Be more specific in definition of pseudo-semistabilizer

main.tex

@@ -113,12 +113,27 @@ bounds on $\chern_0(E)$ of potential destabilizers $E$ of $F$.
 Characters}
 
 \begin{dfn}[Pseudo-semistabilizers]
-	Given a Chern Character $v$, and a given stability condition $\sigma$, 
+	Given a Chern Character $v$ on a Picard rank 1 surface, and a given stability
+	condition $\sigma_{\alpha,\beta}$,
 	a pseudo-semistabilizing $u$ is a `potential' Chern character:
 	\[
 		u = \left(r, c\ell, d \frac{1}{2} \ell^2\right)
 	\]
-	which has the same tilt slope as $v$: $\mu_{\sigma}(u) = \mu_{\sigma}(v)$.
+	which has the same tilt slope as $v$: $\nu_{\alpha,\beta}(u) = \nu_{\alpha,\beta}(v)$.
+
+	\noindent
+	Furthermore the following Bogomolov-Gieseker inequalities are satisfied:
+	\begin{itemize}
+		\item $\Delta(u) \geq 0$
+		\item $\Delta(v-u) \geq 0$
+		\item $\Delta(u) + \Delta(v-u) \leq \Delta(v)$
+	\end{itemize}
+	\noindent
+	And finally these two conditions are satisfied:
+	\begin{itemize}
+		\item $\chern_1^{\beta}(u) \geq 0$
+		\item $\chern_1^{\beta}(v-u) \geq 0$
+	\end{itemize}
 
 	Note $u$ does not need to be a Chern character of an actual sub-object of some
 	object in the stability condition's heart with Chern character $v$.
@@ -128,6 +143,14 @@ 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.
 
+\begin{lemma}[ Sanity check for Pseudo-semistabilizers ]
+	Given a Picard rank 1 surface, and a given stability
+	condition $\sigma_{\alpha,\beta}$,
+	if $E\hookrightarrow F\twoheadrightarrow G$ is a semistabilizing sequence in
+	$\firsttilt\beta$ for $F$.
+	Then $\chern(E)$ is a pseudo-semistabilizer of $\chern(F)$
+\end{lemma}
+
 Considering the stability conditions with two parameters $\alpha, \beta$ on
 Picard rank 1 surfaces.
 We can draw 2 characteristic curves for any given Chern character $v$ with
@@ -606,7 +629,7 @@ Fix a Chern character $v$ with positive rank, $\Delta(v) \geq 0$,
 and $\beta_{-}(v) \in \QQ$.
 The goal is to find all Chern characters $u=(r,c\ell,d\ell^2)$ which satisfy the
 conditions of lemma \ref{lem:pseudo_wall_numerical_tests} using
-$P=(\beta_{-},0)$, as well as the Bogomolov inequalities:
+$P=(\beta_{-},0)$, $\chern_1^{\beta_{-}}(v-u)\geq 0$, as well as the Bogomolov inequalities:
 $\Delta(u),\Delta(v-u) \geq 0$ and $\Delta(u)+\Delta(v-u) \leq \Delta(v)$.
 We want to restrict our attention to pseudo-walls left of $V_v$ (condition (a) of
 lemma), because this is the side of $V_v$ containing the chamber for Gieseker