Overhaul expressions for bgmlv1 condition (\Delta(u,v) >= 0)

main.tex

@@ -10,6 +10,7 @@
 \usepackage{sagetex}
 \usepackage{minted}
 \usepackage{subcaption}
+\usepackage{cancel}
 \usepackage[]{breqn}
 \usepackage[
 backend=biber,
@@ -642,6 +643,7 @@ algorithm will be presented making use of the later theorems in this article,
 with the goal of cutting down the run time.
 
 \subsection*{Problem statement}
+\label{subsect:problem-statement}
 
 Fix a Chern character $v$ with positive rank, $\Delta(v) \geq 0$,
 and $\beta_{-}(v) \in \QQ$.
@@ -732,9 +734,11 @@ As opposed to only eliminating possible values of $\chern_0(E)$ for which all
 corresponding $\chern_1^{\beta}(E)$ fail one of the inequalities (which is what
 was implicitly happening before).
 
-First, let us fix a Chern character for $F$,
-$\chern(F) = (R,C\ell,D\ell^2)$, and consider the possible Chern characters
-$\chern(E) = (r,c\ell,d\ell^2)$ of some semistabilizer $E$.
+First, let us fix a Chern character for $F$, and some semistabilizer $E$:
+\begin{align}
+	v &:= \chern(F) = (R,C\ell,D\ell^2) \\
+	u &:= \chern(E) = (r,c\ell,d\ell^2)
+\end{align}
  
 \begin{sagesilent}
 # Requires extra package:
@@ -748,8 +752,8 @@ u = Chern_Char(*var("r c d", domain="real"))
 Δ = lambda v: v.Q_tilt()
 \end{sagesilent}
 
-Recall from eqn \ref{eqn-tilt-cat-cond} that $\chern_1^{\beta}$ has fixed
-bounds in terms of $\chern(F)$, and so we can write:
+Recall from eqn \ref{eqn-tilt-cat-cond} that $\chern_1^{\beta}(u)$ has fixed
+bounds in terms of $\chern_1^{\beta}(v)$, and so we can write:
 
 \begin{sagesilent}
 ts = stability.Tilt
@@ -766,8 +770,8 @@ c_in_terms_of_q = c_lower_bound + q
 
 \begin{equation}
 	\label{eqn-cintermsofm}
-	c=\chern_1(E) = \sage{c_in_terms_of_q}
-	\qquad 0 \leq q \leq \chern_1^{\beta}(F)
+	c=\chern_1(u) = \sage{c_in_terms_of_q}
+	\qquad 0 \leq q := \chern_1^{\beta}(u) \leq \chern_1^{\beta}(v)
 \end{equation}
 
 Furthermore, $\chern_1 \in \ZZ$ so we only need to consider
@@ -778,128 +782,99 @@ and we shall be varying $\chern_0(E) = r$ to see when certain inequalities fail.
 
 \subsection{Numerical Inequalities}
 
+\subsubsection{Size of pseudo-wall: $\chern_2^P(u)>0$ }
+\label{subsect-d-bound-radiuscond}
+
+This condition refers to consequence 4 from
+lemma \ref{lem:pseudo_wall_numerical_tests}.
+
+In the case where $P$ was chosen to be the base of $\Theta_v^-$: $P=(\beta_{-},0)$.
+This condition amounts to:
+
+\begin{align}
+\label{eqn:radius-cond-betamin}
+	\chern_2^{\beta_{-}}(u) &\geq 0 \\
+	d &\geq \beta_{-}q + \frac{1}{2} \beta_{-}^2r
+\end{align}
+
 \subsubsection{
 	\texorpdfstring{
-		$\Delta(E) + \Delta(G) \leq \Delta(F)$
+		$\Delta(u,v-u) \geq 0$
 	}{
-		Δ(E) + Δ(G) ≤ Δ(F)
+		Δ(u,v-u) ≤ 0
 	}
 }
 \label{subsect-d-bound-bgmlv1}
 
