main.tex

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\usepackage{amsmath}
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\usepackage{amsthm}
\usepackage{graphicx}
\usepackage{hyperref}
\usepackage{color}
\usepackage{sagetex}
\usepackage{minted}
\usepackage{subcaption}
\usepackage{cancel}
\usepackage[]{breqn}
\usepackage[
backend=biber,
style=alphabetic,
sorting=ynt
]{biblatex}
\addbibresource{references.bib}

\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\PP}{\mathbb{P}}
\newcommand{\chern}{\operatorname{ch}}
\newcommand{\lcm}{\operatorname{lcm}}
\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]
\newtheorem{corrolary}{Corrolary}[section]
\newtheorem{lemmadfn}{Lemma/Definition}[section]
\newtheorem{dfn}{Definition}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{fact}{Fact}[section]
\newtheorem{example}{Example}[section]

\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

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
\end{sagesilent}

\title{Explicit Formulae for Bounds on the Ranks of Tilt Destabilizers and
Practical Methods for Finding Pseudowalls}

\author{Luke Naylor}

\maketitle

\tableofcontents

\section{Introduction}
\label{sec:intro}

It is well known that for any rational $\beta_0$,
the vertical line $\{\sigma_{\alpha,\beta_0} \colon \alpha \in \RR_{>0}\}$ only
intersects finitely many walls
\cite[Thm 1.1]{LoJason2014Mfbs}
\cite[Prop 4.2]{alma9924569879402466}
\cite[Lemma 5.20]{MinaHiroYana_SomeModSp}.
A consequence of this is that if
$\beta_{-}$ is rational, then there can only be finitely many circular walls to the
left of the vertical wall $\beta = \mu$.
On the other hand, when $\beta_{-}$ is not rational, \cite{yanagida2014bridgeland}
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
\cite{SchmidtBenjamin2020Bsot} and done explicitly by Benjamin Schmidt within a
SageMath \cite{sagemath} library which computes pseudowalls
\cite{SchmidtGithub2020}.
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 $v$ on a Picard rank 1 surface, and a given stability
	condition $\sigma_{\alpha,\beta}$,
	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 $v$: $\nu_{\alpha,\beta}(u) = \nu_{\alpha,\beta}(v)$.

	\noindent
	Furthermore the following Bogomolov-Gieseker inequalities are satisfied:
	\begin{itemize}
		\item $\Delta(u) \geq 0$
		\item $\Delta(v-u) \geq 0$
		\item $\Delta(u) + \Delta(v-u) \leq \Delta(v)$
	\end{itemize}
	\noindent
	And finally these two conditions are satisfied:
	\begin{itemize}
		\item $\chern_1^{\beta}(u) \geq 0$
		\item $\chern_1^{\beta}(v-u) \geq 0$
	\end{itemize}

	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.

\begin{lemma}[ Sanity check for Pseudo-semistabilizers ]
	Given a Picard rank 1 surface, and a given stability
	condition $\sigma_{\alpha,\beta}$,
	if $E\hookrightarrow F\twoheadrightarrow G$ is a semistabilizing sequence in
	$\firsttilt\beta$ for $F$.
	Then $\chern(E)$ is a pseudo-semistabilizer of $\chern(F)$
\end{lemma}

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 $i=2$).

\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}