From f3e74d657c57d76c444088748918f365124a990b Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Sun, 11 Jun 2023 18:29:21 +0100
Subject: [PATCH] Add explanation for characteristic curves
---
main.tex | 44 +++++++++++++++++++++++++++++++++++++-------
1 file changed, 37 insertions(+), 7 deletions(-)
diff --git a/main.tex b/main.tex
index 6b05075..edfd643 100644
--- a/main.tex
+++ b/main.tex
@@ -118,6 +118,22 @@ Considering the stability conditions with two parameters $\alpha, \beta$ on
Picard rank 1 surfaces.
We can draw 2 characteristic curves for any given Chern character $v$ with
$\Delta(v) \geq 0$ and positive rank.
+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 $i=2$).
+
+
+\minorheading{Relevance of $\chern_1^{\alpha, \beta}=0$ vertical line}
+
+By definition of the first tilt $\firsttilt\beta$, objects of Chern character
+$v$ can only be in $\firsttilt\beta$ on the left of the vertical line, and
+objects of Chern character $-v$ can only be in $\firsttilt\beta$ on the right.
+In fact, if there is a Gieseker semistable coherent sheaf $E$ of Chern character $v$,
+then $E \in \firsttilt\beta$ if and only if $\beta<\mu(E)$ (left of the vertical
+line), and $E[1] \in \firsttilt\beta$ if and only if $\beta\geq\mu(E)$.
+Because of this, when using these characteristic curves, we shall only
+consider positive rank, as negative rank objects are implicitly considered on
+the right hand side of the vertical line.
\begin{sagesilent}
def charact_curves(v):
@@ -145,12 +161,6 @@ v1 = Chern_Char(3, 2, -2)
v2 = Chern_Char(3, 2, 2/3)
\end{sagesilent}
-%\begin{figure}
-% \centering
-% \sageplot[width=\textwidth]{charact_curves(v1)}
-% \caption{}
-% \label{fig:charact_curves_vis}
-%\end{figure}
\begin{figure}
\centering
\begin{subfigure}{.49\textwidth}
@@ -163,7 +173,9 @@ v2 = Chern_Char(3, 2, 2/3)
\begin{subfigure}{.49\textwidth}
\centering
\sageplot[width=\textwidth]{charact_curves(v2)}
- \caption{$\Delta(v)=0$}
+ \caption{
+ $\Delta(v)=0$: hyperbola collapses
+ }
\label{fig:charact_curves_vis_bgmlv0}
\end{subfigure}
\caption{
@@ -174,6 +186,24 @@ v2 = Chern_Char(3, 2, 2/3)
\end{figure}
+\minorheading{Relevance of $\chern_2^{\alpha, \beta}=0$ hyperbola}
+
+Since $\chern_2^{\alpha, \beta}$ is the numerator of the tilt slope
+$\nu_{\alpha, \beta}$.
+The second characteristic curve, where this is 0, firstly divides the
+$\alpha$-$\beta$-half-plane into regions where the signs 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 the hyperbola $\chern_2^{\alpha, \beta}(v)=0$, then
+$\mu_{\alpha,\beta}(v)=0$ and for any $u$, $u$ is a pseudo-semistabilizer of $v$
+iff $\mu_{\alpha,\beta}(u)=0$, and hence $\chern_2^{\alpha, \beta}(u)=0$.
+
+
+
+
+
+
\begin{sagesilent}
v = Chern_Char(3, 2, -2)
u = Chern_Char(1, 0, 0)
--
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