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luke naylor latex documents
research
Max Destabilizer Rank
Commits
740b76c2
Commit
740b76c2
authored
8 months ago
by
Luke Naylor
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Iron out work up to dfn/lemma of e_{v,q}
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9626ee07
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tex/bounds-on-semistabilisers.tex
+47
-59
47 additions, 59 deletions
tex/bounds-on-semistabilisers.tex
with
47 additions
and
59 deletions
tex/bounds-on-semistabilisers.tex
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−
59
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740b76c2
...
...
@@ -141,7 +141,7 @@ $\chern_1^{\sage{extravagant.betaminus}}(F) = \sage{extravagant.twisted.ch[1]}$.
Using the above Theorem
\ref
{
thm:loose-bound-on-r
}
, we get that the ranks of
tilt semistabilisers for
$
v
$
are bounded above by
$
\sage
{
extravagant.loose
_
bound
}$
.
However, when computing all tilt semistabilisers for
$
v
$
on
$
\PP
^
2
$
, the maximum
rank that appears turns out to be
$
\sage
{
extravagant.actual
_
rmax
}$
.
rank that appears turns out to be
$
\sage
{
round
(
extravagant.actual
_
rmax
,
1
)
}$
.
\end{example}
...
...
@@ -811,49 +811,54 @@ from plots_and_expressions import q_sol, bgmlv_v, psi
line bundle
$
L
$
with
$
c
_
1
(
L
)
$
generating
$
\neronseveri
(
X
)
$
and
$
m
\coloneqq\ell
^
2
$
.
Let
$
v
$
be a fixed Chern character on this surface and
$
R
\coloneqq\chern
_
0
(
v
)
\leq
\frac
{
1
}{
2
}
\lcm
(
m,
2
n
^
2
)
{
\chern
_
1
^{
\beta
_{
-
}
(
v
)
}
(
v
)
}^
2
$
.
Then the ranks of the pseudo-semistabilisers
for
$
v
$
,
$
R
\coloneqq\chern
_
0
(
v
)
$
.
Then the ranks of the pseudo-semistabilisers
$
u
$
of
$
v
$
,
which are solutions to problem
\ref
{
problem:problem-statement-2
}
,
are bounded above by the following expression.
\begin{equation*}
\sage
{
main
_
theorem1.corollary
_
r
_
bound
}
\end{equation*}
are bounded above as follows.
\
noindent
If
$
R >
\frac
{
1
}{
2
}
\lcm
(
m,
2
n
^
2
)
{
\chern
_
1
^{
\beta
_{
-
}
(
v
)
}
(
v
)
}^
2
$
,
then these ranks of pseudo-semistabilisers can instead bounded above by the
following.
\begin{equation*
}
\frac
{
\Delta
(v)
\lcm
(m,2n
^
2)
}{
2m
}
\end{
equatio
n*}
\
begin{align*}
r
&
\leq
\sage
{
main
_
theorem1.corollary
_
r
_
bound
}
&
\text
{
if
}
R <
\frac
{
\Delta
(v)
\lcm
(m,2n
^
2)
}{
2m
}
\\
r
&
\leq
\frac
{
\Delta
(v)
\lcm
(m,2n
^
2)
}{
2m
}
&
\text
{
if
}
R
\geq
\frac
{
\Delta
(v)
\lcm
(m,2n
^
2)
}{
2m
}
\end{
alig
n*}
\end{corollary}
\begin{proof}
The ranks of the pseudo-semistabilisers for
$
v
$
are bounded above by the
maximum over
$
q
\in
[
0
,
\chern
_
1
^{
\beta
}
(
F
)]
$
of the expression in Theorem
maximum over
$
q
\in
[
0
,
\chern
_
1
^{
\beta
_{
-
}
}
(
v
)]
$
of the expression in Theorem
\ref
{
thm:rmax
_
with
_
uniform
_
eps
}
.
Noticing that the expression is a maximum of two quadratic functions in
$
q
$
:
Noticing that the expression is a maximum of two quadratic functions in
$
q
$
(
$
\beta
_
0
=
\beta
_{
-
}
(
v
)
$
in this context):
\begin{equation*}
f
_
1(q)
\coloneqq\sage
{
main
_
theorem1.r
_
upper
_
bound1
}
\qquad
f
_
2(q)
\coloneqq\sage
{
main
_
theorem1.r
_
upper
_
bound2
}
\end{equation*}
These have their minimums at
$
q
=
0
$
and
$
q
=
\chern
_
1
^{
\beta
}
(
v
)
$
respectively.
It suffices to find their intersection in
$
q
\in
[
0
,
\chern
_
1
^{
\beta
}
(
v
)]
$
, if it exists,
and evaluating on of the
$
f
_
i
$
there.
The intersection exists, provided that
$
f
_
1
(
\chern
_
1
^{
\beta
}
(
v
))
\geq
f
_
2
(
\chern
_
1
^{
\beta
}
(
v
))=
R
$
,
or equivalently,
$
R
\leq
\frac
{
1
}{
2
}
\lcm
(
m,
2
n
^
2
)
{
\chern
_
1
^{
\beta
}
(
F
)
}^
2
$
.
If this is not the case, then
$
\min
(
f
_
1
, f
_
2
)
$
reaches its maximum over the domain
$
[
0
,
\chern
_
1
^{
\beta
_{
-
}}
(
v
)]
$
at
$
q
=
\chern
_
1
^{
\beta
_{
-
}}
(
v
)
$
, and so is
bounded by
$
f
_
1
(
\chern
_
1
^{
\beta
_{
-
}
(
v
)
}
)
$
.
