Second Bogomolov inequality

main.tex

@@ -230,6 +230,53 @@ This can be rearranged to express a bound on $d$ as follows:
 \end{dmath}
 
 
+\noindent
+In the case $\beta = \beta_{-}$ (or $\beta_{+}$) this can be simplified.
+
+\subsection{$\Delta(E) \geq 0$}
+
+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:
+	bgmlv2 = Δ(u)
+\end{sagesilent}
+
+\begin{equation}
+	\sage{0 <= bgmlv2.expand() }
+\end{equation}
+
+
+\noindent
+Expressing $c$ in terms of $q$ as defined in (eqn \ref{eqn-cintermsofm})
+we get the following:
+
+\begin{sagesilent}
+	bgmlv2_with_q = bgmlv2.expand().subs(c == c_in_terms_of_q)
+\end{sagesilent}
+
+\begin{equation}
+	\sage{0 <= bgmlv2_with_q}
+\end{equation}
+
+
+\noindent
+This can be rearranged to express a bound on $d$ as follows:
+
+\begin{sagesilent}
+	bgmlv2_d_ineq = (
+		((0 <= bgmlv2_with_q)/2/r + d) # Rearrange for d
+		.expand()
+	)
+
+	bgmlv2_d_lowerbound = bgmlv2_d_ineq.rhs() # Keep hold of lower bound for d
+\end{sagesilent}
+
+\begin{equation}
+	\sage{bgmlv2_d_ineq}
+\end{equation}
+
+
 \noindent
 In the case $\beta = \beta_{-}$ (or $\beta_{+}$) this can be simplified.