Newer
Older
\sage{0 <= bgmlv3.expand() }
\end{equation}
\noindent
Expressing $c$ in terms of $q$ as defined in (eqn \ref{eqn-cintermsofm})
we get the following:
\begin{sagesilent}
bgmlv3_with_q = (
bgmlv3
.expand()
.subs(c == c_in_terms_of_q)
)
\end{sagesilent}
\begin{equation}
\sage{0 <= bgmlv3_with_q}
\end{equation}
\noindent
This can be rearranged to express a bound on $d$ as follows:
\begin{sagesilent}
var("r_alt",domain="real") # r_alt = r - R temporary substitution
bgmlv3_with_q_reparam = (
bgmlv3_with_q
.subs(r == r_alt + R)
/r_alt # This operation assumes r_alt > 0
).expand()
bgmlv3_d_ineq = (
((0 <= bgmlv3_with_q_reparam)/2 + d) # Rearrange for d
.subs(r_alt == r - R) # Resubstitute r back in
.expand()
)
# Check that this equation represents a bound for d
bgmlv3_d_upperbound = bgmlv3_d_ineq.rhs() # Keep hold of lower bound for d
\end{sagesilent}
\begin{sagesilent}
# Seperate out the terms of the lower bound for d
bgmlv3_d_upperbound_without_hyp = (
bgmlv3_d_upperbound
.subs(1/(R-r) == 0)
)
bgmlv3_d_upperbound_const_term = (
bgmlv3_d_upperbound_without_hyp
.subs(r==0)
)
bgmlv3_d_upperbound_linear_term = (
bgmlv3_d_upperbound_without_hyp
- bgmlv3_d_upperbound_const_term
).expand()
bgmlv3_d_upperbound_exp_term = (
bgmlv3_d_upperbound
- bgmlv3_d_upperbound_without_hyp
).expand()
# Verify the simplified forms of the terms that will be mentioned in text
var("chb1v chb2v",domain="real") # symbol to represent ch_1^\beta(v)
var("psi phi", domain="real") # symbol to represent ch_1^\beta(v) and
# ch_2^\beta(v)
(
# keep hold of this alternative expression:
bgmlv3_d_upperbound_const_term_alt := (
)
)
.subs(phi == v.twist(beta).ch[2]) # subs real val of ch_1^\beta(v)
.expand()
)
# Keep hold of this alternative expression:
bgmlv3_d_upperbound_exp_term_alt :=
(
+ (C - q)^2/2
- D*R
)/(r-R)
)
.subs(phi == v.twist(beta).ch[2]) # subs real val of ch_1^\beta(v)
.expand()
)
assert bgmlv3_d_upperbound_exp_term == (
(
# Keep hold of this alternative expression:
bgmlv3_d_upperbound_exp_term_alt2 :=
(
(psi - q)^2/2/(r-R)
)
)
.subs(psi == v.twist(beta).ch[1]) # subs real val of ch_1^\beta(v)
.expand()
\bgroup
\def\psi{\chern_1^{\beta}(F)}
\def\phi{\chern_2^{\beta}(F)}
\begin{dmath}
\label{eqn-bgmlv3_d_upperbound}
d \leq
\sage{bgmlv3_d_upperbound_linear_term}
+ \sage{bgmlv3_d_upperbound_const_term_alt}
+ \sage{bgmlv3_d_upperbound_exp_term_alt2}
\end{dmath}
\egroup
\noindent
Viewing equation \ref{eqn-bgmlv3_d_upperbound} as an upper bound for $d$ give:
as a function of $r$, the terms can be rewritten as follows.
The constant term in $r$ is
$\chern^{\beta}_2(F) + \beta q$.
The linear term in $r$ is
$\sage{bgmlv3_d_upperbound_linear_term}$.
Finally, there is an hyperbolic term in $r$ which tends to 0 as $r \to \infty$,
\bgroup
\def\psi{\chern_1^{\beta}(F)}
$\sage{bgmlv3_d_upperbound_exp_term_alt2}$.
\egroup
In the case $\beta = \beta_{-}$ (or $\beta_{+}$) we have
$\chern^{\beta}_2(F) = 0$,
so some of these expressions simplify, and in particular, the constant and
linear terms match those of the other bounds in the previous subsections.
