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luke naylor latex documents
research
Max Destabilizer Rank
Commits
b5a118f4
Commit
b5a118f4
authored
1 year ago
by
Luke Naylor
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Adjust settings section to more general surface
parent
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main.tex
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b5a118f4
...
...
@@ -167,8 +167,10 @@ on $\chern_0(E)$ of potential destabilizers $E$ of $F$.
%satisfies $\chern^\beta_1 \geq 0$.
Throughout this article, as noted in the introduction, we will be exclusively
working over one of the following two surfaces: principally polarized abelian
surfaces and
$
\PP
^
2
$
.
working over surfaces
$
X
$
with Picard rank 1, with a choice of ample line bundle
$
L
$
such that
$
\ell\coloneqq
c
_
1
(
L
)
$
generates
$
NS
(
X
)
$
.
We take
$
m
\coloneqq
\ell
^
2
$
as this will be the main quantity which will
affect the results.
\begin{definition}
[Pseudo-semistabilizers]
\label
{
dfn:pseudo-semistabilizer
}
...
...
@@ -177,9 +179,12 @@ surfaces and $\PP^2$.
condition
$
\sigma
_{
\alpha
,
\beta
}$
,
a
\textit
{
pseudo-semistabilizing
}
$
u
$
is a `potential' Chern character:
\[
u
=
\left
(
r, c
\ell
, d
\frac
{
1
}{
2
}
\ell
^
2
\right
)
u
=
\left
(
r, c
\ell
,
\frac
{
e
}{
\lcm
(
m,
2
)
}
\ell
^
2
\right
)
\qquad
r,c,e
\in
\ZZ
\]
which has the same tilt slope as
$
v
$
:
$
\nu
_{
\alpha
,
\beta
}
(
u
)
=
\nu
_{
\alpha
,
\beta
}
(
v
)
$
.
which has the same tilt slope as
$
v
$
:
$
\nu
_{
\alpha
,
\beta
}
(
u
)
=
\nu
_{
\alpha
,
\beta
}
(
v
)
$
.
\noindent
Furthermore the following inequalities are satisfied:
...
...
@@ -197,6 +202,11 @@ At this point, and in this document, we do not care about whether
pseudo-semistabilizers are even Chern characters of actual elements of
$
\bddderived
(
X
)
$
, some other sources may have this extra restriction too.
Later, Chern characters will be written
$
(
r,c
\ell
,d
\ell
^
2
)
$
because operations
(such as multiplication) are more easily defined in terms of the coefficients of
the
$
\ell
^
i
$
. However, at the end, it will become important again that
$
d
\in
\frac
{
1
}{
\lcm
(
m,
2
)
}
\ZZ
$
.
\begin{definition}
[Pseudo-walls]
\label
{
dfn:pseudo-wall
}
Let
$
u
$
be a pseudo-semistabilizer of
$
v
$
, for some stability condition.
...
...
@@ -222,19 +232,24 @@ $\bddderived(X)$, some other sources may have this extra restriction too.
\begin{equation*}
\chern
(E) = -
\chern
(
\homol
^{
-1
}_{
\coh
}
(E)) +
\chern
(
\homol
^{
0
}_{
\coh
}
(E))
\end{equation*}
Therefore,
$
\chern
(
E
)
$
is of the form
$
(
r,c
\ell
,d
\frac
{
1
}{
2
}
\ell
^
2
)
$
Therefore,
$
\chern
(
E
)
$
is of the form
$
(
r,c
\ell
,
\frac
{
e
}{
\lcm
(
m,
2
)
}
\ell
^
2
)
$
provided that this is true for any coherent sheaf.
For any coherent sheaf
$
H
$
, we have the following:
\begin{equation*}
\chern
(H) =
\left
(c
_
0(H), c
_
1(H), - c
_
2(H) +
\frac
{
1
}{
2
}
{
c
_
1(H)
}^
2
\right
)
\end{equation*}
Given that
$
\ell
$
generates the Neron-Severi group,
$
c
_
1
(
H
)
$
can then be
written as a multiple of
$
\ell
$
.
Furthermore, for
$
\PP
^
2
$
and principally polarized abelian surfaces,
$
\ell
^
2
=
1
$
or
$
2
$
.
This fact along with
$
c
_
2
$
being an integer on surfaces implies that
$
\chern
(
H
)
$
(and hence
$
\chern
(
E
)
$
too) is of the required form
$
(
r,c
\ell
,d
\frac
{
1
}{
2
}
\ell
^
2
)
$
for some
$
r,c,d
\in
\ZZ
$
.
written
$
c
\ell
$
.
\begin{equation*}
\chern
(H) =
\left
(
c
_
0(H), c
\ell
,
\left
(-
\frac
{
c
_
2(H)
}{
\ell
^
2
}
+
\frac
{
c
^
2
}{
2
}
\right
)
\ell
^
2
\right
)
\end{equation*}
This fact along with
$
c
_
0
$
,
$
c
_
2
$
being an integers on surfaces, and
$
m
\coloneqq
\ell
^
2
$
implies that
$
\chern
(
H
)
$
(hence
$
\chern
(
E
)
$
too) is of the required form.
Since all the objects in the sequence are in
$
\firsttilt\beta
$
, we have
...
...
@@ -826,6 +841,7 @@ corresponding $\chern_1^{\beta}(E)$ fail one of the inequalities (which is what
was implicitly happening before).
% NOTE FUTURE: surface specialization
First, let us fix a Chern character for
$
F
$
, and some pseudo-semistabilizer
$
u
$
which is a solution to problem
\ref
{
problem:problem-statement-1
}
or
...
...
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