diff --git a/main.tex b/main.tex
index 634c09041854b6dd7a108eb17a695eff0f5939c1..f26a6c4cc02142a3d13e8e09c9e077dbd5d5997f 100644
--- a/main.tex
+++ b/main.tex
@@ -1361,7 +1361,7 @@ considering equations
\bgroup
\let\originalepsilon\epsilon
-\renewcommand\epsilon{{\originalepsilon_{F}}}
+\renewcommand\epsilon{{\originalepsilon_{v}}}
\begin{sagesilent}
var("epsilon")
@@ -1549,8 +1549,8 @@ Next, we seek to find a larger $\epsilon$ to use in place of $\epsilon_F$ in the
proof of theorem \ref{thm:rmax_with_uniform_eps}:
\begin{lemmadfn}[
- Finding better alternatives to $\epsilon_F$:
- $\epsilon_{q,1}$ and $\epsilon_{q,2}$
+ Finding a better alternative to $\epsilon_v$:
+ $\epsilon_{v,q}$
]
\label{lemdfn:epsilon_q}
Suppose $d \in \frac{1}{2}\ZZ$ satisfies the condition in
@@ -1564,39 +1564,23 @@ proof of theorem \ref{thm:rmax_with_uniform_eps}:
\noindent
Then we have:
- \begin{equation*}
+ \begin{equation}
+ \label{eqn:epsilon_q_lemma_prop}
d - \frac{(\aa r + 2\bb)\aa}{2n^2}
- \geq \epsilon_{q,2} \geq \epsilon_{q,1} > 0
-\end{equation*}
+ \geq \epsilon_{v,q} \geq \epsilon_v > 0
+ \end{equation}
-Where $\epsilon_{q,1}$ and $\epsilon_{q,2}$ are defined as follows:
+ \noindent
+ Where $\epsilon_{v,q}$ is defined as follows:
-\begin{equation*}
- \epsilon_{q,1} :=
- \frac{k_{q,1}}{2n^2}
- \qquad
- \epsilon_{q,2} :=
- \frac{k_{q,2}}{2n^2}
-\end{equation*}
-\begin{align*}
- \text{where }
- &k_{q,1} \text{ is the least }
- k\in\ZZ_{>0}\: s.t.:\:
- k \equiv -\aa\bb \mod n
-\\
- &k_{q,2} \text{ is the least }
- k\in\ZZ_{>0}\: s.t.:\:
- k \equiv \aa\bb (\aa\aa^{'}-2)
- \mod n\gcd(n,\aa^2)
-\end{align*}
+ \begin{equation*}
+ \epsilon_{v,q} :=
+ \frac{k_{q}}{2n^2}
+ \end{equation*}
+ with $k_{v,q}$ being the least $k\in\ZZ_{>0}$ satisfying $k \equiv -\aa\bb \mod n$
\end{lemmadfn}
-It is worth noting that $\epsilon_{q,2}$ is potentially larger than
-$\epsilon_{q,1}$
-but calculating it involves a $\gcd$, a modulo reduction, and a modulo $n$
-inverse, for each $q$ considered.
-
\begin{proof}
Consider the following:
@@ -1621,9 +1605,9 @@ Consider the following:
&\equiv k &&
\mod n^2
\\ &\Longrightarrow&
- \aa^2 \aa^{'}\bb - 2\aa\bb
+ \aa^2 \aa^{-1}\bb - 2\aa\bb
&\equiv k &&
- \mod \gcd(n^2, \aa^2 n)
+ \mod n
\label{eqn:better_eps_problem_k_mod_gcd2n2_a2mn}
\\ &\Longrightarrow&
-\aa\bb
@@ -1637,12 +1621,11 @@ eqn \ref{eqn:finding_better_eps_problem}.
Since such a $k$ must also satisfy eqn \ref{eqn:better_eps_problem_k_mod_n},
we can pick the smallest $k_{q,1} \in \ZZ_{>0}$ which satisfies this new condition
(a computation only depending on $q$ and $\beta$, but not $r$).
-We are then guaranteed that the gap $\frac{k}{2n^2}$ is at least
-$\epsilon_{q,1}$.
-Furthermore, $k$ also satisfies
-eqn \ref{eqn:better_eps_problem_k_mod_gcd2n2_a2mn}
-so we can also pick the smallest $k_{q,2} \in \ZZ_{>0}$ satisfying this condition,
-which also guarantees that the gap $\frac{k}{2n^2}$ is at least $\epsilon_{q,2}$.
+We are then guaranteed that $k_{v,q}$ is less than any $k$ satisfying eqn
+\ref{eqn:finding_better_eps_problem}, giving the first inequality in eqn
+\ref{eqn:epsilon_q_lemma_prop}.
+Furthermore, $k_{v,q}\geq 1$ gives the second part of the inequality:
+$\epsilon_{v,q}\geq\epsilon_v$, with equality when $k_{v,q}=1$.
\end{proof}
@@ -1653,14 +1636,14 @@ which also guarantees that the gap $\frac{k}{2n^2}$ is at least $\epsilon_{q,2}$
rational and expressed in lowest terms.