\subsubsection{All Bounds on $d$ together}
%% RECAP ON INEQUALITIES TOGETHER
%%%% RATIONAL BETA MINUS
\minorheading{Special Case: Rational $\beta_{-}$}
Suppose we take $\beta = \beta_{-}$ (so that $\chern^{\beta}_2(F)=0$)
in the previous subsections, to find all circular walls to the left of the
vertical wall (TODO as discussed in ref).
% redefine \beta (especially coming from rendered SageMath expressions)
% to be \beta_{-} for the rest of this subsubsection
\bgroup
\let\originalbeta\beta
\renewcommand\beta{{\originalbeta_{-}}}
\bgroup
% redefine \psi in sage expressions (placeholder for ch_1^\beta(F)
\def\psi{\chern_1^{\beta}(F)}
d &>&
\frac{1}{2}\beta^2 r
&+ \beta q,
\phantom{+}& % to keep terms aligned
&\qquad\text{when\:} r > 0
\label{eqn:radiuscond_d_bound_betamin}
\\
d &\leq&
\frac{1}{2}{\beta}^2r
&+ \beta q
+&
\frac{
q(\chern_1^{\beta}(v)-q)
}
{R-2r},
&\qquad\text{when\:} 0 < r < \frac{R}{2}
\label{eqn:bgmlv1_d_bound_betamin}
\sage{bgmlv2_d_upperbound_linear_term}
&+ \sage{bgmlv2_d_upperbound_const_term}
+& \sage{bgmlv2_d_upperbound_exp_term},
&\qquad\text{when\:} r > 0
\label{eqn:bgmlv2_d_bound_betamin}
\sage{bgmlv3_d_upperbound_linear_term}
&+ \sage{bgmlv3_d_upperbound_const_term_alt.subs(phi == 0)}
% ^ ch_2^\beta(F)=0 for beta_{-}
+& \sage{bgmlv3_d_upperbound_exp_term_alt2},
\label{eqn:bgmlv3_d_bound_betamin}
Furthermore, we get an extra bound for $d$ resulting from the condition that the
radius of the circular wall must be positive. As discussed in (TODO ref), this
is equivalent to $\chern^{\beta}_2(E) > 0$, which yields:
positive_radius_condition = (
(
(0 > - u.twist(beta).ch[2])
+ d # rearrange for d
.subs(solve(q == u.twist(beta).ch[1], c)[0]) # express c in term of q
.expand()
)
\begin{equation}
\label{eqn:positive_rad_d_bound_betamin}
\begin{sagesilent}
def beta_min(chern):
ts = stability.Tilt()
return min(
map(
lambda soln: soln.rhs(),
solve(
(ts.degree(chern))
.expand()
.subs(ts.alpha == 0),
beta
)
)
)
v_example = Chern_Char(3,2,-2)
q_example = 7/3
def plot_d_bound(
v_example,
q_example,
ymax=5,
ymin=-2,
xmax=20,
aspect_ratio=None
):
# Equations to plot imminently representing the bounds on d:
eq1 = (
(
beta^2*r/2
+ beta*q
+ q*(chb1v - q)/(R-2*r)
)
.subs(chb1v == v_example.twist(beta_min(v_example)).ch[1])
.subs(beta = beta_min(v_example))
.subs(q == q_example)
.subs(R == v_example.ch[0])
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)
eq2 = (
bgmlv2_d_upperbound
.subs(R == v_example.ch[0])
.subs(C == v_example.ch[1])
.subs(D == v_example.ch[2])
.subs(beta = beta_min(v_example))
.subs(q == q_example)
)
eq3 = (
bgmlv3_d_upperbound
.subs(R == v_example.ch[0])
.subs(C == v_example.ch[1])
.subs(D == v_example.ch[2])
.subs(beta = beta_min(v_example))
.subs(q == q_example)
)
eq4 = (
positive_radius_condition.rhs()
.subs(q == q_example)
.subs(beta = beta_min(v_example))
)
example_bounds_on_d_plot = (
plot(
eq3,
(r,v_example.ch[0],xmax),
color='green',
linestyle = "dashed",
legend_label=r"upper bound: $\Delta(v-u) \geq 0$",
)
+ plot(
eq2,
(r,0,xmax),
color='blue',
linestyle = "dashed",
legend_label=r"upper bound: $\Delta(u) \geq 0$"
)
+ plot(
eq4,
(r,0,xmax),
color='orange',
