Start subsection on third bogomolov inequality

main.tex

@@ -341,6 +341,100 @@ Notice that for $\beta = \beta_{-}$ (or $\beta_{+}$), that is when
 $\chern^{\beta}_2(F)=0$, the constant and linear terms match up with the ones
 for the bound found for $d$ in subsection \ref{subsect-d-bound-bgmlv1}.
 
+\subsection{$\Delta(G) \geq 0$}
+\label{subsect-d-bound-bgmlv3}
+
+This condition expressed in terms of $R,C,D,r,c,d$ looks as follows:
+
+\begin{sagesilent}
+	# Third Bogomolov-Gieseker form expression that must be non-negative:
+	bgmlv3 = Δ(v-u)
+\end{sagesilent}
+
+\begin{equation}
+	\sage{0 <= bgmlv3.expand() }
+\end{equation}
+
+
+\noindent
+Expressing $c$ in terms of $q$ as defined in (eqn \ref{eqn-cintermsofm})
+we get the following:
+
+\begin{sagesilent}
+	bgmlv3_with_q = bgmlv3.expand().subs(c == c_in_terms_of_q)
+\end{sagesilent}
+
+\begin{equation}
+	\sage{0 <= bgmlv3_with_q}
+\end{equation}
+
+
+\noindent
+This can be rearranged to express a bound on $d$ as follows:
+
+\begin{sagesilent}
+	var("r_alt",domain="real") # r_alt = r - R temporary substitution
+
+	bgmlv3_with_q_reparam = (
+		bgmlv3_with_q.subs(r == r_alt + R)
+		/r_alt # This operation assumes r_alt > 0
+	).expand()
+	bgmlv3_d_ineq = (
+		((0 <= bgmlv3_with_q_reparam)/2 + d) # Rearrange for d
+		.subs(r_alt == r - R) # Resubstitute r back in
+		.expand()
+	)
+	# Check that this equation represents a bound for d
+	assert bgmlv3_d_ineq.lhs() == d, f"Inequality is of the form: {bgmlv3_d_ineq}"
+	bgmlv3_d_upperbound = bgmlv3_d_ineq.rhs() # Keep hold of lower bound for d
+\end{sagesilent}
+
+\begin{dmath}
+	\label{eqn-bgmlv3_d_upperbound}
+	\sage{bgmlv3_d_ineq}
+\end{dmath}
+
+\begin{sagesilent}
+	# Seperate out the terms of the lower bound for d
+
+	bgmlv3_d_upperbound_without_hyp = bgmlv3_d_upperbound.subs(1/(R-r) == 0)
+	bgmlv3_d_upperbound_const_term = bgmlv3_d_upperbound_without_hyp.subs(r==0)
+	bgmlv3_d_upperbound_linear_term = (
+		bgmlv3_d_upperbound_without_hyp
+		- bgmlv3_d_upperbound_const_term
+	).expand()
+	bgmlv3_d_upperbound_exp_term = (
+		bgmlv3_d_upperbound
+		- bgmlv3_d_upperbound_without_hyp
+	).expand()
+
+	# Verify the simplified forms of the terms that will be mentioned in text
+	assert bgmlv3_d_upperbound_const_term == ( 
+		v.twist(beta_min).ch[2]
+		+ beta_min*q
+	).expand()
+\end{sagesilent}
+
+Hyperbolic term:
+\begin{equation}
+	\sage{bgmlv3_d_upperbound_exp_term}
+\end{equation}
+
+\noindent
+Viewing equation \ref{eqn-bgmlv3_d_upperbound} as an upper bound for $d$ give:
+as a function of $r$, the terms can be rewritten as follows.
+The constant term in $r$ is
+$\chern^{\beta}_2(F) + \beta q$.
+The linear term in $r$ is
+$\sage{bgmlv3_d_upperbound_linear_term}$.
+Finally, there's an hyperbolic term in $r$ which tends to 0 as $r \to \infty$,
+and can be written:
+$?$.
+In the case $\beta = \beta_{-}$ (or $\beta_{+}$) we have
+$\chern^{\beta}_2(F) = 0$,
+so some of these expressions simplify, and in particular, the constant and
+linear terms match those of the other bounds in the previous subsections.
+
 \section{Conclusion}
 
 \newpage