Skip to content
GitLab
Explore
Sign in
Primary navigation
Search or go to…
Project
M
Max Destabilizer Rank
Manage
Activity
Members
Labels
Plan
Issues
Issue boards
Milestones
Wiki
Code
Merge requests
Repository
Branches
Commits
Tags
Repository graph
Compare revisions
Snippets
Build
Pipelines
Jobs
Pipeline schedules
Artifacts
Deploy
Releases
Package Registry
Container Registry
Model registry
Operate
Environments
Terraform modules
Monitor
Incidents
Analyze
Value stream analytics
Contributor analytics
CI/CD analytics
Repository analytics
Model experiments
Help
Help
Support
GitLab documentation
Compare GitLab plans
Community forum
Contribute to GitLab
Provide feedback
Keyboard shortcuts
?
Snippets
Groups
Projects
Show more breadcrumbs
luke naylor latex documents
research
Max Destabilizer Rank
Commits
2af1f8cb
Commit
2af1f8cb
authored
1 year ago
by
Luke Naylor
Browse files
Options
Downloads
Patches
Plain Diff
Define beta +- in zero/positive rank cases
parent
2636c4d3
No related branches found
No related tags found
No related merge requests found
Pipeline
#29359
passed
1 year ago
Stage: test
Changes
1
Pipelines
1
Hide whitespace changes
Inline
Side-by-side
Showing
1 changed file
main.tex
+31
-2
31 additions, 2 deletions
main.tex
with
31 additions
and
2 deletions
main.tex
+
31
−
2
View file @
2af1f8cb
...
...
@@ -356,11 +356,27 @@ degenerate_characteristic_curves
\label
{
fig:charact
_
curves
_
vis
}
\end{figure}
\begin{definition}
[
$
\beta
_{
\pm
}$
]
\label
{
dfn:beta
_
pm
}
Given a formal Chern character
$
v
$
with positive rank, we define
$
\beta
_{
\pm
}
(
v
)
$
to be
the
$
\beta
$
-coordinate of where
$
\Theta
_
v
^{
\pm
}$
meets the
$
\beta
$
-axis:
\[
\beta
_
\pm
(
R,C
\ell
,D
\ell
^
2
)
=
\frac
{
C
\pm
\sqrt
{
C
^
2
-
2
RD
}}{
R
}
\]
\noindent
In particular, this means
$
\beta
_
\pm
(
v
)
$
are the two roots of the quadratic
equation
$
\chern
_
2
^{
\beta
}
(
v
)=
0
$
.
This definition will be extended to the rank 0 case in definition
\ref
{
dfn:beta
_
-
_
rank0
}
.
\end{definition}
\subsubsection
{
Rank Zero Case
}
\begin{sagesilent}
from rank
_
zero
_
case import Theta
_
v
_
plot
\end{sagesilent}
\subsubsection
{
Rank Zero Case
}
\begin{fact}
[Geometry of Characteristic Curves in Rank 0 Case]
The following facts can be deduced from the formulae for
$
\chern
_
i
^{
\alpha
,
\beta
}
(
v
)
$
as well as the restrictions on
$
v
$
, when
$
\chern
_
0
(
v
)=
0
$
and
$
\chern
_
1
(
v
)
>
0
$
:
...
...
@@ -387,7 +403,7 @@ Indeed:
\begin{align*}
\mu\left
(
\varepsilon
, C
\ell
, D
\ell
^
2
\right
) =
\frac
{
C
}{
\varepsilon
}
&
\longrightarrow
+
\infty
\\
\text
{
as
}
\:
0<
\varepsilon
&
\longrightarrow
0
\text
{
as
}
\:
\:
0<
\varepsilon
&
\longrightarrow
0
\end{align*}
So we can view
$
V
_
v
$
as moving off infinitely to the right, with
$
\Theta
_
v
^
+
$
even further.
But also, considering the base point of
$
\Theta
_
v
^
-
$
:
...
...
@@ -404,6 +420,19 @@ For this reason, I will refer to the whole of $\Theta_v$ in the rank zero case
as
$
\Theta
_
v
^
-
$
to be able to use the same terminology in both positive rank
and rank zero cases.
\begin{definition}
[Extending
$
\beta
_
-
$
to rank 0 case]
\label
{
dfn:beta
_
-
_
rank0
}
Given a formal Chern character
$
v
$
with rank 0 and
$
\chern
_
1
(
v
)
>
0
$
, we define
$
\beta
_
-(
v
)
$
to be the
$
\beta
$
-coordinate of point where
$
\Theta
_
v
$
meets the
$
\beta
$
-axis:
\[
\beta
_
-(
0
,C
\ell
,D
\ell
^
2
)
=
\frac
{
D
}{
C
}
\]
\noindent
If
$
\beta
_
+
$
were also to be generalised to the rank 0 case, we would consider
its value to be
$
+
\infty
$
due to the discussion above.
\end{definition}
\subsection
{
Relevance of
$
V
_
v
$}
\label
{
subsect:relevance-of-V
_
v
}
...
...
This diff is collapsed.
Click to expand it.
Preview
0%
Loading
Try again
or
attach a new file
.
Cancel
You are about to add
0
people
to the discussion. Proceed with caution.
Finish editing this message first!
Save comment
Cancel
Please
register
or
sign in
to comment