linestyle = "dotted",
legend_label=r"lower bound: $\mathrm{ch}_2^{\beta_{-}}(u)>0$"
)
+ plot(
eq1,
(r,0,v_example.ch[0]/2),
linestyle = "dashed",
legend_label=r"upper bound: $\Delta(u,v) \geq 0$"
)
)
example_bounds_on_d_plot.ymin(ymin)
example_bounds_on_d_plot.ymax(ymax)
example_bounds_on_d_plot.axes_labels(['$r$', '$d$'])
if aspect_ratio:
example_bounds_on_d_plot.set_aspect_ratio(aspect_ratio)
return example_bounds_on_d_plot
\end{sagesilent}
\begin{figure}
\centering
\centering
\sageplot[width=\linewidth]{plot_d_bound(v_example, 0, ymin=-0.5)}
\caption{$q = 0$ (all bounds other than green coincide on line)}
\label{fig:d_bounds_xmpl_min_q}
\end{subfigure}%
\centering
\sageplot[width=\linewidth]{plot_d_bound(v_example, 4, ymin=-3, ymax=3)}
\caption{$q = \chern^{\beta}(F)$ (all bounds other than blue coincide on line)}
\label{fig:d_bounds_xmpl_max_q}
\end{subfigure}
\caption{
Bounds on $d\coloneqq\chern_2(E)$ in terms of $r\coloneqq\chern_0(E)$ for fixed, extreme,
values of $q\coloneqq\chern_1^{\beta}(E)$.
Where $\chern(F) = (3,2,-2)$.
}
\label{fig:d_bounds_xmpl_extrm_q}
\end{figure}
Recalling that $q \coloneqq \chern^{\beta}_1(E) \in [0, \chern^{\beta}_1(F)]$,
it is worth noting that the extreme values of $q$ in this range lead to the
tightest bounds on $d$, as illustrated in figure
(\ref{fig:d_bounds_xmpl_extrm_q}).
In fact, in each case, one of the weak upper bounds coincides with one of the
weak lower bounds, (implying no possible destabilizers $E$ with
$\chern_0(E)=\vcentcolon r>R\coloneqq\chern_0(F)$ for these $q$-values).
This indeed happens in general since the right hand sides of
(eqn \ref{eqn:bgmlv2_d_bound_betamin}) and
(eqn \ref{eqn:positive_rad_d_bound_betamin}) match when $q=0$.
In the other case, $q=\chern^{\beta}_1(F)$, it is the right hand sides of
(eqn \ref{eqn:bgmlv3_d_bound_betamin}) and
(eqn \ref{eqn:positive_rad_d_bound_betamin}) which match.
The more generic case, when $0 < q\coloneqq\chern_1^{\beta}(E) < \chern_1^{\beta}(F)$
for the bounds on $d$ in terms of $r$ is illustrated in figure
(\ref{fig:d_bounds_xmpl_gnrc_q}).
The question of whether there are pseudo-destabilizers of arbitrarily large
rank, in the context of the graph, comes down to whether there are points
$(r,d) \in \ZZ \oplus \frac{1}{2} \ZZ$ (with large $r$)
% TODO have a proper definition for pseudo-destabilizers/walls
that fit above the yellow line (ensuring positive radius of wall) but below the
blue and green (ensuring $\Delta(E), \Delta(G) > 0$).
These lines have the same assymptote at $r \to \infty$
(eqns \ref{eqn:bgmlv2_d_bound_betamin},
\ref{eqn:bgmlv3_d_bound_betamin},
\ref{eqn:positive_rad_d_bound_betamin}).
As mentioned in the introduction (sec \ref{sec:intro}), the finiteness of these
solutions is entirely determined by whether $\beta$ is rational or irrational.
Some of the details around the associated numerics are explored next.
\begin{figure}
\centering
\sageplot[
width=\linewidth
]{plot_d_bound(v_example, 2, ymax=4, ymin=-2, aspect_ratio=1)}
\caption{
Bounds on $d\coloneqq\chern_2(E)$ in terms of $r\coloneqq \chern_0(E)$ for a fixed
value $\chern_1^{\beta}(F)/2$ of $q\coloneqq\chern_1^{\beta}(E)$.
