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Commit 0254d611 authored by Luke Naylor's avatar Luke Naylor
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Sort out section about bounds on r in problem 1

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......@@ -229,6 +229,7 @@ was implicitly happening before in the proof of Theorem
\end{proof}
\subsection{Bounds on \texorpdfstring{`$d$'}{d}-values for Solutions of Problems}
\label{subsec:bounds-on-d}
Let $v$ be a Chern character with $\Delta(v)\geq 0$,
$\chern_0(v)>0$, or $\chern_0(v)=0$ and $\chern_1(v)>0$,
......@@ -603,7 +604,7 @@ the lower bound on $d$ is equal to one of the upper bounds on $d$
\begin{theorem}[Problem \ref{problem:problem-statement-1} upper Bound on $r$]
\label{lem:prob1:r_bound}
Let $u$ be a solution to problem \ref{problem:problem-statement-1}
Let $u$ be a solution to Problem \ref{problem:problem-statement-1}
and $q\coloneqq\chern_1^{B}(u)$.
Then $r\coloneqq\chern_0(u)$ is bounded above by the following expression:
\begin{equation}
......@@ -612,9 +613,15 @@ the lower bound on $d$ is equal to one of the upper bounds on $d$
\end{theorem}
\begin{proof}
Recall that $d\coloneqq\chern_2(u)$ has two upper bounds in terms of $r$: in
Lemma \ref{lem:fixed-q-semistabs-criterion} gives us that any solution $u$
must be of the form in Equation \ref{eqn:u-coords} and
satisfy Equations \ref{lem:eqn:cond-for-fixed-q} as well as the three
conditions $\chern^P_2(u)>0$, $\Delta(u) \geq 0$, and $\Delta(v-u) \geq 0$.
Subsection \ref{subsec:bounds-on-d} equates these latter three conditions
(provided Equations \ref{lem:eqn:cond-for-fixed-q})
to upper bounds on $d$ given by
Equations \ref{eqn:prob1:bgmlv2} and \ref{eqn:prob1:bgmlv3};
and one lower bound: in Equation \ref{eqn:prob1:radiuscond}.
and one lower bound given by Equation \ref{eqn:prob1:radiuscond}.
Solving for the lower bound in Equation \ref{eqn:prob1:radiuscond} being
less than the upper bound in Equation \ref{eqn:prob1:bgmlv2} yields:
......@@ -622,18 +629,20 @@ the lower bound on $d$ is equal to one of the upper bounds on $d$
r<\sage{problem1.positive_intersection_bgmlv2}
\end{equation}
\noindent
Similarly, but with the upper bound in Equation \ref{eqn:prob1:bgmlv3}, gives:
\begin{equation}
r<\sage{problem1.positive_intersection_bgmlv3}
\end{equation}
\noindent
Therefore, $r$ is bounded above by the minimum of these two expressions which
can then be factored into the expression given in the Lemma.
\end{proof}
The above Lemma \ref{lem:prob1:r_bound} gives an upper bound on $r$ in terms of $q$.
But given that $0 \leq q \leq \chern_1^{B}(v)$, we can take the maximum of this
But given that $0 < q < \chern_1^{\beta_0}(v)$, we can take the maximum of this
bound, over $q$ in this range, to get a simpler (but weaker) bound in the
following Lemma \ref{lem:prob1:convenient_r_bound}.
......
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