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\usepackage{hyperref}
\usepackage{color}
\usepackage{sagetex}
\usepackage{minted}
\usepackage{subcaption}
\usepackage{cancel}
\usepackage{mathtools}
\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}}
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\newtheorem{fact}{Fact}[section]
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\newtheorem{problem}{Problem Statement}

\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{Tighter Bounds for Ranks of Tilt Semistabilizers on Picard Rank 1 Surfaces
\\[1em] \large
Practical Methods for Narrowing Down Possible Walls}


\author{Luke Naylor}

\maketitle

\tableofcontents

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

The theory of Bridgeland stability conditions \cite{BridgelandTom2007SCoT} on
complexes of sheaves was developed as a generalisation of stability for vector
bundles. The definition is most analoguous to Mumford stability, but is more
aware of the features that sheaves can have on spaces of dimension greater
than 1. Whilst also asymptotically matching up with Gieseker stability.
For K3 surfaces, explicit stability conditions were defined in
\cite{Bridgeland_StabK3}, and later shown to also be valid on other surfaces.

The moduli spaces of stable objects of some fixed Chern character $v$ is
studied, as well as how they change as we vary the Bridgeland stability
condition. They in fact do not change over whole regions of the stability
space (called chambers), but do undergo changes as we cross `walls' in the
stability space. These are where there is some stable object $F$ of $v$ which
has a subobject who's slope overtakes the slope of $v$, making $F$ unstable
after crossing the wall.

% NOTE: SURFACE SPECIALIZATION
% (come back to these when adjusting to general Picard rank 1)
In this document we concentrate on two surfaces: Principally polarized abelian
surfaces and the projective surface $\PP^2$. Although this can be generalised
for Picard rank 1 surfaces, the formulae will need adjusting.
The Bridgeland stability conditions (defined in \cite{Bridgeland_StabK3}) are
given by two parameters $\alpha \in \RR_{>0}$, $\beta \in \RR$, which will be
illustrated throughout this article with diagrams of the upper half plane.

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 semistabilizing sequence
$E \hookrightarrow F \twoheadrightarrow G$ in $\mathcal{B}^\beta$
we have the following conditions.
There are some Bogomolov-Gieseker inequalities:
$0 \leq \Delta(E), \Delta(G)$.
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}

Throughout this article, as noted in the introduction, we will be exclusively
working over one of the following two surfaces: principally polarized abelian
surfaces and $\PP^2$.

\begin{dfn}[Pseudo-semistabilizers]
% NOTE: SURFACE SPECIALIZATION
	Given a Chern Character $v$, 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 ]
% NOTE: SURFACE SPECIALIZATION
	Given a 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}