From bd0d95ecceec50ce3b159d0acbc226e34ccb88d4 Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Sun, 7 Jul 2024 23:49:56 +0100
Subject: [PATCH] Iron out final theorem bound of chapter
---
tex/bounds-on-semistabilisers.tex | 52 ++++++++++++++++++-------------
1 file changed, 30 insertions(+), 22 deletions(-)
diff --git a/tex/bounds-on-semistabilisers.tex b/tex/bounds-on-semistabilisers.tex
index a0bea7b..781bdf7 100644
--- a/tex/bounds-on-semistabilisers.tex
+++ b/tex/bounds-on-semistabilisers.tex
@@ -730,9 +730,9 @@ proof of Theorem
\begin{sagesilent}
-from plots_and_expressions import main_theorem1
+from plots_and_expressions import main_theorem1, betamin_subs
\end{sagesilent}
-\begin{theorem}[Bound on $r$ \#1]
+\begin{theorem}[Bound on $r$ \#1 for Problem \ref{problem:problem-statement-2}]
\label{thm:rmax_with_uniform_eps}
Let $X$ be a smooth projective surface with Picard rank 1 and choice of ample
line bundle $L$ with $c_1(L)$ generating $\neronseveri(X)$ and
@@ -747,8 +747,8 @@ from plots_and_expressions import main_theorem1
\begin{align*}
\min
\left(
- \sage{main_theorem1.r_upper_bound1}, \:\:
- \sage{main_theorem1.r_upper_bound2}
+ \sage{main_theorem1.r_upper_bound1.subs(betamin_subs)}, \:\:
+ \sage{main_theorem1.r_upper_bound2.subs(betamin_subs)}
\right)
\end{align*}
\noindent
@@ -760,8 +760,8 @@ Both $d$ and the lower bound in
(Equation \ref{eqn:positive_rad_condition_in_terms_of_q_beta})
are elements of $\frac{1}{\lcm(m,2n^2)}\ZZ$.
So, if any of the two upper bounds on $d$ come to within
-$\frac{1}{\lcm(m,2n^2)}$ of this lower bound, then there are no solutions for
-$d$.
+$\epsilon_v \coloneqq \frac{1}{\lcm(m,2n^2)}$ of this lower bound,
+then there are no solutions for $d$.
Hence any corresponding $r$ cannot be a rank of a
pseudo-semistabiliser for $v$.
@@ -778,8 +778,8 @@ assymptote_gap_condition1, assymptote_gap_condition2, k
\begin{align}
- &\sage{assymptote_gap_condition1.subs(k==1)} \\
- &\sage{assymptote_gap_condition2.subs(k==1)}
+ \epsilon_v =&\sage{assymptote_gap_condition1.subs(k==1)} \\
+ \epsilon_v =&\sage{assymptote_gap_condition2.subs(k==1)}
\end{align}
\noindent
@@ -805,7 +805,7 @@ This is equivalent to:
from plots_and_expressions import q_sol, bgmlv_v, psi
\end{sagesilent}
-\begin{corollary}[Bound on $r$ \#2]
+\begin{corollary}[Global bound on $r$ \#2 for Problem \ref{problem:problem-statement-2}]
\label{cor:direct_rmax_with_uniform_eps}
Let $X$ be a smooth projective surface with Picard rank 1 and choice of ample
line bundle $L$ with $c_1(L)$ generating $\neronseveri(X)$ and
@@ -1042,29 +1042,37 @@ $\epsilon_{v,q}\geq\epsilon_v$, with equality when $k_{v,q}=1$.
\begin{sagesilent}
from plots_and_expressions import main_theorem2
\end{sagesilent}
-\begin{theorem}[Bound on $r$ \#3]
+\begin{theorem}[Bound on $r$ \#3 for Problem \ref{problem:problem-statement-2}]
\label{thm:rmax_with_eps1}
- Let $v$ be a fixed Chern character, with $\frac{a_v}{n}=\beta\coloneqq\beta(v)$
- rational and expressed in lowest terms.
- Then the ranks $r$ of the pseudo-semistabilisers $u$ for $v$ with,
- which are solutions to problem \ref{problem:problem-statement-2},
- $\chern_1^\beta(u) = q = \frac{b_q}{n}$
- are bounded above by the following expression:
+ Let $X$ be a smooth projective surface with Picard rank 1 and choice of ample
+ line bundle $L$ with $c_1(L)$ generating $\neronseveri(X)$ and
+ $m\coloneqq\ell^2$.
+ Let $v$ be a fixed Chern character on this surface with positive rank
+ (or rank 0 and $c_1(v)>0$), and $\Delta(v)\geq 0$.
+ Then the ranks of the pseudo-semistabilisers $u$ for $v$,
+ which are solutions to Problem \ref{problem:problem-statement-2},
+ with $\chern_1^{\beta_{-}(v)}(u) = q$
+ are bounded above by the following expression.
\begin{align*}
\min
\left(
- \sage{main_theorem2.r_upper_bound1}, \:\:
- \sage{main_theorem2.r_upper_bound2}
+ \sage{main_theorem2.r_upper_bound1.subs(betamin_subs)}, \:\:
+ \sage{main_theorem2.r_upper_bound2.subs(betamin_subs)}
\right)
\end{align*}
- Where $k_{v,q}$ is defined as in definition/Lemma \ref{lemdfn:epsilon_q},
+ Where $k_{v,q}$ is defined as in Definition/Lemma \ref{lemdfn:epsilon_q},
and $R = \chern_0(v)$
-
- Furthermore, if $\aa \not= 0$ then
- $r \equiv \aa^{-1}b_q \pmod{n}$.
\end{theorem}
+\begin{proof}
+ Following the same proof as Theorem \ref{thm:rmax_with_uniform_eps},
+ $\epsilon_{v,q} = \frac{k_{v,q}}{\lcm(m, 2n^2)}$ can be used instead of
+ $\epsilon_{v} = \frac{1}{\lcm(m, 2n^2)}$ as it satisfies the same required
+ property, as per Definition/Lemma \ref{lemdfn:epsilon_q}.
+
+\end{proof}
+
Although the general form of this bound is quite complicated, it does simplify a
lot when $m$ is small.
--
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