From 7672ddc0db139e59d2a9f620d699c197048d6045 Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Tue, 14 Jan 2025 23:37:06 +0000
Subject: [PATCH] Add pic(T)=Zl condition to all ppas
---
tex/bounds-on-semistabilisers.tex | 1 +
tex/computing-solutions.tex | 4 +++-
tex/setting-and-problems.tex | 3 ++-
3 files changed, 6 insertions(+), 2 deletions(-)
diff --git a/tex/bounds-on-semistabilisers.tex b/tex/bounds-on-semistabilisers.tex
index cca0478..5998a9d 100644
--- a/tex/bounds-on-semistabilisers.tex
+++ b/tex/bounds-on-semistabilisers.tex
@@ -1147,6 +1147,7 @@ lot when $m$ is small.
\begin{corollary}[Third bound on $r$ on $\PP^2$ and principally polarised abelian surfaces]
\label{cor:rmax_with_eps1}
Suppose we are working over $\PP^2$ or a principally polarised abelian surface
+ with $\mathrm{Pic}(\ppas) = \ZZ\ell$
(or any other surfaces with $m=\ell^2=1$ or $2$).
Let $v$ be a fixed Chern character, with $\beta_{-}\coloneqq\beta_{-}(v)=\frac{a_v}{n}$
rational and expressed in lowest terms.
diff --git a/tex/computing-solutions.tex b/tex/computing-solutions.tex
index f6e2aa7..64a1c71 100644
--- a/tex/computing-solutions.tex
+++ b/tex/computing-solutions.tex
@@ -261,7 +261,9 @@ the latter having already been discussed in that same section.
In order to see a difference between the different patches, we use the Chern
character $v=(45,54\ell,-41\frac{\ell^2}{2})$ for a smooth projective surface $X$
with a generator $\ell$ for $NS(X)$ such that $\ell^2=1$ or 2 (such as a
-principally polarised surface or $\mathbb{P}^2$).
+principally polarised surface
+with $\mathrm{Pic}(\ppas) = \ZZ\ell$
+or $\mathbb{P}^2$).
This example was chosen for the large rank $\chern_0(v)=45$,
but also the large Bogomolov discriminant $\Delta(v)=4761\ell^2$, which are both
indicators of the size of the bounds on the pseudo-semistabiliser ranks.
diff --git a/tex/setting-and-problems.tex b/tex/setting-and-problems.tex
index 2844832..b24d44e 100644
--- a/tex/setting-and-problems.tex
+++ b/tex/setting-and-problems.tex
@@ -33,7 +33,8 @@ affect the results.
We could introduce a slightly stronger definition including an extra condition on $e$
in terms of $r$ and $c$ to ensure that $u$ could arise from integral Chern classes.
However, this will not affect finiteness questions considered later and this also
- condition turns out to be vacuous for principally polarised abelian surfaces.
+ condition turns out to be vacuous for principally polarised abelian surfaces
+ with $\mathrm{Pic}(\ppas) = \ZZ\ell$.
\end{remark}
\begin{remark}
Note $u$ does not need to be a Chern character of an actual sub-object of some
--
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