Adjust Schmidt section to more general surface

main.tex

@@ -701,9 +701,8 @@ is best seen with the following graph:
 
 
 This is where the rationality of $\beta_{-}$ comes in. If
-$\beta_{-} = \frac{a_v}{n}$ for some $a_v,n \in \ZZ$. 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).
+$\beta_{-} = \frac{a_v}{n}$ for some $a_v,n \in \ZZ$.
+Then $\chern^{\beta_-}_2(E) \in \frac{1}{\lcm(m,2n^2)}\ZZ$.
 In particular, since $\chern_2^{\beta_-}(E) > 0$ (by using 	$P=(\beta_-,0)$ in
 lemma \ref{lem:pseudo_wall_numerical_tests}) we must also have
 $\chern^{\beta_-}_2(E) \geq \frac{1}{\lcm(m,2n^2)}$, which then in turn gives a
@@ -724,7 +723,7 @@ from examples import recurring
 \begin{example}[$v=(3, 2\ell, -2)$ on $\PP^2$]
 \label{exmpl:recurring-first}
 Taking $\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so
-that $m=2$, $\beta_-=\sage{recurring.betaminus}$,
+that $m=1$, $\beta_-=\sage{recurring.betaminus}$,
 giving $n=\sage{recurring.n}$ and
 $\chern_1^{\sage{recurring.betaminus}}(F) = \sage{recurring.twisted.ch[1]}$.
 
@@ -742,7 +741,7 @@ from examples import extravagant
 \begin{example}[extravagant example: $v=(29, 13\ell, -3/2)$ on $\PP^2$]
 \label{exmpl:extravagant-first}
 Taking $\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so
-that $m=2$, $\beta_-=\sage{extravagant.betaminus}$,
+that $m=1$, $\beta_-=\sage{extravagant.betaminus}$,
 giving $n=\sage{extravagant.n}$ and
 $\chern_1^{\sage{extravagant.betaminus}}(F) = \sage{extravagant.twisted.ch[1]}$.