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Commit a00becb5 authored by Luke Naylor's avatar Luke Naylor
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Adjust Schmidt section to more general surface

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