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luke naylor latex documents
research
Max Destabilizer Rank
Commits
1e395091
Commit
1e395091
authored
1 year ago
by
Luke Naylor
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Generalise main theorems to other picard rank 1 surfaces
parent
24056de0
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Pipeline
#28835
passed
1 year ago
Stage: test
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main.tex
+48
-4
48 additions, 4 deletions
main.tex
plots_and_expressions.ipynb
+106
-11
106 additions, 11 deletions
plots_and_expressions.ipynb
with
154 additions
and
15 deletions
main.tex
+
48
−
4
View file @
1e395091
...
...
@@ -1514,8 +1514,7 @@ $\epsilon_{v,q}\geq\epsilon_v$, with equality when $k_{v,q}=1$.
\end{proof}
\begin{sagesilent}
from plots
_
and
_
expressions import r
_
upper
_
bound1, r
_
upper
_
bound2
from plots
_
and
_
expressions import main
_
theorem2
\end{sagesilent}
\begin{theorem}
[Bound on
$
r
$
\#
3]
\label
{
thm:rmax
_
with
_
eps1
}
...
...
@@ -1533,8 +1532,8 @@ from plots_and_expressions import r_upper_bound1, r_upper_bound2
\begin{align*}
\min
\left
(
\sage
{
r
_
upper
_
bound1
.rhs()
}
,
\:\:
\sage
{
r
_
upper
_
bound2
.rhs()
}
\sage
{
main
_
theorem2.
r
_
upper
_
bound1
}
,
\:\:
\sage
{
main
_
theorem2.
r
_
upper
_
bound2
}
\right
)
\end{align*}
\egroup
...
...
@@ -1545,6 +1544,51 @@ from plots_and_expressions import r_upper_bound1, r_upper_bound2
$
r
\equiv
\aa
^{
-
1
}
b
_
q
\pmod
{
n
}$
.
\end{theorem}
Although the general form of this bound is quite complicated, it does simplify a
lot when
$
m
$
is small.
\begin{sagesilent}
from plots
_
and
_
expressions import main
_
theorem2
_
corollary
\end{sagesilent}
\begin{corollary}
[Bound on
$
r
$
\#
3 on
$
\PP
^
2
$
and Principally polarized abelian surfaces]
\label
{
thm:rmax
_
with
_
eps1
}
Suppose we are working over
$
\PP
^
2
$
or a principally polarized abelian surface
(or any other surfaces with
$
m
=
1
$
or
$
2
$
).
Let
$
v
$
be a fixed Chern character, with
$
\frac
{
a
_
v
}{
n
}
=
\beta\coloneqq\beta
(
v
)
$
rational and expressed in lowest terms.
Then the ranks
$
r
$
of the pseudo-semistabilizers
$
u
$
for
$
v
$
with,
which are solutions to problem
\ref
{
problem:problem-statement-2
}
,
$
\chern
_
1
^
\beta
(
u
)
=
q
=
\frac
{
b
_
q
}{
n
}$
are bounded above by the following expression:
\bgroup
\def\kappa
{
k
_{
v,q
}}
\def\psi
{
\chern
_
1
^{
\beta
}
(F)
}
\begin{align*}
\min
\left
(
\sage
{
main
_
theorem2
_
corollary.r
_
upper
_
bound1
}
,
\:\:
\sage
{
main
_
theorem2
_
corollary.r
_
upper
_
bound2
}
\right
)
\end{align*}
\egroup
Where
$
R
=
\chern
_
0
(
v
)
$
and
$
k
_{
v,q
}$
is the least
$
k
\in\ZZ
_{
>
0
}$
satisfying
\begin{equation*}
k
\equiv
-
\aa\bb
\pmod
{
n
}
\end{equation*}
\noindent
Furthermore, if
$
\aa
\not
=
0
$
then
$
r
\equiv
\aa
^{
-
1
}
b
_
q
\pmod
{
n
}$
.
\end{corollary}
\begin{proof}
This is a specialisation of theorem
\ref
{
thm:rmax
_
with
_
eps1
}
, where we can
drastically simplify the
$
\lcm
$
and
$
\gcd
$
terms by noting that
$
m
$
divides both
$
2
$
and
$
2
n
^
2
$
, and that
$
a
_
v
$
is coprime to
$
n
$
.
\end{proof}
\begin{example}
[
$
v
=(
3
,
2
\ell
,
-
2
)
$
on
$
\PP
^
2
$
]
\label
{
exmpl:recurring-third
}
...
...
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