ point([-2, sqrt(2)], size=50, rgbcolor="black", zorder=50)
+ point([pbeta, sqrt(pbeta^2-2)], size=50, rgbcolor="black", zorder=50)
+ text("P",[pbeta+0.1, sqrt(pbeta^2-2)], **black_text_args)
+ circle((-2,0),sqrt(2), linestyle="dashed", rgbcolor="purple")
# dummy lines to add legends (circumvent bug in implicit_plot)
+ line([(2,0),(2,0)] , rgbcolor = "purple", linestyle="dotted",
legend_label=r"pseudo-wall")
+ line([(2,0),(2,0)] , rgbcolor = "black",
legend_label=r"$\Theta_v^-$")
+ line([(2,0),(2,0)] , rgbcolor = "red", legend_label=r"$\Theta_u$ case 1")
+ line([(2,0),(2,0)] , rgbcolor = "blue", legend_label=r"$\Theta_u$ case 2")
+ line([(2,0),(2,0)] , rgbcolor = "green", legend_label=r"$\Theta_u$ case 3")
p.set_legend_options(loc="upper right", font_size="x-large",
font_family="serif")
p.xmax(coords_range[0][2])
p.xmin(coords_range[0][1])
p.ymax(coords_range[1][2])
p.ymin(coords_range[1][1])
return p
def correct_hyperbola_intersection_plot():
var("alpha beta", domain="real")
coords_range = (beta, -2.5, 0.5), (alpha, 0, 3)
delta2 = 1/2
pbeta=-1.5
text_args = {"fontsize":"large", "clip":True}
black_text_args = {"rgbcolor":"black", **text_args}
implicit_plot( beta^2 - alpha^2 == 2,
*coords_range , rgbcolor = "black", legend_label=r"a")
+ implicit_plot((beta+delta2)^2 - alpha^2 == (delta2-2)^2-2,
*coords_range , rgbcolor = "green")
+ point([-2, sqrt(2)], size=50, rgbcolor="black", zorder=50)
+ point([pbeta, sqrt(pbeta^2-2)], size=50, rgbcolor="black", zorder=50)
+ text("P",[pbeta+0.1, sqrt(pbeta^2-2)], **black_text_args)
+ circle((-2,0),sqrt(2), linestyle="dashed", rgbcolor="purple")
# dummy lines to add legends (circumvent bug in implicit_plot)
+ line([(2,0),(2,0)] , rgbcolor = "purple", linestyle="dotted",
legend_label=r"pseudo-wall")
+ line([(2,0),(2,0)] , rgbcolor = "black",
# vertical characteristic lines
+ line([(0,0),(0,coords_range[1][2])],
rgbcolor="black", linestyle="dashed",
+ line([(-delta2,0),(-delta2,coords_range[1][2])],
rgbcolor="green", linestyle="dashed",
+ line([(-delta2,0),(-delta2-coords_range[1][2],coords_range[1][2])],
rgbcolor="green", linestyle="dotted",
legend_label=r"$\Theta_u^-$ assymptote")
+ line([(0,0),(-coords_range[1][2],coords_range[1][2])],
rgbcolor="black", linestyle="dotted",
legend_label=r"$\Theta_v^-$ assymptote")
p.set_legend_options(loc="upper right", font_size="x-large",
font_family="serif")
p.xmax(coords_range[0][2])
p.xmin(coords_range[0][1])
p.ymax(coords_range[1][2])
p.ymin(coords_range[1][1])
p.axes_labels([r"$\beta$", r"$\alpha$"])
return p
\end{sagesilent}
\begin{figure}
\begin{subfigure}[t]{0.48\textwidth}
\centering
\sageplot[width=\textwidth]{hyperbola_intersection_plot()}
\caption{Three ways the characteristic hyperbola for $u$ can intersect the left
branch of the characteristic hyperbola for $v$}
\label{fig:hyperbol-intersection}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.48\textwidth}
\centering
\sageplot[width=\textwidth]{correct_hyperbola_intersection_plot()}
\caption{Closer look at characteristic curves for valid case}
\label{fig:correct-hyperbol-intersection}
\end{subfigure}
\end{figure}
Fixing attention on the only valid case (2), illustrated in Fig
\ref{fig:correct-hyperbol-intersection}. We must have $\Theta_u^-$ taking a
base-point to the right $\Theta_v$, but then, further up, crossing over to the
left side. The latter fact implies that the assymptote for $\Theta_u^-$ must be
to the left of the one for $\Theta_v^-$. Given that they are parallel and
intersect the $\beta$-axis at $\beta=\mu(u)$ and $\beta=\mu(v)$ respectively. We
must have $\mu(u)<\mu(v)$, that is, $V_u$ is strictly to the left of $V_v$.
Finally, the fact that it is the left branch of the hyperbola for $u$ implies
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