From c5bd40565d5476d8982d6e7e9913cfa4946d679d Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Tue, 30 May 2023 15:42:14 +0100
Subject: [PATCH] Refine statement of theorem: bound on r #3
---
main.tex | 15 ++++++++++-----
1 file changed, 10 insertions(+), 5 deletions(-)
diff --git a/main.tex b/main.tex
index d2b3567..8df39d3 100644
--- a/main.tex
+++ b/main.tex
@@ -1188,10 +1188,11 @@ which also guarantees that the gap $\frac{k}{2mn^2}$ is at least $\epsilon_{q,2}
\begin{theorem}[Bound on $r$ \#3]
\label{thm:rmax_with_eps1}
- Let $v = (R,C,D)$ be a fixed Chern character. Then the ranks of the
- pseudo-semistabilizers for $v$ with
- $\chern_1^\beta = q = \frac{a_q}{n}$
- are bounded above by the following expression (with $i=1$ or 2).
+ Let $v$ be a fixed Chern character, with $\frac{a_F}{n}=\beta:=\beta(v)$
+ rational and expressed in lowest terms.
+ Then the ranks $r$ of the pseudo-semistabilizers $u$ for $v$ with
+ $\chern_1^\beta(u) = q = \frac{b_q}{n}$
+ are bounded above by the following expression (with $i=1$ or $2$).
\begin{sagesilent}
eps_k_i_subs = Delta == (2*m*n^2)/delta
@@ -1208,7 +1209,11 @@ eps_k_i_subs = Delta == (2*m*n^2)/delta
\right)
\end{align*}
\egroup
- Where $\epsilon_{q,i}$ is defined as in definition/lemma \ref{lemdfn:epsilon_q}.
+ Where $k_{q,i}$ is defined as in definition/lemma \ref{lemdfn:epsilon_q},
+ and $R = \chern_0(v)$
+
+ Furthermore, if $\aa \not= 0$ then
+ $r \equiv \aa^{-1}b_q (\mod n)$.
\end{theorem}
--
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