Remove all references to the Delta(u,v-u)>=0 condition

main.tex

@@ -169,17 +169,11 @@ surfaces and $\PP^2$.
 	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:
+	Furthermore the following 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 $\chern_1^{\beta}(v-u) \geq 0$
+		\item $0 \leq \chern_1^{\beta}(u) \leq \chern_1^{\beta}(v)$
 	\end{itemize}
 
 	Note $u$ does not need to be a Chern character of an actual sub-object of some
@@ -498,7 +492,7 @@ Fixing attention on the only possible case (2), illustrated in Fig
 \ref{fig:correct-hyperbol-intersection}.
 $P$ is on the left of $V_{\pm u}$ (first part of consequence 2), so $u$ must
 have positive rank (consequence 1)
-to ensure that $\chern_1^{\beta{P}} \geq 0$ (since the pseudo-wall passed over
+to ensure that $\chern_1^{\beta(P)} \geq 0$ (since the pseudo-wall passed over
 $P$).
 Furthermore, $P$ being on the left of $V_u$ implies
 $\chern_1^{\beta{P}}(u) \geq 0$,
@@ -845,85 +839,6 @@ This condition amounts to:
 	d &\geq \beta_{-}q + \frac{1}{2} \beta_{-}^2r
 \end{align}
 
-\subsubsection{
-	\texorpdfstring{
-		$\Delta(u,v-u) \geq 0$
-	}{
-		Δ(u,v-u) ≤ 0
-	}
-}
-\label{subsect-d-bound-bgmlv1}
-
-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}
-\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}
-
-Which rearranges to (using additivity of $\chern_2^{\beta}$):
-
-\begin{equation}
-\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 $\chern_1^{\beta}(u),\chern_1^{\beta}(v-u) \geq 0$.
-
-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.
-
-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 $P\coloneqq(\beta_{-},0)$, 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-2}) hold.
-
-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}
-
-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{
 		$\Delta(E) \geq 0$
@@ -1203,17 +1118,6 @@ vertical wall (TODO as discussed in ref).
 	\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}
@@ -1279,18 +1183,6 @@ def plot_d_bound(
 ):
 
   # Equations to plot imminently representing the bounds on d:
-  eq1 = (
-    (
-      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 = (
     bgmlv2_d_upperbound
     .subs(R == v_example.ch[0])
@@ -1337,13 +1229,6 @@ def plot_d_bound(
 			linestyle = "dotted",
       legend_label=r"lower bound: $\mathrm{ch}_2^{\beta_{-}}(u)>0$"
     )
-    + plot(
-      eq1,
-      (r,0,v_example.ch[0]/2),
-      color='red',
-			linestyle = "dashed",
-      legend_label=r"upper bound: $\Delta(u,v) \geq 0$"
-    )
   )
   example_bounds_on_d_plot.ymin(ymin)
   example_bounds_on_d_plot.ymax(ymax)