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luke naylor latex documents
research
Max Destabilizer Rank
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0254d611
Commit
0254d611
authored
8 months ago
by
Luke Naylor
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Sort out section about bounds on r in problem 1
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8 months ago
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tex/bounds-on-semistabilisers.tex
+13
-4
13 additions, 4 deletions
tex/bounds-on-semistabilisers.tex
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and
4 deletions
tex/bounds-on-semistabilisers.tex
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View file @
0254d611
...
...
@@ -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
p
roblem
\ref
{
problem:problem-statement-1
}
Let
$
u
$
be a solution to
P
roblem
\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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