From b3f3fc6acdef8ead3c0c8423061c47d9418979df Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Wed, 19 Jul 2023 16:02:21 +0100
Subject: [PATCH] Specialize Picard rank 1 to ppas and p2
---
main.tex | 14 +++++++++++---
1 file changed, 11 insertions(+), 3 deletions(-)
diff --git a/main.tex b/main.tex
index e64982b..5e0cd80 100644
--- a/main.tex
+++ b/main.tex
@@ -93,7 +93,7 @@ stability space. These are where there is some stable object $F$ of $v$ which
has a subobject who's slope overtakes the slope of $v$, making $F$ unstable
after crossing the wall.
-% NOTE: SURFACE SPACIALIZATION
+% NOTE: SURFACE SPECIALIZATION
% (come back to these when adjusting to general Picard rank 1)
In this document we concentrate on two surfaces: Principally polarized abelian
surfaces and the projective surface $\PP^2$. Although this can be generalised
@@ -154,8 +154,13 @@ on $\chern_0(E)$ of potential destabilizers $E$ of $F$.
\section{Characteristic Curves of Stability Conditions Associated to Chern
Characters}
+Throughout this article, as noted in the introduction, we will be exclusively
+working over one of the following two surfaces: principally polarized abelian
+surfaces and $\PP^2$.
+
\begin{dfn}[Pseudo-semistabilizers]
- Given a Chern Character $v$ on a Picard rank 1 surface, and a given stability
+% NOTE: SURFACE SPECIALIZATION
+ Given a Chern Character $v$, and a given stability
condition $\sigma_{\alpha,\beta}$,
a pseudo-semistabilizing $u$ is a `potential' Chern character:
\[
@@ -186,13 +191,15 @@ pseudo-semistabilizers are even Chern characters of actual elements of
$\bddderived(X)$, some other sources may have this extra restriction too.
\begin{lemma}[ Sanity check for Pseudo-semistabilizers ]
- Given a Picard rank 1 surface, and a given stability
+% NOTE: SURFACE SPECIALIZATION
+ Given a stability
condition $\sigma_{\alpha,\beta}$,
if $E\hookrightarrow F\twoheadrightarrow G$ is a semistabilizing sequence in
$\firsttilt\beta$ for $F$.
Then $\chern(E)$ is a pseudo-semistabilizer of $\chern(F)$
\end{lemma}
+% NOTE: SURFACE SPECIALIZATION
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
@@ -658,6 +665,7 @@ rank that appears turns out to be $\sage{extravagant.actual_rmax}$.
\section{B.Schmidt's Wall Finding Method}
+% NOTE: SURFACE SPECIALIZATION
The proof for the previous theorem was hinted at in
\cite{SchmidtBenjamin2020Bsot}, but the value appears explicitly in
\cite{SchmidtGithub2020}. The latter reference is a SageMath \cite{sagemath}
--
GitLab