Adjust settings section to more general surface

main.tex

@@ -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