Add remark on the numerical restrictions for semistabs

tex/setting-and-problems.tex

@@ -28,12 +28,12 @@ $L$ 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]
+\begin{definition}[Pseudo-semistabilisers]
 \label{dfn:pseudo-semistabilizer}
 % NOTE: SURFACE SPECIALIZATION
 	Given a Chern Character $v$, and a given stability
 	condition $\sigma_{\alpha,\beta}$,
-	a \textit{pseudo-semistabilizing} $u$ is a `potential' Chern character:
+	a \textit{pseudo-semistabilising} $u$ is a `potential' Chern character:
 	\[
 		u = \left(r, c\ell, \frac{e}{\lcm(m,2)} \ell^2\right)
 		\qquad
@@ -55,7 +55,7 @@ affect the results.
 \end{definition}
 
 At this point, and in this document, we do not care about whether
-pseudo-semistabilizers are even Chern characters of actual elements of
+pseudo-semistabilisers 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
@@ -65,25 +65,25 @@ $d \in \frac{1}{\lcm(m,2)}\ZZ$.
 
 \begin{definition}[Pseudo-walls]
 \label{dfn:pseudo-wall}
-	Let $u$ be a pseudo-semistabilizer of $v$, for some stability condition.
+	Let $u$ be a pseudo-semistabiliser of $v$, for some stability condition.
 	Then the \textit{pseudo-wall} associated to $u$ is the set of all stablity
-	conditions where $u$ is a pseudo-semistabilizer of $v$.
+	conditions where $u$ is a pseudo-semistabiliser of $v$.
 \end{definition}
 
 % TODO possibly reference forwards to Bertram's nested wall Theorem section to 
-% cover that being a pseudo-semistabilizer somewhere implies also on whole circle
+% cover that being a pseudo-semistabiliser somewhere implies also on whole circle
 
-\begin{lemma}[Sanity check for Pseudo-semistabilizers]
+\begin{lemma}[Sanity check for Pseudo-semistabilisers]
 \label{lem:sanity-check-for-pseudo-semistabilizers}
 	Given a stability
 	condition $\sigma_{\alpha,\beta}$,
-	if $E\hookrightarrow F\twoheadrightarrow G$ is a semistabilizing sequence in
+	if $E\hookrightarrow F\twoheadrightarrow G$ is a semistabilising sequence in
 	$\firsttilt\beta$ for $F$.
-	Then $\chern(E)$ is a pseudo-semistabilizer of $\chern(F)$
+	Then $\chern(E)$ is a pseudo-semistabiliser of $\chern(F)$
 \end{lemma}
 
 \begin{proof}
-	Suppose $E\hookrightarrow F\twoheadrightarrow G$ is a semistabilizing
+	Suppose $E\hookrightarrow F\twoheadrightarrow G$ is a semistabilising
 	sequence with respect to a stability condition $\sigma_{\alpha,\beta}$.
 	\begin{equation*}
 		\chern(E) = -\chern(\homol^{-1}_{\coh}(E)) + \chern(\homol^{0}_{\coh}(E))
@@ -114,12 +114,12 @@ $d \in \frac{1}{\lcm(m,2)}\ZZ$.
 	$0 \leq \chern_1^{\beta}(E) \leq \chern_1^{\beta}(F)$.
 
 
-	$E \hookrightarrow F \twoheadrightarrow G$ being a semistabilizing sequence
+	$E \hookrightarrow F \twoheadrightarrow G$ being a semistabilising sequence
 	means	$\nu_{\alpha,\beta}(E) = \nu_{\alpha,\beta}(F) = \nu_{\alpha,\beta}(F)$.
 	% MAYBE: justify this harder
 	But also, that this is an instance of $F$ being semistable, so $E$ must also
 	be semistable
-	(otherwise the destabilizing subobject would also destabilize $F$).
+	(otherwise the destabilising subobject would also destabilise $F$).
 	Similarly $G$ must also be semistable too.
 	$E$ and $G$ being semistable implies they also satisfy the Bogomolov
 	inequalities:
@@ -130,10 +130,10 @@ $d \in \frac{1}{\lcm(m,2)}\ZZ$.
 
 \end{proof}
 
-\subsection{Characteristic Curves for Pseudo-semistabilizers}
+\subsection{Characteristic Curves for Pseudo-semistabilisers}
 
 These characteristic curves introduced are convenient tools to think about the
-numerical conditions that can be used to test for pseudo-semistabilizers, and
+numerical conditions that can be used to test for pseudo-semistabilisers, and
 for solutions to the problems
 (\ref{problem:problem-statement-1},\ref{problem:problem-statement-2})
 tackled in this article (to be introduced later).
@@ -143,9 +143,9 @@ a list of numerical inequalities on it's solutions $u$.
 
