From 6f3c280427e03e269290efcc33a78ec7422fa99b Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Tue, 13 Jun 2023 17:35:06 +0100
Subject: [PATCH] Start rewording characteristic curves

---
 main.tex | 53 +++++++++++++++++++++++------------------------------
 1 file changed, 23 insertions(+), 30 deletions(-)

diff --git a/main.tex b/main.tex
index d6fe72d..6d09e13 100644
--- a/main.tex
+++ b/main.tex
@@ -151,17 +151,18 @@ as well as the restrictions on $v$:
 \end{fact}
 
 
-\minorheading{Relevance of $\chern_1^{\alpha, \beta}=0$ vertical line}
+\minorheading{Relevance of the vertical line $V_v$}
 
 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.
+$v$ 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
+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
+$V_v$), and $E[1] \in \firsttilt\beta$ if and only if $\beta\geq\mu(E)$.
+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 $V_v$.
 
 \begin{sagesilent}
 def charact_curves(v):
@@ -216,37 +217,29 @@ v2 = Chern_Char(3, 2, 2/3)
 \end{figure}
 
 
-\minorheading{Relevance of $\chern_2^{\alpha, \beta}=0$ hyperbola}
+\minorheading{Relevance of the hyperbola $\Theta_v$}
 
 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
-for any $u$, $u$ is a pseudo-semistabilizer of $v$
-iff $\mu_{\alpha,\beta}(u)=0$, 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 the pseudo-wall where $u$ pseudo-semistabilizes $v$.
+$\nu_{\alpha, \beta}$. The curve $\Theta_v$, where this is 0, firstly divides the
+$(\alpha$-$\beta)$-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
+$\nu_{\alpha,\beta}(u)=0$, 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
+the pseudo-wall where $u$ pseudo-semistabilizes $v$. This is done by finding the
+intersection of $\Theta_v$ and $\Theta_u$.
 %TODO ref forwards
 
-Commenting on the geometry of the hyperbola, it always has left and right
-branches (as opposed to up and down), or degenerates to 2 lines. This is a
-consequence of $\Delta(v)\geq 0$. Furthermore the assymptotes are angled at $\pm
-45^\circ$, crossing through the base of the first characteristic curve
-$\chern_1^{\alpha,\beta}=0$ (vertical line).
-
 \subsection{Bertram's nested wall theorem}
 
 Although Bertram's nested wall theorem can be proved more directly, it's also
 important for the content of this document to understand the connection with
 these characteristic curves.
 Emanuele Macri noticed in (TODO ref) that any circular wall of $v$ reaches a critical
-point on the second critical curve for $v$ ($\chern_2^{\alpha, \beta}(v)=0$),
-this is a consequence of
+point on $\Theta_v$ (TODO ref). This is a consequence of
 $\frac{\delta}{\delta\beta} \chern_2^{\alpha,\beta} = -\chern_1^{\alpha,\beta}$.
 This fact, along with the hindsight knowledge that non-vertical walls are
 circles with centers on the $\beta$-axis, gives an alternative view to see that
-- 
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