From 4d2777abfe1124846b34427cbb8695913af2c3df Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Thu, 7 Sep 2023 18:23:29 +0100
Subject: [PATCH] Introduce rank 0 case up to main lemma
---
main.tex | 40 ++++++++++++++++++++++++++++++++++------
1 file changed, 34 insertions(+), 6 deletions(-)
diff --git a/main.tex b/main.tex
index b28f254..3247b76 100644
--- a/main.tex
+++ b/main.tex
@@ -289,8 +289,8 @@ 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
+ V_v &\colon \:\: \chern_1^{\alpha, \beta}(v) = 0 \\
+ \Theta_v &\colon \:\: \chern_2^{\alpha, \beta}(v) = 0
\end{align*}
\end{definition}
@@ -366,26 +366,49 @@ The following facts can be deduced from the formulae for $\chern_i^{\alpha, \bet
as well as the restrictions on $v$, when $\chern_0(v)=0$ and $\chern_1(v)>0$:
-\begin{minipage}{0.49\textwidth}
+\begin{minipage}{0.59\textwidth}
\begin{itemize}
\item $V_v = \emptyset$
\item $\Theta_v$ is a vertical line at $\beta=\frac{D}{C}$
where $v=\left(0,C\ell,D\ell^2\right)$
\end{itemize}
\end{minipage}
-\begin{minipage}{0.49\textwidth}
+\hfill
+\begin{minipage}{0.39\textwidth}
\sageplot[width=\textwidth]{Theta_v_plot}
%\caption{$\Delta(v)>0$}
%\label{fig:charact_curves_rank0}
\end{minipage}
\end{fact}
+We can view the characteristic curves for $\left(0,C\ell, D\ell^2\right)$ with $C>0$ as
+the limiting behaviour of those of $\left(\varepsilon, C\ell, D\ell^2\right)$.
+Indeed:
+\begin{align*}
+ \mu\left(\varepsilon, C\ell, D\ell^2\right) = \frac{C}{\varepsilon} &\longrightarrow +\infty
+ \\
+ \text{as} \: 0<\varepsilon &\longrightarrow 0
+\end{align*}
+So we can view $V_v$ as moving off infinitely to the right, with $\Theta_v^+$ even further.
+But also, considering the base point of $\Theta_v^-$:
+\begin{align*}
+ \beta_{-}\left(\varepsilon, C\ell, D\ell^2\right) = \frac{C - \sqrt{C^2-2D\varepsilon}}{\varepsilon}
+ &\longrightarrow \frac{D}{C}
+ \\
+ \text{as} \:\: 0<\varepsilon &\longrightarrow 0
+ &\text{(via L'H\^opital)}
+\end{align*}
+
+So we can view $\Theta_v^-$ as approaching the vertical line that $\Theta_v$ becomes.
+For this reason, I will refer to the whole of $\Theta_v$ in the rank zero case
+as $\Theta_v^-$ to be able to use the same terminology in both positive rank
+and rank zero cases.
\subsection{Relevance of $V_v$}
\label{subsect:relevance-of-V_v}
-By definition of the first tilt $\firsttilt\beta$, objects of Chern character
+For the positive rank case, by definition of the first tilt $\firsttilt\beta$, objects of Chern character
$v$ can only be in $\firsttilt\beta$ on the left of $V_v$, where
$\chern_1^{\alpha,\beta}(v)>0$, and objects of Chern character $-v$ can only be
in $\firsttilt\beta$ on the right, where $\chern_1^{\alpha,\beta}(-v)>0$. In
@@ -396,6 +419,11 @@ Because of this, when using these characteristic curves, only positive ranks are
considered, as negative rank objects are implicitly considered on the right hand
side of $V_v$.
+In the rank zero case, this still applies if we consider $V_v$ to be
+`infinitely to the right' ($\mu(v) = +\infty$). Precisely, Gieseker semistable
+coherent sheaves $E$ of Chern character $v$ are contained in
+$\firsttilt{\beta}$ for all $\beta$
+
\subsection{Relevance of $\Theta_v$}
@@ -406,7 +434,7 @@ $(\alpha$-$\beta)$-half-plane into regions where the signs of tilt slopes of
objects of Chern character $v$ (or $-v$) are fixed. Secondly, it gives more of a
fixed target for some $u=(r,c\ell,d\frac{1}{2}\ell^2)$ to be a
pseudo-semistabilizer of $v$, in the following sense: If $(\alpha,\beta)$, is on
-$\Theta_v$, then for any $u$, $u$ is a pseudo-semistabilizer of $v$ iff
+$\Theta_v$, then for any $u$, $u$ can only be a pseudo-semistabilizer of $v$ if
$\nu_{\alpha,\beta}(u)=0$, and hence $\chern_2^{\alpha, \beta}(u)=0$. In fact,
this allows us to use the characteristic curves of some $v$ and $u$ (with
$\Delta(v), \Delta(u)\geq 0$ and positive ranks) to determine the location of
--
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