 The next Lemma is a key to making this translation and revolves around the
 geometry and configuration of the characteristic curves involved in a
-semistabilizing sequence.
+semistabilising sequence.
 
-\begin{lemma}[Numerical tests for left-wall pseudo-semistabilizers]
+\begin{lemma}[Numerical tests for left-wall pseudo-semistabilisers]
 \label{lem:pseudo_wall_numerical_tests}
 Let $v$ and $u$ be Chern characters with $\Delta(v),
 \Delta(u)\geq 0$, and $v$ has non-negative rank (and $\chern_1(v)>0$ if rank 0).
@@ -153,10 +153,10 @@ Let $P$ be a point on $\Theta_v^-$.
 
 \noindent
 The following conditions:
-\begin{enumerate}
-\item $u$ is a pseudo-semistabilizer of $v$ at some point on $\Theta_v^-$ above
+\begin{enumerate}[label=\alph*]
+\item $u$ is a pseudo-semistabiliser of $v$ at some point on $\Theta_v^-$ above
 	$P$
-\item $u$ destabilizes $v$ going `inwards', that is,
+\item $u$ destabilises $v$ going `inwards', that is,
 	$\nu_{\alpha,\beta}(u)<\nu_{\alpha,\beta}(v)$ outside the pseudo-wall, and
 	$\nu_{\alpha,\beta}(u)>\nu_{\alpha,\beta}(v)$ inside.
 \end{enumerate}
@@ -166,7 +166,9 @@ are equivalent to the following more numerical conditions:
 \begin{enumerate}
 	\item $u$ has positive rank
 	\item $\beta(P)<\mu(u)<\mu(v)$, i.e. $V_u$ is strictly between $P$ and $V_v$.
-	\item $\chern_1^{\beta(P)}(u) \leq \chern_1^{\beta(P)}(v)$, $\Delta(v-u) \geq 0$
+		\label{lem:ps-wall-num-test:num-cond-slope}
+	\item $\chern_1^{\beta(P)}(u) \leq \chern_1^{\beta(P)}(v)$
+	\item	$\Delta(v-u) \geq 0$
 	\item $\chern_2^{P}(u)>0$
 \end{enumerate}
 \end{lemma}
@@ -178,9 +180,9 @@ $\Delta(u),\Delta(v) \geq 0$, 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-semistabilizer of $v$.
+pseudo-semistabiliser of $v$.
 Firstly, consequence 3 is part of the definition for $u$ being a
-pseudo-semistabilizer at a point with same $\beta$ value of $P$ (since the
+pseudo-semistabiliser at a point with same $\beta$ value of $P$ (since the
 pseudo-wall surrounds $P$).
 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,
@@ -248,8 +250,8 @@ This implies that the characteristic curves for $u$ and $v$ are in the
 configuration illustrated in Fig \ref{fig:correct-hyperbol-intersection}.
 We then have $\nu(u)=\nu(v)$ along a circle to the left of $V_u$ reaching it's
 apex at $Q$, and encircling $P$. This along with consequence 3 implies that $u$
-is a pseudo-semistabilizer at the point on the circle with $\beta=\beta(P)$.
-Therefore, it's also a pseudo-semistabilizer further along the circle at $Q$
+is a pseudo-semistabiliser at the point on the circle with $\beta=\beta(P)$.
+Therefore, it's also a pseudo-semistabiliser further along the circle at $Q$
 (supposition a).
 Finally, consequence 4 along with $P$ being to the left of $V_u$ implies
 $\nu_P(u) > 0$ giving supposition b.
@@ -258,10 +260,53 @@ The case with rank 0 can be handled the same way.
 
