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\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\chern}{\operatorname{ch}}
\newcommand{\firsttilt}[1]{\mathcal{B}^{#1}}
\newcommand{\bddderived}{\mathcal{D}^{b}}

\begin{document}

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

\author{Luke Naylor}

\maketitle

\section{Introduction}

There are theoretical results [ref] that show that for any $\beta_0 \in \QQ$,
the vertical line $\{\sigma_{\alpha,\beta_0} \colon \alpha \in \QQ_{>0}\}$ only
intersects finitely many walls. A consequence of this is that if
$\beta_{-} \in \QQ$, then there can only be finitely many circular walls to the
left of the vertical wall $\beta = \mu$.
On the other hand, when $\beta_{-} \not\in \QQ$, [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 elements of
$\bddderived(X)$.

Since real walls are a subset of pseudowalls, the $\beta_{-} \not\in \QQ$ case
follows immediately from the corresponding case for real walls.
However, the $\beta_{-} \in \QQ$ case involves showing that the following
conditions only admit finitely many solutions (despite the fact that the same
conditions admit infinitely many solutions when $\beta_{-} \not\in \QQ$).


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's a condition ensuring that the radius of the circular wall is
strictly positive: $\chern^\beta_2(E) > 0$.


\section{Section 1}

\section{Section 2}

\section{Section 3}

\section{Conclusion}

\end{document}