Where $\chern(F) = (3,2,-2)$.
}
\label{fig:d_bounds_xmpl_gnrc_q}
\end{figure}
\subsection{Bounds on Semistabilizer Rank \texorpdfstring{$r$}{r}}
Now, the inequalities from the above (TODO REF) will be used to find, for
each given $q=\chern^{\beta}_1(E)$, how large $r$ needs to be in order to leave
no possible solutions for $d$. At that point, there are no Chern characters
$(r,c,d)$ that satisfy all inequalities to give a pseudowall.
\subsubsection{All Semistabilizers Left of Vertical Wall for Rational Beta min}
The strategy here is similar to what was shown in (sect
\ref{sec:twisted-chern}).
One specialization here is to use that $\ell\coloneqq c_1(H)$ generates $NS(X)$, so that
in fact, any Chern character can be written as
$\left(r,c\ell,\frac{e}{2}\ell^2\right)$ for $r,c,e\in\ZZ$.
var("a_v b_q n") # Define symbols introduce for values of beta and q
beta_value_expr = (beta == a_v/n)
q_value_expr = (q == b_q/n)
\end{sagesilent}
\renewcommand{\aa}{{a_v}}
\newcommand{\bb}{{b_q}}
Suppose $\beta = \frac{\aa}{n}$ for some coprime $n \in \NN,\aa \in \ZZ$.
Then fix a value of $q$:
\begin{equation}
\in
\frac{1}{n} \ZZ
\cap [0, \chern_1^{\beta}(F)]
\end{equation}
as noted at the beginning of this section (\ref{sec:refinement}).
Substituting the current values of $q$ and $\beta$ into the condition for the
radius of the pseudo-wall being positive
(eqn \ref{eqn:positive_rad_d_bound_betamin}) we get:
\begin{equation}
\label{eqn:positive_rad_condition_in_terms_of_q_beta}
\sage{positive_radius_condition.subs([q_value_expr,beta_value_expr]).factor()}
\in
\frac{1}{2n^2}\ZZ
\end{equation}
# placeholder for the specific values of k (start with 1):
var("kappa", domain="real")
assymptote_gap_condition1 = (kappa/(2*n^2) < bgmlv2_d_upperbound_exp_term)
assymptote_gap_condition2 = (kappa/(2*n^2) < bgmlv3_d_upperbound_exp_term_alt2)
)
assert r_upper_bound1.lhs() == r
r_upper_bound2 = (
* (r-R) * 2*n^2 / kappa + R
)
assert r_upper_bound2.lhs() == r
\end{sagesilent}
\begin{theorem}[Bound on $r$ \#1]
\label{thm:rmax_with_uniform_eps}
Let $v = (R,C\ell,D\ell^2)$ be a fixed Chern character. Then the ranks of the
pseudo-semistabilizers for $v$ with
$\chern_1^\beta = q$
are bounded above by the following expression.
\bgroup
\def\psi{\chern_1^{\beta}(F)}
\sage{r_upper_bound1.subs(kappa==1).rhs()}, \:\:
\sage{r_upper_bound2.subs(kappa==1).rhs()}
Taking the maximum of this expression over
$q \in [0, \chern_1^{\beta}(F)]\cap \frac{1}{n}\ZZ$
would give an upper bound for the ranks of pseudo-semistabilizers for $v$.
Both $d$ and the lower bound in
(eqn \ref{eqn:positive_rad_condition_in_terms_of_q_beta})
are elements of $\frac{1}{2n^2}\ZZ$.
So, if any of the two upper bounds on $d$ come to within
$\frac{1}{2n^2}$ of this lower bound, then there are no solutions for
Hence any corresponding $r$ cannot be a rank of a
pseudo-semistabilizer for $v$.
To avoid this, we must have,
considering equations
\ref{eqn:bgmlv2_d_bound_betamin},
\ref{eqn:bgmlv3_d_bound_betamin},
\ref{eqn:positive_rad_d_bound_betamin}.
