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Commit 740b76c2 authored by Luke Naylor's avatar Luke Naylor
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Iron out work up to dfn/lemma of e_{v,q}

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......@@ -141,7 +141,7 @@ $\chern_1^{\sage{extravagant.betaminus}}(F) = \sage{extravagant.twisted.ch[1]}$.
Using the above Theorem \ref{thm:loose-bound-on-r}, we get that the ranks of
tilt semistabilisers for $v$ are bounded above by $\sage{extravagant.loose_bound}$.
However, when computing all tilt semistabilisers for $v$ on $\PP^2$, the maximum
rank that appears turns out to be $\sage{extravagant.actual_rmax}$.
rank that appears turns out to be $\sage{round(extravagant.actual_rmax, 1)}$.
\end{example}
......@@ -811,49 +811,54 @@ from plots_and_expressions import q_sol, bgmlv_v, psi
line bundle $L$ with $c_1(L)$ generating $\neronseveri(X)$ and
$m\coloneqq\ell^2$.
Let $v$ be a fixed Chern character on this surface and
$R\coloneqq\chern_0(v) \leq \frac{1}{2}\lcm(m,2n^2){\chern_1^{\beta_{-}(v)}(v)}^2$.
Then the ranks of the pseudo-semistabilisers for $v$,
$R\coloneqq\chern_0(v)$.
Then the ranks of the pseudo-semistabilisers $u$ of $v$,
which are solutions to problem \ref{problem:problem-statement-2},
are bounded above by the following expression.
\begin{equation*}
\sage{main_theorem1.corollary_r_bound}
\end{equation*}
are bounded above as follows.
\noindent
If $R > \frac{1}{2}\lcm(m,2n^2){\chern_1^{\beta_{-}(v)}(v)}^2$,
then these ranks of pseudo-semistabilisers can instead bounded above by the
following.
\begin{equation*}
\frac{\Delta(v)\lcm(m,2n^2)}{2m}
\end{equation*}
\begin{align*}
r &\leq \sage{main_theorem1.corollary_r_bound}
&\text{if } R < \frac{\Delta(v)\lcm(m,2n^2)}{2m}
\\
r &\leq \frac{\Delta(v)\lcm(m,2n^2)}{2m}
&\text{if } R \geq \frac{\Delta(v)\lcm(m,2n^2)}{2m}
\end{align*}
\end{corollary}
\begin{proof}
The ranks of the pseudo-semistabilisers for $v$ are bounded above by the
maximum over $q\in [0, \chern_1^{\beta}(F)]$ of the expression in Theorem
maximum over $q\in [0, \chern_1^{\beta_{-}}(v)]$ of the expression in Theorem
\ref{thm:rmax_with_uniform_eps}.
Noticing that the expression is a maximum of two quadratic functions in $q$:
Noticing that the expression is a maximum of two quadratic functions in $q$
($\beta_0=\beta_{-}(v)$ in this context):
\begin{equation*}
f_1(q)\coloneqq\sage{main_theorem1.r_upper_bound1} \qquad
f_2(q)\coloneqq\sage{main_theorem1.r_upper_bound2}
\end{equation*}
These have their minimums at $q=0$ and $q=\chern_1^{\beta}(v)$ respectively.
It suffices to find their intersection in
$q\in [0, \chern_1^{\beta}(v)]$, if it exists,
and evaluating on of the $f_i$ there.
The intersection exists, provided that
$f_1(\chern_1^{\beta}(v)) \geq f_2(\chern_1^{\beta}(v))=R$,
or equivalently,
$R \leq \frac{1}{2}\lcm(m,2n^2){\chern_1^{\beta}(F)}^2$.
If this is not the case, then
$\min(f_1, f_2)$ reaches its maximum over the domain
$[0, \chern_1^{\beta_{-}}(v)]$
at $q=\chern_1^{\beta_{-}}(v)$, and so is
bounded by $f_1(\chern_1^{\beta_{-}(v)})$.
In the case where
$R \leq \frac{1}{2}\lcm(m,2n^2){\chern_1^{\beta}(F)}^2$,
These have their minimums at $q=0$ and $q=\chern_1^{\beta_{-}}(v)$ respectively,
with values 0 and $R>0$ respectively.
So provided that
$f_2\left(\chern^{\beta_{-}}_1(v)\right) < f_1\left(\chern^{\beta_{-}}_1(v)\right)$,
the maximum is achieved at their intersection.
Otherwise, the maximum is achieved at
$\chern^{\beta_{-}}_1(v)$.
So we can say that
\begin{align*}
r &\leq
f_{1}(q_{\mathrm{max}})
&\text{if } f_2\left(\chern^{\beta_{-}}_1(v)\right) <
f_1\left(\chern^{\beta_{-}}_1(v)\right)
\\ &&
\text{where $q_{\mathrm{max}}$ is the $q$-value where the $f_i$ intersect}
\\
r &\leq f_1\left(\chern^{\beta_{-}}(v)\right)
&\text{if } f_2\left(\chern^{\beta_{-}}_1(v)\right) \geq
f_1\left(\chern^{\beta_{-}}_1(v)\right)
\end{align*}
\noindent
In the first case,
solving for $f_1(q)=f_2(q)$ yields
\begin{equation*}
q=\sage{q_sol.expand()}
......@@ -862,8 +867,10 @@ And evaluating $f_1$ at this $q$-value gives:
\begin{equation*}
\sage{main_theorem1.corollary_intermediate}
\end{equation*}
Finally, noting that $\Delta(v)=(\chern_1^{\beta_{-}(v)}(v))^2\ell^2$, we get the bound as
stated in the corollary.
\noindent
Finally, noting that $\Delta(v)=\left(\chern_1^{\beta_{-}(v)}(v)\right)^2\ell^2$,
we get the bounds as stated in the statement of the Corollary.
\end{proof}
......@@ -876,7 +883,8 @@ giving $n=\sage{recurring.n}$.
Using the above corollary \ref{cor:direct_rmax_with_uniform_eps}, we get that
the ranks of tilt semistabilisers for $v$ are bounded above by
$\sage{recurring.corrolary_bound} \approx \sage{float(recurring.corrolary_bound)}$,
$\sage{recurring.corrolary_bound} \approx
\sage{round(float(recurring.corrolary_bound), 1)}$,
which is much closer to real maximum 25 than the original bound 144.
\end{example}
......@@ -889,7 +897,8 @@ giving $n=\sage{extravagant.n}$.
Using the above corollary \ref{cor:direct_rmax_with_uniform_eps}, we get that
the ranks of tilt semistabilisers for $v$ are bounded above by
$\sage{extravagant.corrolary_bound} \approx \sage{float(extravagant.corrolary_bound)}$,
$\sage{extravagant.corrolary_bound} \approx
\sage{round(float(extravagant.corrolary_bound), 1)}$,
which is much closer to real maximum $\sage{extravagant.actual_rmax}$ than the
original bound 215296.
\end{example}
......@@ -913,28 +922,7 @@ $q$, then iterate through values of $r$ within the bounds (dependent on $q$),
which would then determine $c$, and then find the corresponding possible values
for $d$.
Firstly, we only need to consider $r$-values for which $c\coloneqq\chern_1(E)$ is
integral:
\begin{sagesilent}
from plots_and_expressions import c_in_terms_of_q
\end{sagesilent}
\begin{equation*}
c =
\sage{c_in_terms_of_q.subs([q_value_expr,beta_value_expr])}
\in \ZZ
\end{equation*}
\noindent
That is, $r \equiv -\aa^{-1}\bb$ mod $n$ ($\aa$ is coprime to
$n$, and so invertible mod $n$).
\noindent
Let $\aa^{'}$ be an integer representative of $\aa^{-1}$ in $\ZZ/n\ZZ$.
Next, we seek to find a larger $\epsilon$ to use in place of $\epsilon_F$ in the
Next, we seek to find a larger $\epsilon$ to use in place of $\epsilon_v$ in the
proof of Theorem \ref{thm:rmax_with_uniform_eps}:
\begin{lemmadfn}[
......
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