diff --git a/tex/setting-and-problems.tex b/tex/setting-and-problems.tex
index 823b5c719cb05bf451c7479a5f6a846d8a969d23..da5240c3cb77751d4286d66809aa9382ac03beb1 100644
--- a/tex/setting-and-problems.tex
+++ b/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}