diff --git a/main.tex b/main.tex
index aac27e695e09ba3f9d5303048ff5c74856bcae5d..9bc843685b57fa9fb0b0a6dabe7dfa9548f45f3d 100644
--- a/main.tex
+++ b/main.tex
@@ -11,6 +11,12 @@
\usepackage{minted}
\usepackage{subcaption}
\usepackage[]{breqn}
+\usepackage[
+backend=biber,
+style=alphabetic,
+sorting=ynt
+]{biblatex}
+\addbibresource{references.bib}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\ZZ}{\mathbb{Z}}
@@ -66,13 +72,17 @@ Practical Methods for Finding Pseudowalls}
\section{Introduction}
\label{sec:intro}
-[ref] shows that for any rational $\beta_0$,
+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. A consequence of this is that if
+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, [ref] showed that there are
-infinitely many walls.
+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
@@ -102,11 +112,13 @@ 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 $F$.
+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
@@ -444,7 +456,8 @@ def correct_hyperbola_intersection_plot():
Fixing attention on the only possible case (2), illustrated in Fig
\ref{fig:correct-hyperbol-intersection}.
-$P$ is on the left of $V_{\pm u}$ (first part of consequence 2), so $u$ must have positive rank (consequence 1)
+$P$ is on the left of $V_{\pm u}$ (first part of consequence 2), so $u$ must
+have positive rank (consequence 1)
to ensure that $\chern_1^{\beta{P}} \geq 0$ (since the pseudo-wall passed over
$P$).
Furthermore, $P$ being on the left of $V_u$ implies
@@ -1972,6 +1985,9 @@ Goals:
\item Relate to numerical condition described by Yanagida/Yoshioka
\end{itemize}
+\newpage
+\printbibliography
+
\newpage
\section{Appendix - SageMath code}
diff --git a/references.bib b/references.bib
new file mode 100644
index 0000000000000000000000000000000000000000..2b8a8bb15313982afd664b0545578f4379dbf0bf
--- /dev/null
+++ b/references.bib
@@ -0,0 +1,124 @@
+@article{SchmidtBenjamin2020Bsot,
+ publisher = {Univ Press Inc},
+ title = {Bridgeland stability on threefolds: Some wall crossings},
+ volume = {29},
+ year = {2020},
+ author = {Schmidt, Benjamin},
+ address = {PROVIDENCE},
+ keywords = {Mathematics ; Physical Sciences ; Science & Technology},
+ language = {eng},
+ number = {2},
+ pages = {247-283},
+ abstract = {Following up on the construction of Bridgeland stability condition on P-3 by Macri, we develop techniques to study concrete wall crossing behavior for the first time on a threefold. In some cases, such as complete intersections of two hypersurfaces of the same degree or twisted cubics, we show that there are two chambers in the stability manifold where the moduli space is given by a smooth projective irreducible variety, respectively, the Hilbert scheme. In the case of twisted cubics, we compute all walls and moduli spaces on a path between those two chambers. This allows us to give a new proof of the global structure of the main component, originally due to Ellingsrud, Piene, and Stromme. In between slope stability and Bridgeland stability there is the notion of tilt stability that is defined similarly to Bridgeland stability on surfaces. Beyond just P-3, we develop tools to use computations in tilt stability to compute wall crossings in Bridgeland stability.},
+ copyright = {Copyright 2020 Elsevier B.V., All rights reserved.},
+ issn = {1056-3911},
+ journal = {Journal of algebraic geometry},
+}
+@misc{SchmidtGithub2020,
+ author = {Schmidt, Benjamin},
+ title = {stability\_conditions, SageMath library},
+ year = {2020},
+ publisher = {GitHub},
+ journal = {GitHub repository},
+ howpublished = {\url{https://github.com/benjaminschmidt/stability_conditions}},
+ commit = {cf448d4}
+}
+@misc{NaylorDoc2023,
+ author = {Naylor, Luke},
+ title = {Article behind poster},
+ year = {2023},
+ journal = {Git repository},
+ howpublished = {\url{git.ecdf.ed.ac.uk/personal-latex-documents/research/max-destabilizer-rank}},
+}
+@misc{NaylorRust2023,
+ author = {Naylor, Luke},
+ title = {Pseudo-wall finder, try by scanning QR code at top-right},
+ year = {2023},
+ publisher = {GitLab},
+ journal = {GitLab repository},
+ howpublished = {\url{https://gitlab.com/pseudowalls/tilt.rs}},
+}
+@article{BogGiestypeineq,
+ author = {Bayer, Arend and Macri, Emanuele and Toda, Yukinobu},
+ copyright = {info:eu-repo/semantics/restrictedAccess},
+ language = {eng},
+ title = {Bridgeland Stability conditions on threefolds I: BG type inequalities},
+ year = {2013},
+}
+@article{yanagida2014bridgeland,
+ title={Bridgeland’s stabilities on abelian surfaces},
