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Commit b5a118f4 authored by Luke Naylor's avatar Luke Naylor
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Adjust settings section to more general surface

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......@@ -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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