\bgroup
\let\originalepsilon\epsilon
\renewcommand\epsilon{{\originalepsilon_{v}}}
\begin{sagesilent}
var("epsilon")
var("chbv") # symbol to represent \chern_1^{\beta}(v)
# Tightness conditions:
bounds_too_tight_condition1 = (
bgmlv2_d_upperbound_exp_term
< epsilon
)
bounds_too_tight_condition2 = (
bgmlv3_d_upperbound_exp_term_alt.subs(chbv==0)
< epsilon
)
\end{sagesilent}
\bgroup
\def\psi{\chern_1^{\beta}(F)}
\begin{equation}
\min\left(
\sage{bgmlv2_d_upperbound_exp_term},
\sage{bgmlv3_d_upperbound_exp_term_alt2}
\noindent
This is equivalent to:
\bgroup
\def\psi{\chern_1^{\beta}(F)}
\label{eqn:thm-bound-for-r-impossible-cond-for-r}
r \leq
r_upper_bound1.subs(kappa==1).rhs()
r_upper_bound2.subs(kappa==1).rhs()
\egroup % end scope where epsilon redefined
q_sol = solve(
r_upper_bound1.subs(kappa==1).rhs()
== r_upper_bound2.subs(kappa==1).rhs()
, q)[0].rhs()
r_upper_bound_all_q = (
r_upper_bound1.rhs()
.expand()
.subs(q==q_sol)
.subs(psi**2 == Delta/nu^2)
.subs(1/psi**2 == nu^2/Delta)
)
\end{sagesilent}
Luke Naylor
committed
\begin{corrolary}[Bound on $r$ \#2]
\label{cor:direct_rmax_with_uniform_eps}
Luke Naylor
committed
Let $v$ be a fixed Chern character and
Luke Naylor
committed
Then the ranks of the pseudo-semistabilizers for $v$
are bounded above by the following expression.
\bgroup
\let\originalDelta\Delta
\renewcommand\Delta{{\originalDelta(v)}}
\begin{equation*}
\sage{r_upper_bound_all_q.expand()}
\end{equation*}
Luke Naylor
committed
\egroup
\end{corrolary}
\begin{proof}
\bgroup
\def\psi{\chern_1^{\beta}(F)}
\let\originalDelta\Delta
The ranks of the pseudo-semistabilizers for $v$ are bounded above by the
maximum over $q\in [0, \chern_1^{\beta}(F)]$ of the expression in theorem
\ref{thm:rmax_with_uniform_eps}.
Noticing that the expression is a maximum of two quadratic functions in $q$:
\begin{equation*}
f_1(q)\coloneqq\sage{r_upper_bound1.subs(kappa==1).rhs()} \qquad
f_2(q)\coloneqq\sage{r_upper_bound2.subs(kappa==1).rhs()}
\end{equation*}
These have their minimums at $q=0$ and $q=\chern_1^{\beta}(F)$ respectively.
It suffices to find their intersection in
$q\in [0, \chern_1^{\beta}(F)]$, if it exists,
and evaluating on of the $f_i$ there.
The intersection exists, provided that
$f_1(\chern_1^{\beta}(F))>f_2(\chern_1^{\beta}(F))=R$,
or equivalently,
$R \leq n^2{\chern_1^{\beta}(F)}^2$.
Setting $f_1(q)=f_2(q)$ yields
$q=\sage{q_sol.expand()}$.
And evaluating $f_1$ at this $q$-value gives:
$\sage{r_upper_bound_all_q.expand().subs([nu==1,Delta==psi^2])}$.
Finally, noting that $\originalDelta(v)=\psi^2\ell^2$, we get the bound as
\begin{example}[$v=(3, 2\ell, -2)$ on $\PP^2$]
\label{exmpl:recurring-second}
Just like in example \ref{exmpl:recurring-first}, take
$\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so
that $m=2$, $\beta=\sage{recurring.b}$,
giving $n=\sage{recurring.b.denominator()}$.
\begin{sagesilent}
recurring.n = recurring.b.denominator()
recurring.bgmlv = recurring.chern.Q_tilt()
corrolary_bound = (
r_upper_bound_all_q.expand()
.subs(Delta==recurring.bgmlv)
.subs(nu==1) ## \ell^2=1 on P^2
.subs(R==recurring.chern.ch[0])
.subs(n==recurring.n)
)
\end{sagesilent}
Using the above corrolary \ref{cor:direct_rmax_with_uniform_eps}, we get that
the ranks of tilt semistabilizers for $v$ are bounded above by
$\sage{corrolary_bound} \approx \sage{float(corrolary_bound)}$,
which is much closer to real maximum 25 than the original bound 144.