 \end{proof}
 
+\begin{remark}
+	Given a fixed positive $v$ with $\Delta(v)\geq 0$,
+	for any $u$ satisfying the numerical conditions of this Lemma,
+	with some given fixed rank ($\chern_0(u)$),
+	condition \ref{lem:ps-wall-num-test:num-cond-slope}
+	gives us bounds for $\chern_1(u)$. For any fixed values for
+	$\chern_0(u)$ and $\chern_1(u)$, the last three conditions will give bounds
+	for $\chern_2(u)$.
+	These bounds exist regardless of any extra assumptions about $v$.
+	However any bounds on $\chern_0(u)$ are less immediate from these numerical
+	conditions.
+	The semistabiliser $E$ for the largest `left' wall for $v$ must destabilise a Gieseker
+	stable sheaf $F$ with $\chern(F)=v$. We would have a short exact sequence in
+	the heart of a stability condition on the wall given by
+	\[
+		0 \to E \hookrightarrow F \twoheadrightarrow G \to 0
+	\]
+	for some $G$ in the heart.
+	Considering cohomology over $\coh(X)$, and using the fact that $F$ is a sheaf,
+	we get an exact sequence in $\coh(X)$ given by:
+	\[
+		0 \to
+		\cancelto{0}{\cohom^{-1}(E)} \to
+		0 \to
+		\cohom^{-1}(G) \to
+		\cohom^{0}(E) \to
+		\cohom^{0}(F) \to
+		\cohom^{0}(G) \to
+		0
+	\]
+	So we do have that $E$ must be a sheaf, but not necessarily a subsheaf of $F$,
+	and so $\chern_0(E)$ is not necessarily smaller than $\chern_0(F)$.
+	Furthermore, for smaller walls, the semistabiliser may not even be a sheaf.
+
+	When choosing the base-point of $\Theta_v^{-}$: $P=(\beta_{-}(v), 0)$, we
+	will see in Part \ref{part:inf-walls} that there can be infinitely many walls
+	when $\beta_{-}(v)$ is irrational,
+	hence infinitely many $u$ satisfying the above.
+	Therefore any construction of a bound on the ranks of possible $u$
+	must rely on the rationality of $\beta(P)$ in this case.
+\end{remark}
+
+
 \section{The Problem: Finding Pseudo-walls}
 
 As hinted in the introduction (\ref{sec:intro}), the main motivation of the
-results in this article are not only the bounds on pseudo-semistabilizer
+results in this article are not only the bounds on pseudo-semistabiliser
 ranks;
 but also applications for finding a list (comprehensive or subset) of
 pseudo-walls.
@@ -277,34 +322,34 @@ are trying to solve for.
 
 Fix a Chern character $v$ with non-negative rank (and $\chern_1(v)>0$ if rank 0),
 and $\Delta(v) \geq 0$.
-The goal is to find all pseudo-semistabilizers $u$
+The goal is to find all pseudo-semistabilisers $u$
 which give circular pseudo-walls containing some fixed point
 $P\in\Theta_v^-$.
-With the added restriction that $u$ `destabilizes' $v$ moving `inwards', that is,
+With the added restriction that $u$ `destabilises' $v$ moving `inwards', that is,
 $\nu(u)>\nu(v)$ inside the circular pseudo-wall.
 \end{problem}
 This will give all pseudo-walls between the chamber corresponding to Gieseker
 stability and the stability condition corresponding to $P$.
 The purpose of the final `direction' condition is because, up to that condition,
-semistabilizers are not distinguished from their corresponding quotients:
+semistabilisers are not distinguished from their corresponding quotients:
 Suppose $E\hookrightarrow F\twoheadrightarrow G$, then the tilt slopes
 $\nu_{\alpha,\beta}$
 are strictly increasing, strictly decreasing, or equal across the short exact
 sequence (consequence of the see-saw principle).
-In this case, $\chern(E)$ is a pseudo-semistabilizer of $\chern(F)$, if and
-only if $\chern(G)$ is a pseudo-semistabilizer of $\chern(F)$.
-The numerical inequalities in the definition for pseudo-semistabilizer cannot
+In this case, $\chern(E)$ is a pseudo-semistabiliser of $\chern(F)$, if and
+only if $\chern(G)$ is a pseudo-semistabiliser of $\chern(F)$.
+The numerical inequalities in the definition for pseudo-semistabiliser cannot
 tell which of $E$ or $G$ is the subobject.
 However, what can be distinguished is the direction across the wall that
-$\chern(E)$ or $\chern(G)$ destabilizes $\chern(F)$
-(they will each destabilize in the opposite direction to the other).
-The `inwards' semistabilizers are preferred because we are moving from a
+$\chern(E)$ or $\chern(G)$ destabilises $\chern(F)$
+(they will each destabilise in the opposite direction to the other).
+The `inwards' semistabilisers are preferred because we are moving from a
 typically more familiar chamber
 (the stable objects of Chern character $v$ in the outside chamber will only be
 Gieseker stable sheaves).
 
 Also note that this last restriction does not remove any pseudo-walls found,
-and if we do want to recover `outwards' semistabilizers, we can simply take
+and if we do want to recover `outwards' semistabilisers, we can simply take
 $v-u$ for each solution $u$ of the problem.
 
 
@@ -313,7 +358,7 @@ $v-u$ for each solution $u$ of the problem.
 
 Fix a Chern character $v$ with non-negative rank (and $\chern_1(v)>0$ if rank 0),
 $\Delta(v) \geq 0$, and $\beta_{-}(v) \in \QQ$.
-The goal is to find all pseudo-semistabilizers $u$ which give circular
+The goal is to find all pseudo-semistabilisers $u$ which give circular
 pseudo-walls on the left side of $V_v$.
 \end{problem}