Start introduction

main.tex

@@ -8,10 +8,15 @@
 \usepackage{hyperref}
 \usepackage{color}
 
+\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 Maximal Ranks of Tilt Destabilizers}
+\title{Explicit Formulae for Bounds on the Ranks of Tilt Destabilizers and
+Practical Methods for Finding Pseudowalls}
 
 \author{Luke Naylor}
 
@@ -19,6 +24,39 @@
 
 \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}