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luke naylor latex documents
research
Max Destabilizer Rank
Commits
b642eedf
Commit
b642eedf
authored
7 months ago
by
Luke Naylor
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Complete one outstanding TODO
parent
5bdbcba6
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tex/setting-and-problems.tex
+20
-7
20 additions, 7 deletions
tex/setting-and-problems.tex
with
20 additions
and
7 deletions
tex/setting-and-problems.tex
+
20
−
7
View file @
b642eedf
...
...
@@ -157,12 +157,11 @@ are equivalent to the following more numerical conditions:
\item
$
\beta
(
P
)
<
\mu
(
u
)
<
\mu
(
v
)
$
, i.e.
$
V
_
u
$
is strictly between
$
P
$
and
$
V
_
v
$
.
\label
{
lem:ps-wall-num-test:num-cond-slope
}
\item
$
\chern
_
1
^{
\beta
(
P
)
}
(
u
)
<
\chern
_
1
^{
\beta
(
P
)
}
(
v
)
$
\item
$
\Delta
(
v
-
u
)
\geq
0
$
\item
$
\Delta
(
v
-
u
)
\geq
0
$
\item
$
\chern
_
2
^{
P
}
(
u
)
>
0
$
\end{enumerate}
\end{lemma}
\noindent\textbf
{
\color
{
red
}
TODO ADJUST TO SLIGHTLY STRONGER CONDITION 3
}
\begin{proof}
[Proof for
$
\chern
_
0
(
v
)
>
0
$
case.]
Let
$
u,v
$
be Chern characters with
$
\Delta
(
u
)
,
\Delta
(
v
)
\geq
0
$
, and
$
v
$
has positive rank.
...
...
@@ -170,9 +169,23 @@ $\Delta(u),\Delta(v) \geq 0$, and $v$ has positive rank.
For the forwards implication, assume that the suppositions of the Lemma are
satisfied. Let
$
Q
$
be the point on
$
\Theta
_
v
^
-
$
(above
$
P
$
) where
$
u
$
is a
pseudo-semistabiliser of
$
v
$
.
Firstly, consequence
3
is part of the definition for
$
u
$
being a
Firstly, consequence
4
is part of the definition for
$
u
$
being a
pseudo-semistabiliser at a point with same
$
\beta
$
value of
$
P
$
(since the
pseudo-wall surrounds
$
P
$
).
Also,
$
u
$
is a pseudo-semistabiliser along the pseudo-wall
which surround
$
P
$
, so by definition of pseudo-semistabilisers,
${
0
\leq
\chern
_
1
^
\beta
(
u
)
\leq
\chern
_
1
^
\beta
(
v
)
}$
for all
$
(
\alpha
,
\beta
)
$
on the pseudo-wall.
In particular, this holds for
$
\beta
$
in a neighbourhood of
$
\beta
(
P
)
$
,
so we can conclude
${
0
<
\chern
_
1
^{
\beta
(
P
)
}
(
u
)
<
\chern
_
1
^{
\beta
(
P
)
}
(
v
)
}$
(consequence 5).
Notice that
$
0
<
\chern
_
1
^{
\beta
(
P
)
}
(
u
)
$
follows from consequences 1 and 2, this is why it is not included in consequence 5.
If
$
u
$
were to have 0 rank, it's tilt slope would be decreasing as
$
\beta
$
increases, contradicting supposition b. So
$
u
$
must have strictly non-zero rank,
and we can consider it's characteristic curves (or that of
$
-
u
$
in case of
...
...
@@ -218,8 +231,8 @@ have positive rank (consequence 1)
to ensure that
$
\chern
_
1
^{
\beta
(
P
)
}
\geq
0
$
(since the pseudo-wall passed over
$
P
$
).
Furthermore,
$
P
$
being on the left of
$
V
_
u
$
implies
$
\chern
_
1
^{
\beta
{
P
}
}
(
u
)
\geq
0
$
,
and therefore
$
\chern
_
2
^{
P
}
(
u
)
>
0
$
(consequence
4
) to satisfy supposition b.
$
\chern
_
1
^{
\beta
(
P
)
}
(
u
)
>
0
$
,
and therefore
$
\chern
_
2
^{
P
}
(
u
)
>
0
$
(consequence
5
) to satisfy supposition b.
Next considering the way the hyperbolae intersect, we must have
$
\Theta
_
u
^
-
$
taking a
base-point to the right
$
\Theta
_
v
$
, but then, further up, crossing over to the
left side. The latter fact implies that the assymptote for
$
\Theta
_
u
^
-
$
must be
...
...
@@ -229,9 +242,9 @@ must have $\mu(u)<\mu(v)$ (second part of consequence 2),
that is,
$
V
_
u
$
is strictly to the left of
$
V
_
v
$
.
Conversely, suppose that the consequences 1-
4
are satisfied. Consequence 2
Conversely, suppose that the consequences 1-
5
are satisfied. Consequence 2
implies that the assymptote for
$
\Theta
_
u
^
-
$
is to the left of
$
\Theta
_
v
^
-
$
.
Consequence 4, along with
$
\beta
(
P
)
<
\mu
(
u
)
$
, implies that
$
P
$
must be in the
Consequence
s
4
and 5
, along with
$
\beta
(
P
)
<
\mu
(
u
)
$
, implies that
$
P
$
must be in the
region left of
$
\Theta
_
u
^
-
$
. These two facts imply that
$
\Theta
_
u
^
-
$
is to the
right of
$
\Theta
_
v
^
-
$
at
$
\alpha
=
\alpha
(
P
)
$
, but crosses to the left side as
$
\alpha
\to
+
\infty
$
, intersection at some point
$
Q
$
above
$
P
$
.
...
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