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Commit 24056de0 authored by Luke Naylor's avatar Luke Naylor
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Adjust tighter bounds section to more general surface

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......@@ -838,7 +838,6 @@ As opposed to only eliminating possible values of $\chern_0(E)$ for which all
corresponding $\chern_1^{\beta}(E)$ fail one of the inequalities (which is what
was implicitly happening before).
% NOTE FUTURE: surface specialization
First, let us fix a Chern character for $F$, and some pseudo-semistabilizer
$u$ which is a solution to problem
......@@ -850,10 +849,10 @@ Take $\beta = \beta(P)$ where $P\in\Theta_v^-$ is the choice made in problem
\begin{align}
\chern(F) =\vcentcolon\: v \:=& \:(R,C\ell,D\ell^2)
&& \text{where $R,C,2D\in \ZZ$}
&& \text{where $R,C\in \ZZ$ and $D\in \frac{1}{\lcm(m,2)}\ZZ$}
\\
u \coloneqq& \:(r,c\ell,d\ell^2)
&& \text{where $r,c,2d\in \ZZ$}
&& \text{where $r,c\in \ZZ$ and $d\in \frac{1}{\lcm(m,2)}\ZZ$}
\end{align}
......@@ -1120,10 +1119,10 @@ for the bounds on $d$ in terms of $r$ is illustrated in figure
(\ref{fig:d_bounds_xmpl_gnrc_q}).
The question of whether there are pseudo-destabilizers of arbitrarily large
rank, in the context of the graph, comes down to whether there are points
$(r,d) \in \ZZ \oplus \frac{1}{2} \ZZ$ (with large $r$)
$(r,d) \in \ZZ \oplus \frac{1}{\lcm(m,2)} \ZZ$ (with large $r$)
% TODO have a proper definition for pseudo-destabilizers/walls
that fit above the yellow line (ensuring positive radius of wall) but below the
blue and green (ensuring $\Delta(E), \Delta(G) > 0$).
blue and green (ensuring $\Delta(u), \Delta(v-u) > 0$).
These lines have the same assymptote at $r \to \infty$
(eqns \ref{eqn:bgmlv2_d_bound_betamin},
\ref{eqn:bgmlv3_d_bound_betamin},
......
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