From ee88d79f6967fb0227ea2cb960283df26e7770af Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Tue, 25 Jul 2023 23:03:48 +0100
Subject: [PATCH] Refactor extravagant example into notebook
---
main.tex | 33 ++++++---------------------------
plots_and_expressions.ipynb | 10 +++++-----
2 files changed, 11 insertions(+), 32 deletions(-)
diff --git a/main.tex b/main.tex
index c6618e6..c3c66cc 100644
--- a/main.tex
+++ b/main.tex
@@ -767,29 +767,19 @@ rank that appears turns out to be 25. This will be a recurring example to
illustrate the performance of later theorems about rank bounds
\end{example}
-\begin{example}[extravagant example: $v=(29, 13\ell, -3/2)$ on $\PP^2$]
-\label{exmpl:extravagant-first}
\begin{sagesilent}
-extravagant = Object()
-extravagant.chern = Chern_Char(29, 13, -3/2)
-extravagant.b = beta_minus(extravagant.chern)
-extravagant.twisted = extravagant.chern.twist(extravagant.b)
-extravagant.actual_rmax = 49313
+from plots_and_expressions import extravagant
\end{sagesilent}
+
+\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 $m=2$, $\beta_-=\sage{extravagant.b}$,
giving $n=\sage{extravagant.b.denominator()}$ and
$\chern_1^{\sage{extravagant.b}}(F) = \sage{extravagant.twisted.ch[1]}$.
-\begin{sagesilent}
-n = extravagant.b.denominator()
-m = 2
-loose_bound = (
- m*n^2*extravagant.twisted.ch[1]^2
-) / gcd(m, 2*n^2)
-\end{sagesilent}
Using the above theorem \ref{thm:loose-bound-on-r}, we get that the ranks of
-tilt semistabilizers for $v$ are bounded above by $\sage{loose_bound}$.
+tilt semistabilizers for $v$ are bounded above by $\sage{extravagant.loose_bound}$.
However, when computing all tilt semistabilizers for $v$ on $\PP^2$, the maximum
rank that appears turns out to be $\sage{extravagant.actual_rmax}$.
\end{example}
@@ -1675,20 +1665,9 @@ $\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so
that $m=2$, $\beta=\sage{extravagant.b}$,
giving $n=\sage{extravagant.b.denominator()}$.
-\begin{sagesilent}
-extravagant.n = extravagant.b.denominator()
-extravagant.bgmlv = extravagant.chern.Q_tilt()
-corrolary_bound = (
- r_upper_bound_all_q.expand()
- .subs(Delta==extravagant.bgmlv)
- .subs(nu==1) ## \ell^2=1 on P^2
- .subs(R==extravagant.chern.ch[0])
- .subs(n==extravagant.n)
-)
-\end{sagesilent}
Using the above corollary \ref{cor:direct_rmax_with_uniform_eps}, we get that
the ranks of tilt semistabilizers for $v$ are bounded above by
-$\sage{corrolary_bound} \approx \sage{float(corrolary_bound)}$,
+$\sage{extravagant.corrolary_bound} \approx \sage{float(extravagant.corrolary_bound)}$,
which is much closer to real maximum $\sage{extravagant.actual_rmax}$ than the
original bound 215296.
\end{example}
diff --git a/plots_and_expressions.ipynb b/plots_and_expressions.ipynb
index 84ad3f6..acefe54 100644
--- a/plots_and_expressions.ipynb
+++ b/plots_and_expressions.ipynb
@@ -362,7 +362,7 @@
"# RENDERED TO LATEX: recurring.twisted.ch[1]\n",
"n = recurring.b.denominator()\n",
"m = 2\n",
- "loose_bound = (\n",
+ "recurring.loose_bound = (\n",
" m*n^2*recurring.twisted.ch[1]^2\n",
") / gcd(m, 2*n^2)\n",
"# RENDERED TO LATEX: loose_bound\n",
@@ -376,7 +376,7 @@
"# RENDERED TO LATEX: extravagant.twisted.ch[1]\n",
"n = extravagant.b.denominator()\n",
"m = 2\n",
- "recurring.loose_bound = (\n",
+ "extravagant.loose_bound = (\n",
" m*n^2*extravagant.twisted.ch[1]^2\n",
") / gcd(m, 2*n^2)"
]
@@ -405,7 +405,7 @@
}
],
"source": [
- "loose_bound"
+ "recurring.loose_bound"
]
},
{
@@ -1690,7 +1690,7 @@
"source": [
"extravagant.n = extravagant.b.denominator()\n",
"extravagant.bgmlv = extravagant.chern.Q_tilt()\n",
- "corrolary_bound = (\n",
+ "extravagant.corrolary_bound = (\n",
" r_upper_bound_all_q.expand()\n",
" .subs(Delta==extravagant.bgmlv)\n",
" .subs(nu==1) ## \\ell^2=1 on P^2\n",
@@ -1723,7 +1723,7 @@
}
],
"source": [
- "float(corrolary_bound)"
+ "float(extravagant.corrolary_bound)"
]
},
{
--
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