-This condition expressed in terms of $R,C,D,r,c,d$ looks as follows:
-
-\begin{sagesilent}
-# First Bogomolov-Gieseker form expression that must be non-negative:
-bgmlv1 = Δ(v) - Δ(u) - Δ(v-u)
-\end{sagesilent}
+Writing the condition in terms of the twisted chern characters
+for $u$ and $v$ at $\beta$
+($(r,\chern_1^{\beta}(u),\chern_2^{\beta}(u))$
+and $(R-r,\chern_1^{\beta}(v-u),\chern_2^{\beta}(v-u))$) yields:
 
 \begin{equation}
-	\sage{0 <= bgmlv1.expand() }
+\label{eqn:bgmlv1-pt1}
+	(R-r)\chern_2^{\beta}(u)
+	\leq
+	\chern_1^{\beta}(u)\chern_1^{\beta}(v-u)
+	- r\chern_2^{\beta}(v-u)
 \end{equation}
 
-
-\noindent
-Expressing $c$ in terms of $q$ as defined in (eqn \ref{eqn-cintermsofm})
-we get the following:
-
-\begin{sagesilent}
-bgmlv1_with_q = (
-	bgmlv1
-	.expand()
-	.subs(c == c_in_terms_of_q)
-)
-\end{sagesilent}
+Which rearranges to (using additivity of $\chern_2^{\beta}$):
 
 \begin{equation}
-	\sage{0 <= bgmlv1_with_q}
+\label{eqn:bgmlv1-pt2}
+	(R-2r)\chern_2^{\beta}(u)
+	\leq
+	\chern_1^{\beta}(u)\chern_1^{\beta}(v-u)
+	- r\chern_2^{\beta}(v)
 \end{equation}
 
+With $u$ satisfying the condition given by equation \ref{eqn-cintermsofm},
+we note that $\chern_1^{\beta}(u),\chern_1^{\beta}(v-u) \geq 0$.
 
-\noindent
-This can be rearranged to express a bound on $d$ as follows:
-
-\begin{sagesilent}
-var("r_alt",domain="real") # r_alt = r - R/2 temporary substitution
-
-bgmlv1_with_q_reparam = (bgmlv1_with_q.subs(r == r_alt + R/2)/r_alt).expand()
-
-bgmlv1_d_ineq = (
-	((0 >= -bgmlv1_with_q_reparam)/4 + d) # Rearrange for d
-	.subs(r_alt == r - R/2) # Resubstitute r back in
-	.expand()
-)
-
-bgmlv1_d_lowerbound = bgmlv1_d_ineq.rhs() # Keep hold of lower bound for d
-\end{sagesilent}
-
-\begin{dmath}
-	\label{eqn-bgmlv1_d_lowerbound}
-	\sage{bgmlv1_d_ineq}
-\end{dmath}
-
-\begin{sagesilent}
-# Separate out the terms of the lower bound for d
-bgmlv1_d_lowerbound_without_hyp = bgmlv1_d_lowerbound.subs(1/(R-2*r) == 0)
+In the special case with $P=(\beta_{-},0)$,
+we have $\chern_2^{\beta_{-}}(v)=0$, and we can assume
+equation $\chern_2^{\beta_{-}}(u)>0$ (eqn \ref{eqn:radius-cond-betamin})
+in the context of our problem.
 
-bgmlv1_d_lowerbound_exp_term = (
-	bgmlv1_d_lowerbound
-	- bgmlv1_d_lowerbound_without_hyp
-).expand()
+Finally, $r>0$ as per the statement of the problem, so the right-hand-side
+of equation \ref{eqn:bgmlv1-pt1} is always greater than, or equal, to zero.
+And so, when $P:=(\beta_{-},0)$, this condition $\Delta(u,v-u) \geq 0$ is
+always satisfied when $2r \geq R$, provided that the other conditions of the
+problem statement (\ref{subsect:problem-statement}) hold.
 
-bgmlv1_d_lowerbound_const_term = bgmlv1_d_lowerbound_without_hyp.subs(r==0)
-
-bgmlv1_d_lowerbound_linear_term = (
-	bgmlv1_d_lowerbound_without_hyp
-	- bgmlv1_d_lowerbound_const_term
-).expand()
-
-# Verify the simplified forms of the terms that will be mentioned in text
-var("chbv",domain="real") # symbol to represent ch_1^\beta(v)
-
-assert bgmlv1_d_lowerbound_const_term == (
-	(
-		# Keep hold of this alternative expression:
-		bgmlv1_d_lowerbound_const_term_alt :=
-		(
-			chbv/2
-			+ beta*q
-		)
-	)
-	.subs(chbv == v.twist(beta).ch[2])
-	.expand()
-)
-
-assert bgmlv1_d_lowerbound_exp_term == (
-	(
-		# Keep hold of this alternative expression:
-		bgmlv1_d_lowerbound_exp_term_alt :=
-		(
-			- R*chbv/2
-			- R*beta*q
-			+ C*q
-			- q^2
-		)/(R-2*r)
-	)
-	.subs(chbv == v.twist(beta).ch[2])
-	.expand()
-)
-\end{sagesilent}
+However, when $2r<R$, this condition does add potentially independent condition
+of the others:
 
+\begin{equation}
+\label{eqn:bgmlv1-pt3}
+	\chern_2^{\beta}(u)
+	\leq
+	\frac{
+		\chern_1^{\beta}(u)\chern_1^{\beta}(v-u)
+		- r\chern_2^{\beta}(v)
+	}
+	{R-2r},
+	\qquad
+	2r<R
+\end{equation}
 
-\noindent
-Viewing equation \ref{eqn-bgmlv1_d_lowerbound} as a lower bound for $d$ given
-as a function of $r$, the terms can be rewritten as follows.
-The constant term in $r$ is
-$\chern^{\beta}_2(F)/2 + \beta q$.
-The linear term in $r$ is
-$\sage{bgmlv1_d_lowerbound_linear_term}$.
-Finally, there is an hyperbolic term in $r$ which tends to 0 as $r \to \infty$,
-and can be written:
-$\frac{R\chern^{\beta}_2(F)/2 + R\beta q - Cq + q^2  }{2r-R}$.
-In the case $\beta = \beta_{-}$ (or $\beta_{+}$) we have
-$\chern^{\beta}_2(F) = 0$,
-so some of these expressions simplify.
+Expressed in terms of $d$ and $q$:
+\begin{equation}
+\label{eqn:bgmlv1-pt4}
+	d
+	\leq
+	\beta_{-}q
+	+\frac{1}{2}{\beta_{-}}^2r
+	+
+	\frac{
+		q(\chern_1^{\beta}(v)-q)
+		- r\chern_2^{\beta}(v)
+	}
+	{R-2r},
+	\qquad
+	2r<R
+\end{equation}
 
 \subsubsection{
 	\texorpdfstring{
@@ -1174,12 +1149,23 @@ vertical wall (TODO as discussed in ref).
 % redefine \psi in sage expressions (placeholder for ch_1^\beta(F)
 \def\psi{\chern_1^{\beta}(F)}
 \begin{align}
-	d &\geq&
-	\sage{bgmlv1_d_lowerbound_linear_term}
-	&+ \sage{bgmlv1_d_lowerbound_const_term_alt.subs(chbv == 0)}
-	+& \sage{bgmlv1_d_lowerbound_exp_term_alt.subs(chbv == 0)},
-	 &\qquad\text{when\:} r > \frac{R}{2}
-	 \label{eqn:bgmlv1_d_bound_betamin}
+	d &>&
+	\frac{1}{2}\beta^2 r
+	&+ \beta q,
+	\phantom{+}& % to keep terms aligned
+	 &\qquad\text{when\:} r > 0
+	\label{eqn:radiuscond_d_bound_betamin}
+\\
+	d &\leq&
+	\frac{1}{2}{\beta}^2r
+	&+ \beta q
+	+&
+	\frac{
+		q(\chern_1^{\beta}(v)-q)
+	}
+	{R-2r},
+	 &\qquad\text{when\:} 0 < r < \frac{R}{2}
+	\label{eqn:bgmlv1_d_bound_betamin}
 \\
 	d &\leq&
 	\sage{bgmlv2_d_upperbound_linear_term}
@@ -1246,12 +1232,15 @@ def plot_d_bound(
 
