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luke naylor latex documents
research
Max Destabilizer Rank
Commits
24056de0
Commit
24056de0
authored
1 year ago
by
Luke Naylor
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Adjust tighter bounds section to more general surface
parent
a00becb5
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main.tex
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24056de0
...
...
@@ -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
,
2
D
\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
,
2
d
\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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