In the case where
$
R
\leq
\frac
{
1
}{
2
}
\lcm
(
m,
2
n
^
2
)
{
\chern
_
1
^{
\beta
}
(
F
)
}^
2
$
,
These have their minimums at
$
q
=
0
$
and
$
q
=
\chern
_
1
^{
\beta
_{
-
}}
(
v
)
$
respectively,
with values 0 and
$
R>
0
$
respectively.
So provided that
$
f
_
2
\left
(
\chern
^{
\beta
_{
-
}}_
1
(
v
)
\right
)
< f
_
1
\left
(
\chern
^{
\beta
_{
-
}}_
1
(
v
)
\right
)
$
,
the maximum is achieved at their intersection.
Otherwise, the maximum is achieved at
$
\chern
^{
\beta
_{
-
}}_
1
(
v
)
$
.
So we can say that
\begin{align*}
r
&
\leq
f
_{
1
}
(q
_{
\mathrm
{
max
}}
)
&
\text
{
if
}
f
_
2
\left
(
\chern
^{
\beta
_{
-
}}_
1(v)
\right
) <
f
_
1
\left
(
\chern
^{
\beta
_{
-
}}_
1(v)
\right
)
\\
&&
\text
{
where
$
q
_{
\mathrm
{
max
}}$
is the
$
q
$
-value where the
$
f
_
i
$
intersect
}
\\
r
&
\leq
f
_
1
\left
(
\chern
^{
\beta
_{
-
}}
(v)
\right
)
&
\text
{
if
}
f
_
2
\left
(
\chern
^{
\beta
_{
-
}}_
1(v)
\right
)
\geq
f
_
1
\left
(
\chern
^{
\beta
_{
-
}}_
1(v)
\right
)
\end{align*}
\noindent
In the first case,
solving for
$
f
_
1
(
q
)=
f
_
2
(
q
)
$
yields
\begin{equation*}
q=
\sage
{
q
_
sol.expand()
}
...
...
@@ -862,8 +867,10 @@ And evaluating $f_1$ at this $q$-value gives:
\begin{equation*}
\sage
{
main
_
theorem1.corollary
_
intermediate
}
\end{equation*}
Finally, noting that
$
\Delta
(
v
)=(
\chern
_
1
^{
\beta
_{
-
}
(
v
)
}
(
v
))
^
2
\ell
^
2
$
, we get the bound as
stated in the corollary.
\noindent
Finally, noting that
$
\Delta
(
v
)=
\left
(
\chern
_
1
^{
\beta
_{
-
}
(
v
)
}
(
v
)
\right
)
^
2
\ell
^
2
$
,
we get the bounds as stated in the statement of the Corollary.
\end{proof}
...
...
@@ -876,7 +883,8 @@ giving $n=\sage{recurring.n}$.
Using the above corollary
\ref
{
cor:direct
_
rmax
_
with
_
uniform
_
eps
}
, we get that
the ranks of tilt semistabilisers for
$
v
$
are bounded above by
$
\sage
{
recurring.corrolary
_
bound
}
\approx
\sage
{
float
(
recurring.corrolary
_
bound
)
}$
,
$
\sage
{
recurring.corrolary
_
bound
}
\approx
\sage
{
round
(
float
(
recurring.corrolary
_
bound
)
,
1
)
}$
,
which is much closer to real maximum 25 than the original bound 144.
\end{example}
...
...
@@ -889,7 +897,8 @@ giving $n=\sage{extravagant.n}$.
Using the above corollary
\ref
{
cor:direct
_
rmax
_
with
_
uniform
_
eps
}
, we get that
the ranks of tilt semistabilisers for
$
v
$
are bounded above by
$
\sage
{
extravagant.corrolary
_
bound
}
\approx
\sage
{
float
(
extravagant.corrolary
_
bound
)
}$
,
$
\sage
{
extravagant.corrolary
_
bound
}
\approx
\sage
{
round
(
float
(
extravagant.corrolary
_
bound
)
,
1
)
}$
,
which is much closer to real maximum
$
\sage
{
extravagant.actual
_
rmax
}$
than the
original bound 215296.
\end{example}
...
...
@@ -913,28 +922,7 @@ $q$, then iterate through values of $r$ within the bounds (dependent on $q$),
which would then determine
$
c
$
, and then find the corresponding possible values
for
$
d
$
.
Firstly, we only need to consider
$
r
$
-values for which
$
c
\coloneqq\chern
_
1
(
E
)
$
is
integral:
\begin{sagesilent}
from plots
_
and
_
expressions import c
_
in
_
terms
_
of
_
q
\end{sagesilent}
\begin{equation*}
c =
\sage
{
c
_
in
_
terms
_
of
_
q.subs([q
_
value
_
expr,beta
_
value
_
expr])
}
\in
\ZZ
\end{equation*}
\noindent
That is,
$
r
\equiv
-
\aa
^{
-
1
}
\bb
$
mod
$
n
$
(
$
\aa
$
is coprime to
$
n
$
, and so invertible mod
$
n
$
).
\noindent
Let
$
\aa
^{
'
}$
be an integer representative of
$
\aa
^{
-
1
}$
in
$
\ZZ
/
n
\ZZ
$
.
Next, we seek to find a larger
$
\epsilon
$
to use in place of
$
\epsilon
_
F
$
in the
Next, we seek to find a larger
$
\epsilon
$
to use in place of
$
\epsilon
_
v
$
in the
proof of Theorem
\ref
{
thm:rmax
_
with
_
uniform
_
eps
}
:
\begin{lemmadfn}
[
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