+ author={Yanagida, Shintarou and Yoshioka, K{\=o}ta},
+ journal={Mathematische Zeitschrift},
+ volume={276},
+ number={1},
+ pages={571--610},
+ year={2014},
+ publisher={Springer},
+ URL = {https://doi-org.ezproxy.is.ed.ac.uk/10.1007/s00209-013-1214-1}
+}
+@article{JardimMarcos2019Waaf,
+ title = {Walls and asymptotics for Bridgeland stability conditions on 3-folds},
+ year = {2019},
+ author = {Jardim, Marcos and Maciocia, Antony},
+ language = {eng},
+ abstract = {\'Epijournal de G\'eom\'etrie Alg\'ebrique, Volume 6 (2022),
+ Article No. 22 We consider Bridgeland stability conditions for three-folds conjectured by
+ Bayer-Macr\`i-Toda in the case of Picard rank one. We study the differential
+ geometry of numerical walls, characterizing when they are bounded, discussing
+ possible intersections, and showing that they are essentially regular. Next, we
+ prove that walls within a certain region of the upper half plane that
+ parametrizes geometric stability conditions must always intersect the curve
+ given by the vanishing of the slope function and, for a fixed value of s, have
+ a maximum turning point there. We then use all of these facts to prove that
+ Gieseker semistability is equivalent to asymptotic semistability along a class
+ of paths in the upper half plane, and to show how to find large families of
+ walls. We illustrate how to compute all of the walls and describe the
+ Bridgeland moduli spaces for the Chern character (2,0,-1,0) on complex
+ projective 3-space in a suitable region of the upper half plane.},
+ copyright = {http://creativecommons.org/licenses/by-sa/4.0},
+}
+@article{LoJason2014Mfbs,
+ author = {Lo, Jason and Qin, Zhenbo},
+ address = {SOMERVILLE},
+ copyright = {Copyright 2015 Elsevier B.V., All rights reserved.},
+ issn = {1093-6106},
+ journal = {The Asian journal of mathematics},
+ keywords = {14D20 ; 14F05 ; 14J60 ; Bridgeland stability ; Derived category ; Mathematics ; Mathematics, Applied ; Physical Sciences ; Polynomial stability ; Science & Technology ; Walls},
+ language = {eng},
+ number = {2},
+ abstract = {For the derived category of bounded complexes of sheaves on a smooth projective surface, Bridge land [Bri2] and Arcara-Bertram [ABL] constructed Bridge land stability conditions (Z(m), P-m) parametrized by m epsilon (0, +infinity). In this paper, we show that the set of mini-walls in (0, +infinity) of a fixed numerical type is locally finite. In addition, we strengthen a result of Bayer [Bay] by proving that the moduli of polynomial Bridge land semistable objects of a fixed numerical type coincides with the moduli of (Z(m), P-m)-semistable objects whenever m is larger than a universal constant depending only on the numerical type. We further identify the moduli of polynomial Bridge land semistable objects with the Gieseker/Simpson moduli spaces and the Uhlenbeck compactification spaces.},
+ pages = {321-344},
+ publisher = {Int Press Boston, Inc},
+ title = {Mini-walls for bridgeland stability conditions on the derived category of sheaves over surfaces},
+ volume = {18},
+ year = {2014},
+}
+@article{alma9924569879402466,
+ author = {Maciocia, Antony},
+ copyright = {info:eu-repo/semantics/openAccess},
+ language = {eng},
+ title = {Computing the Walls Associated to Bridgeland Stability Conditions on Projective Surfaces},
+ year = {2014-03-31},
+}
+@article{MinaHiroYana_SomeModSp,
+ author = {Minamide, Hiroki and Yanagida, Shintarou and Yoshioka, KÅta},
+ title = "{Some Moduli Spaces of Bridgeland’s Stability Conditions}",
+ journal = {International Mathematics Research Notices},
+ volume = {2014},
+ number = {19},
+ pages = {5264-5327},
+ year = {2013},
+ month = {06},
+ abstract = "{We study some moduli spaces of Bridgeland's semi-stable objects on abelian surfaces and K3 surfaces with Picard number 1. In particular, we show that the moduli spaces are isomorphic to the moduli spaces of Gieseker semi-stable sheaves. As an application, we construct ample line bundles on the moduli spaces, and study the ample cone of the moduli spaces by using wall/chamber structure of stability conditions.}",
+ issn = {1073-7928},
+ doi = {10.1093/imrn/rnt126},
+ url = {https://doi.org/10.1093/imrn/rnt126},
+ eprint = {https://academic.oup.com/imrn/article-pdf/2014/19/5264/18895160/rnt126.pdf},
+}
+@manual{sagemath,
+ key = {SageMath},
+ author = {{The Sage Developers}},
+ title = {{S}ageMath, the {S}age {M}athematics {S}oftware {S}ystem ({V}ersion 9.6.0)},
+ note = {{\tt https://www.sagemath.org}},
+ year = {2022}
+}
\ No newline at end of file