Start BG(E) + BG(G) <= BG(F) condition

main.tex

@@ -153,20 +153,23 @@ $\chern(E) = (r,c,d)$ of some semistabilizer $E$.
 
 	v = Chern_Char(*var("R C D", domain="real"))
 	u = Chern_Char(*var("r c d", domain="real"))
-
-	ts = stability.Tilt
-
-	beta_min = min(map(lambda sol: sol.rhs(),
-			solve(ts(alpha=0).degree(v), ts().beta)
-	))
 \end{sagesilent}
 
-Here, we have $\beta_{-} = \sage{beta_min}$.
 Recall [ref] that $\chern_1^{\beta_{-}}$ has fixed bounds in terms of
 $\chern(F)$, and so we can write:
 
+\begin{sagesilent}
+	ts = stability.Tilt
+	beta_min = var("beta", domain="real")
+	c_lower_bound = -(ts(beta=beta_min).rank(u)/ts().alpha).expand() + c
+
+	var("m", domain="real")
+	c_val = c_lower_bound + m
+\end{sagesilent}
+
 \begin{equation}
-	\chern_1(E) = r\beta_{-} + m
+	\label{eqn-cintermsofm}
+	c=\chern_1(E) = \sage{c_val}
 	\qquad 0 \leq m \leq \chern_1^{\beta_{-}}(F)
 \end{equation}
 
@@ -175,13 +178,46 @@ $m \in \frac{1}{n} \ZZ \cap [0, \chern_1^{\beta_{-}}(F)]$.
 For the next subsections, we consider $m$ to be fixed with one of these values,
 and we shall be varying $\chern_0(E) = r$ to see when certain inequalities fail.
 
+
+\subsection{$\Delta(E) + \Delta(G) \leq \Delta(F)$}
+
+This condition expressed in terms of $R,C,D,r,c,d$ looks as follows:
+
 \begin{sagesilent}
-	c_lower_bound = -(ts(beta=beta_min).rank(u)/ts().alpha).expand() + c
+	positive_condition = - u.Q_tilt() - (v-u).Q_tilt() + v.Q_tilt()
+\end{sagesilent}
 
-	var("m", domain="real")
-	c_val = c_lower_bound + m
+\begin{equation}
+	\sage{(0 <= positive_condition.expand() )}
+\end{equation}
+
+Expressing $c$ in terms of $m$ as defined in (eqn \ref{eqn-cintermsofm})
+we get the following:
+
+\begin{sagesilent}
+	positive_condition = positive_condition.expand().subs(c == c_val)
+\end{sagesilent}
+
+\begin{equation}
+	\sage{(0 <= positive_condition) + 2*R*d - 4*d*r}
+\end{equation}
+
+This can be rearranged to express a bound on $d$ as follows:
+
+\begin{sagesilent}
+	var("r_alt",domain="real")
+	nc = (positive_condition.subs(r == r_alt + R/2)/r_alt).expand()
+	nc = ((0 > -nc) + 4*d)/4 # rearrange for d
+	nc = nc.subs(r_alt == r - R/2).expand() # resubs r back in
 \end{sagesilent}
 
+\begin{footnotesize}
+\begin{equation}
+	\sage{nc}
+\end{equation}
+\end{footnotesize}
+
+In the case $\beta = \beta_{-}$ (or $\beta_{+}$) this can be simplified.