First of the recurring examples for 3,2l,-2

main.tex

@@ -16,6 +16,7 @@
 \newcommand{\ZZ}{\mathbb{Z}}
 \newcommand{\RR}{\mathbb{R}}
 \newcommand{\NN}{\mathbb{N}}
+\newcommand{\PP}{\mathbb{P}}
 \newcommand{\chern}{\operatorname{ch}}
 \newcommand{\lcm}{\operatorname{lcm}}
 \newcommand{\firsttilt}[1]{\mathcal{B}^{#1}}
@@ -29,6 +30,7 @@
 \newtheorem{dfn}{Definition}[section]
 \newtheorem{lemma}{Lemma}[section]
 \newtheorem{fact}{Fact}[section]
+\newtheorem{example}{Example}[section]
 
 \begin{document}
 
@@ -40,6 +42,16 @@ from pseudowalls import *
 
 Δ = lambda v: v.Q_tilt()
 mu = stability.Mumford().slope
+
+def beta_minus(v):
+  beta = stability.Tilt().beta
+  solutions = solve(
+    stability.Tilt(alpha=0).degree(v)==0,
+    beta)
+  return min(map(lambda s: s.rhs(), solutions))
+
+class Object(object):
+  pass
 \end{sagesilent}
 
 \title{Explicit Formulae for Bounds on the Ranks of Tilt Destabilizers and
@@ -471,6 +483,7 @@ normal one. So $0 \leq \Delta(E)$ yields:
 \end{equation}
 
 \begin{theorem}[Bound on $r$ - Benjamin Schmidt]
+\label{thm:loose-bound-on-r}
 Given a Chern character $v$ such that $\beta_-:=\beta_{-}(v)\in\QQ$, the rank $r$ of
 any semistabilizer $E$ of some $F \in \firsttilt{\beta_-}$ with $\chern(F)=v$ is
 bounded above by:
@@ -507,6 +520,32 @@ bound for the rank of $E$:
 
 \end{proof}
 
+\begin{example}[$v=(3, 2\ell, -2)$ on $\PP^2$]
+\begin{sagesilent}
+recurring = Object()
+recurring.chern = Chern_Char(3, 2, -2)
+recurring.b = beta_minus(recurring.chern)
+recurring.twisted = recurring.chern.twist(recurring.b)
+\end{sagesilent}
+Taking $\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so
+that $m=2$, $\beta_-=\sage{recurring.b}$,
+giving $n=\sage{recurring.b.denominator()}$ and
+$\chern_1^{\sage{recurring.b}}(F) = \sage{recurring.twisted.ch[1]}$.
+
+\begin{sagesilent}
+n = recurring.b.denominator()
+m = 2
+loose_bound = (
+  m*n^2*recurring.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 $\sage{loose_bound}$.
+However, when computing all tilt semistabilizers for $v$ on $\PP^2$, the maximum
+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}
+
 \section{B.Schmidt's Method}
 
 Goals: