Add simplified bounds for d at beta_{-}

main.tex

@@ -550,27 +550,26 @@ $(r,c,d)$ that satisfy all inequalities to give a pseudowall.
 
 \subsubsection{All circular pseudowalls left of vertical wall}
 
-Suppose we take $\beta = \beta_{-}$ in the previous subsections, to find all
-circular walls to the left of the vertical wall (TODO as discussed in ref).
+Suppose we take $\beta = \beta_{-}$ (so that $\chern^{\beta}_2(F)=0$)
+in the previous subsections, to find all circular walls to the left of the
+vertical wall (TODO as discussed in ref).
 
-\begin{equation*}
-	d \geq
+\begin{align}
+	d &\geq&
 	\sage{bgmlv1_d_lowerbound_const_term_alt.subs(chbv == 0)}
-	+ \sage{bgmlv1_d_lowerbound_linear_term}
-	+ \sage{bgmlv1_d_lowerbound_exp_term_alt.subs(chbv == 0)}
-\end{equation*}
-\begin{equation*}
-	d \geq
-	\sage{bgmlv2_d_lowerbound_const_term}
-	+ \sage{bgmlv2_d_lowerbound_linear_term}
-	+ \sage{bgmlv2_d_lowerbound_exp_term}
-\end{equation*}
-\begin{equation*}
-	d \leq
+	&+ \sage{bgmlv1_d_lowerbound_linear_term}
+	&+ \sage{bgmlv1_d_lowerbound_exp_term_alt.subs(chbv == 0)}
+\\
+	d &\leq&
+	\sage{bgmlv2_d_upperbound_const_term}
+	&+ \sage{bgmlv2_d_upperbound_linear_term}
+	&+ \sage{bgmlv2_d_upperbound_exp_term}
+\\
+	d &\leq&
 	\sage{bgmlv3_d_upperbound_const_term_alt.subs(chbv == 0)}
-	+ \sage{bgmlv3_d_upperbound_linear_term}
-	+ \sage{bgmlv3_d_upperbound_exp_term_alt.subs(chbv == 0)}
-\end{equation*}
+	&+ \sage{bgmlv3_d_upperbound_linear_term}
+	&+ \sage{bgmlv3_d_upperbound_exp_term_alt.subs(chbv == 0)}
+\end{align}
 
 
 \section{Conclusion}