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Commit b642eedf authored by Luke Naylor's avatar Luke Naylor
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Complete one outstanding TODO

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......@@ -157,12 +157,11 @@ are equivalent to the following more numerical conditions:
\item $\beta(P)<\mu(u)<\mu(v)$, i.e. $V_u$ is strictly between $P$ and $V_v$.
\label{lem:ps-wall-num-test:num-cond-slope}
\item $\chern_1^{\beta(P)}(u) < \chern_1^{\beta(P)}(v)$
\item $\Delta(v-u) \geq 0$
\item $\Delta(v-u) \geq 0$
\item $\chern_2^{P}(u)>0$
\end{enumerate}
\end{lemma}
\noindent\textbf{\color{red} TODO ADJUST TO SLIGHTLY STRONGER CONDITION 3}
\begin{proof}[Proof for $\chern_0(v)>0$ case.]
Let $u,v$ be Chern characters with
$\Delta(u),\Delta(v) \geq 0$, and $v$ has positive rank.
......@@ -170,9 +169,23 @@ $\Delta(u),\Delta(v) \geq 0$, and $v$ has positive rank.
For the forwards implication, assume that the suppositions of the Lemma are
satisfied. Let $Q$ be the point on $\Theta_v^-$ (above $P$) where $u$ is a
pseudo-semistabiliser of $v$.
Firstly, consequence 3 is part of the definition for $u$ being a
Firstly, consequence 4 is part of the definition for $u$ being a
pseudo-semistabiliser at a point with same $\beta$ value of $P$ (since the
pseudo-wall surrounds $P$).
Also, $u$ is a pseudo-semistabiliser along the pseudo-wall
which surround $P$, so by definition of pseudo-semistabilisers,
${
0 \leq \chern_1^\beta(u) \leq \chern_1^\beta(v)
}$
for all $(\alpha, \beta)$ on the pseudo-wall.
In particular, this holds for $\beta$ in a neighbourhood of $\beta(P)$,
so we can conclude
${
0 < \chern_1^{\beta(P)}(u) < \chern_1^{\beta(P)}(v)
}$
(consequence 5).
Notice that $0 < \chern_1^{\beta(P)}(u)$
follows from consequences 1 and 2, this is why it is not included in consequence 5.
If $u$ were to have 0 rank, it's tilt slope would be decreasing as $\beta$
increases, contradicting supposition b. So $u$ must have strictly non-zero rank,
and we can consider it's characteristic curves (or that of $-u$ in case of
......@@ -218,8 +231,8 @@ have positive rank (consequence 1)
to ensure that $\chern_1^{\beta(P)} \geq 0$ (since the pseudo-wall passed over
$P$).
Furthermore, $P$ being on the left of $V_u$ implies
$\chern_1^{\beta{P}}(u) \geq 0$,
and therefore $\chern_2^{P}(u) > 0$ (consequence 4) to satisfy supposition b.
$\chern_1^{\beta(P)}(u) > 0$,
and therefore $\chern_2^{P}(u) > 0$ (consequence 5) to satisfy supposition b.
Next considering the way the hyperbolae intersect, we must have $\Theta_u^-$ taking a
base-point to the right $\Theta_v$, but then, further up, crossing over to the
left side. The latter fact implies that the assymptote for $\Theta_u^-$ must be
......@@ -229,9 +242,9 @@ must have $\mu(u)<\mu(v)$ (second part of consequence 2),
that is, $V_u$ is strictly to the left of $V_v$.
Conversely, suppose that the consequences 1-4 are satisfied. Consequence 2
Conversely, suppose that the consequences 1-5 are satisfied. Consequence 2
implies that the assymptote for $\Theta_u^-$ is to the left of $\Theta_v^-$.
Consequence 4, along with $\beta(P)<\mu(u)$, implies that $P$ must be in the
Consequences 4 and 5, along with $\beta(P)<\mu(u)$, implies that $P$ must be in the
region left of $\Theta_u^-$. These two facts imply that $\Theta_u^-$ is to the
right of $\Theta_v^-$ at $\alpha=\alpha(P)$, but crosses to the left side as
$\alpha \to +\infty$, intersection at some point $Q$ above $P$.
......
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