From ca789abdf73fe6ff989b8eecbc4bfe669c739dd5 Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Tue, 13 Jun 2023 16:55:07 +0100
Subject: [PATCH] Introduce new notation for characteristic curves
---
main.tex | 27 +++++++++++++++++++++++++++
1 file changed, 27 insertions(+)
diff --git a/main.tex b/main.tex
index 4398595..d6fe72d 100644
--- a/main.tex
+++ b/main.tex
@@ -28,6 +28,7 @@
\newtheorem{lemmadfn}{Lemma/Definition}[section]
\newtheorem{dfn}{Definition}[section]
\newtheorem{lemma}{Lemma}[section]
+\newtheorem{fact}{Fact}[section]
\begin{document}
@@ -123,6 +124,32 @@ 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}
+
\minorheading{Relevance of $\chern_1^{\alpha, \beta}=0$ vertical line}
--
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