   # Equations to plot imminently representing the bounds on d:
   eq1 = (
-    bgmlv1_d_lowerbound
-    .subs(R == v_example.ch[0])
-    .subs(C == v_example.ch[1])
-    .subs(D == v_example.ch[2])
+    (
+      beta^2*r/2
+      + beta*q
+      + q*(chb1v - q)/(R-2*r)
+    )
+    .subs(chb1v == v_example.twist(beta_min(v_example)).ch[1])
     .subs(beta = beta_min(v_example))
     .subs(q == q_example)
+    .subs(R == v_example.ch[0])
   )
 
   eq2 = (
@@ -1284,28 +1273,28 @@ def plot_d_bound(
       (r,v_example.ch[0],xmax),
       color='green',
 			linestyle = "dashed",
-      legend_label=r"upper bound: $\Delta(G) \geq 0$",
+      legend_label=r"upper bound: $\Delta(v-u) \geq 0$",
     )
     + plot(
       eq2,
       (r,0,xmax),
       color='blue',
 			linestyle = "dashed",
-      legend_label=r"upper bound: $\Delta(E) \geq 0$"
+      legend_label=r"upper bound: $\Delta(u) \geq 0$"
     )
     + plot(
       eq4,
       (r,0,xmax),
       color='orange',
 			linestyle = "dotted",
-      legend_label=r"lower bound: $\mathrm{ch}_2^{\beta_{-}}(E)>0$"
+      legend_label=r"lower bound: $\mathrm{ch}_2^{\beta_{-}}(u)>0$"
     )
     + plot(
       eq1,
-      (r,v_example.ch[0]/2,xmax),
+      (r,0,v_example.ch[0]/2),
       color='red',
-			linestyle = "dotted",
-      legend_label=r"lower bound: $\Delta(E) + \Delta(G) \leq \Delta(F)$"
+			linestyle = "dashed",
+      legend_label=r"upper bound: $\Delta(u,v) \geq 0$"
     )
   )
   example_bounds_on_d_plot.ymin(ymin)
@@ -1328,7 +1317,7 @@ def plot_d_bound(
 \hfill
 \begin{subfigure}{.45\textwidth}
   \centering
-	\sageplot[width=\linewidth]{plot_d_bound(v_example, 4)}
+	\sageplot[width=\linewidth]{plot_d_bound(v_example, 4, ymin=-3)}
 	\caption{$q = \chern^{\beta}(F)$ (all bounds other than blue coincide on line)}
   \label{fig:d_bounds_xmpl_max_q}
 \end{subfigure}
@@ -1376,7 +1365,7 @@ Some of the details around the associated numerics are explored next.
 \centering
 \sageplot[
 	width=\linewidth
-]{plot_d_bound(v_example, 2, ymax=6, ymin=-0.5, aspect_ratio=1)}
+]{plot_d_bound(v_example, 2, ymax=6, ymin=-2, aspect_ratio=1)}
 \caption{
 	Bounds on $d:=\chern_2(E)$ in terms of $r:=\chern_0(E)$ for a fixed
 	value $\chern_1^{\beta}(F)/2$ of $q:=\chern_1^{\beta}(E)$.
@@ -1507,6 +1496,7 @@ considering equations
 
 \begin{sagesilent}
 var("epsilon")
+var("chbv") # symbol to represent \chern_1^{\beta}(v)
 
 # Tightness conditions: