Bound for rank

main.tex

@@ -9,7 +9,10 @@
 \usepackage{color}
 
 \newcommand{\QQ}{\mathbb{Q}}
+\newcommand{\ZZ}{\mathbb{Z}}
 \newcommand{\chern}{\operatorname{ch}}
+\newcommand{\lcm}{\operatorname{lcm}}
+\newcommand{\gcd}{\operatorname{gcd}}
 \newcommand{\firsttilt}[1]{\mathcal{B}^{#1}}
 \newcommand{\bddderived}{\mathcal{D}^{b}}
 \newcommand{\centralcharge}{\mathcal{Z}}
@@ -84,6 +87,8 @@ $\chern^\beta(E) = \chern(E) \cdot \exp(-\beta \ell)$:
 	\chern^\beta_2(E) &= \chern_2(E) - \beta \chern_1(E) + \frac{\beta^2}{2} \chern_0(E)
 \end{align*}
 
+% TODO I think this^ needs adjusting for general Surface with $\ell$
+
 $\chern^\beta_1(E)$ is the imaginary component of the central charge
 $\centralcharge_{\alpha,\beta}(E)$ and any element of $\firsttilt\beta$
 satisfies $\chern^\beta_1 \geq 0$. This, along with additivity gives us, for any
@@ -101,7 +106,7 @@ normal one. So $0 \leq \Delta(E)$ yields:
 
 \begin{equation}
 	\label{eqn-bgmlv-on-E}
-	\chern^\beta_0(E) \chern^\beta_2(E) \leq \left(\chern^\beta_1(E)\right)^2
+	2\chern^\beta_0(E) \chern^\beta_2(E) \leq \left(\chern^\beta_1(E)\right)^2
 \end{equation}
 
 The restrictions on $\chern^\beta_0(E)$ and $\chern^\beta_2(E)$
@@ -109,6 +114,20 @@ is best seen with the following graph:
 
 % TODO: hyperbola restriction graph (shaded)
 
+This is where the $\beta_{-}$ criterion comes in. If $\beta_{-} = \frac{*}{n}$
+for some $*,n \in \ZZ$.
+Then $\chern^\beta_2(E) \in \frac{1}{\lcm(m,2n^2)}\ZZ$ where $m$ is the integer
+which guarantees $\chern_2(E) \in \frac{1}{m}\ZZ$ (determined by the variety).
+In particular, since $\chern_2(E) > 0$ we must also have
+$\chern^\beta_2(E) \geq \frac{1}{\lcm(m,2n^2)}$, which then in turn gives a bound
+for the rank of $E$:
+
+\begin{align}
+	\chern_0(E) &= \chern^\beta_0(E) \\
+	&\leq \frac{\lcm(m,2n^2) \chern^\beta_1(E)^2}{2} \\
+	&\leq \frac{mn^2 \chern^\beta_1(F)^2}{\gcd(m,2n^2)}
+\end{align}
+
 \section{Section 3}
 
 \section{Conclusion}