Then the ranks $r$ of the pseudo-semistabilizers $u$ for $v$ with
$\chern_1^\beta(u) = q = \frac{b_q}{n}$
- are bounded above by the following expression (with $i=1$ or $2$).
+ are bounded above by the following expression:
\begin{sagesilent}
var("delta", domain="real") # placeholder symbol to be replaced by k_{q,i}
\end{sagesilent}
\bgroup
- \def\kappa{k_{q,i}}
+ \def\kappa{k_{v,q}}
\def\psi{\chern_1^{\beta}(F)}
\begin{align*}
\min
@@ -1670,7 +1653,7 @@ var("delta", domain="real") # placeholder symbol to be replaced by k_{q,i}
\right)
\end{align*}
\egroup
- Where $k_{q,i}$ is defined as in definition/lemma \ref{lemdfn:epsilon_q},
+ Where $k_{v,q}$ is defined as in definition/lemma \ref{lemdfn:epsilon_q},
and $R = \chern_0(v)$
Furthermore, if $\aa \not= 0$ then
@@ -1683,12 +1666,11 @@ var("delta", domain="real") # placeholder symbol to be replaced by k_{q,i}
Just like in examples \ref{exmpl:recurring-first} and
\ref{exmpl:recurring-second},
take $\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so that
-$m=2$, $\beta=\sage{recurring.b}$, giving $n=\sage{recurring.b.denominator()}$
+$\beta=\sage{recurring.b}$, giving $n=\sage{recurring.b.denominator()}$
and $\chern_1^{\sage{recurring.b}}(F) = \sage{recurring.twisted.ch[1]}$.
%% TODO transcode notebook code
-Using the above theorem \ref{thm:rmax_with_eps1},
-TODO fill in values
-\end{example}
+The (non-exclusive) upper bounds for $r:=\chern_0(u)$ of a tilt semistabilizer $u$ of $v$
+in terms of the possible values for $q:=\chern_1^{\beta}(u)$ are as follows:
\begin{sagesilent}
import numpy as np
@@ -1705,41 +1687,27 @@ def bound_comparisons(example):
+ n^2*(v_twisted.ch[1] - q_val)^2/k
))
- def k_1(n, a_v, b_q):
+ def k(n, a_v, b_q):
n = int(n)
a_v = int(a_v)
b_q = int(b_q)
k = -a_v*b_q % n
return k if k > 0 else k + n
- def k_2(n, a_v, b_q):
- n = int(n)
- a_v = int(a_v)
- b_q = int(b_q)
- a_v_inv = inverse_mod(a_v, n)
- modulo = n*gcd(n, a_v^2)
- k = (a_v^2*a_v_inv*b_q - 2*a_v*b_q) % modulo
- return k if k > 0 else k + modulo
-
b_qs = list(range(example.twisted.ch[1]*n+1))
qs = list(map(lambda x: x/n,b_qs))
- k_1s = list(map(lambda b_q: k_1(n, a_v, b_q), b_qs))
- k_2s = list(map(lambda b_q: k_2(n, a_v, b_q), b_qs))
+ ks = list(map(lambda b_q: k(n, a_v, b_q), b_qs))
theorem2_bounds = [
theorem_bound(example.twisted, q_val, 1)
for q_val in qs
]
- theorem31_bounds = [
- theorem_bound(example.twisted, q_val, k)
- for q_val, k in zip(qs,k_1s)
- ]
- theorem32_bounds = [
+ theorem3_bounds = [
theorem_bound(example.twisted, q_val, k)
- for q_val, k in zip(qs,k_2s)
+ for q_val, k in zip(qs,ks)
]
- return qs, theorem2_bounds, theorem31_bounds, theorem32_bounds
+ return qs, theorem2_bounds, theorem3_bounds
-qs, theorem2_bounds, theorem31_bounds, theorem32_bounds = bound_comparisons(recurring)
+qs, theorem2_bounds, theorem3_bounds = bound_comparisons(recurring)
\end{sagesilent}
\directlua{ table_width = 3*4+1 }
@@ -1756,20 +1724,20 @@ end}
tex.sprint(cell)
end}
\\
- Thm \ref{thm:rmax_with_eps1} (i=1)
+ Thm \ref{thm:rmax_with_eps1}
\directlua{for i=0,table_width-1 do
- local cell = [[&$\noexpand\sage{theorem31_bounds[]] .. i .. "]}$"
- tex.sprint(cell)
-end}
- \\
- Thm \ref{thm:rmax_with_eps1} (i=2)
-\directlua{for i=0,table_width-1 do
- local cell = [[&$\noexpand\sage{theorem32_bounds[]] .. i .. "]}$"
+ local cell = [[&$\noexpand\sage{theorem3_bounds[]] .. i .. "]}$"
tex.sprint(cell)
end}
\end{tabular}
-\minorheading{Irrational $\beta$}
+It's worth noting that the bounds given by theorem \ref{thm:rmax_with_eps1}
+reach, but do not exceed the actual maximum rank 25 of the
+pseudo-semistabilizers of $v$ in this case.
+As a reminder, the original loose bound from theorem \ref{thm:loose-bound-on-r}
+was 144.
+
+\end{example}
\egroup % end scope where beta redefined to beta_{-}