\end{example}
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\begin{example}[extravagant example: $v=(29, 13\ell, -3/2)$ on $\PP^2$]
\label{exmpl:extravagant-second}
Just like in example \ref{exmpl:extravagant-first}, take
$\ell=c_1(\mathcal{O}(1))$ 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 corrolary \ref{cor:direct_rmax_with_uniform_eps}, we get that
the ranks of tilt semistabilizers for $v$ are bounded above by
$\sage{corrolary_bound} \approx \sage{float(corrolary_bound)}$,
which is much closer to real maximum $\sage{extravagant.actual_rmax}$ than the
original bound 215296.
%% refinements using specific values of q and beta
These bound can be refined a bit more by considering restrictions from the
possible values that $r$ take.
Furthermore, the proof of theorem \ref{thm:rmax_with_uniform_eps} uses the fact
that, given an element of $\frac{1}{2n^2}\ZZ$, the closest non-equal element of
$\frac{1}{2}\ZZ$ is at least $\frac{1}{2n^2}$ away. However this a
conservative estimate, and a larger gap can sometimes be guaranteed if we know
this value of $\frac{1}{2n^2}\ZZ$ explicitly.
The expressions that will follow will be a bit more complicated and have more
parts which depend on the values of $q$ and $\beta$, even their numerators
$\aa,\bb$ specifically. The upcoming theorem (TODO ref) is less useful as a
`clean' formula for a bound on the ranks of the pseudo-semistabilizers, but has a
purpose in the context of writing a computer program to find
pseudo-semistabilizers. Such a program would iterate through possible values of
$q$, 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
\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 $n$).
positive_radius_condition
.rhs()
.subs([q_value_expr,beta_value_expr])
.factor()
.numerator()
\end{sagesilent}
\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
proof of theorem \ref{thm:rmax_with_uniform_eps}:
\begin{lemmadfn}[
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
eqn \ref{eqn:positive_rad_condition_in_terms_of_q_beta}.
That is:
\begin{equation*}
\sage{positive_radius_condition.subs([q_value_expr,beta_value_expr]).factor()}
\end{equation*}
\noindent
Then we have:
\begin{equation}
\label{eqn:epsilon_q_lemma_prop}
d - \frac{(\aa r + 2\bb)\aa}{2n^2}
\geq \epsilon_{v,q} \geq \epsilon_v > 0
\end{equation}
\noindent
Where $\epsilon_{v,q}$ is defined as follows:
\begin{equation*}
\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}
\begin{proof}
- \frac{
(\aa r+2\bb)\aa
}{
2n^2
}
\quad \text{for some } x \in \ZZ
\span \span \span \span \span
\label{eqn:finding_better_eps_problem}
\aa^2 \aa^{-1}\bb - 2\aa\bb
\mod n
\label{eqn:better_eps_problem_k_mod_gcd2n2_a2mn}
\label{eqn:better_eps_problem_k_mod_n}
In our situation, we want to find the least $k$ satisfying
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 $k_{v,q}$ 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 $k_{v,q}=1$.
\begin{theorem}[Bound on $r$ \#3]
\label{thm:rmax_with_eps1}
Let $v$ be a fixed Chern character, with $\frac{a_v}{n}=\beta\coloneqq\beta(v)$
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:
var("delta", domain="real") # placeholder symbol to be replaced by k_{q,i}
\def\kappa{k_{v,q}}
\def\psi{\chern_1^{\beta}(F)}
\begin{align*}
\min
\left(
\sage{r_upper_bound1.rhs()}, \:\:
\sage{r_upper_bound2.rhs()}
\right)
\end{align*}
\egroup
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
$r \equiv \aa^{-1}b_q (\mod n)$.
\begin{example}[$v=(3, 2\ell, -2)$ on $\PP^2$]
\label{exmpl:recurring-third}
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
$\beta=\sage{recurring.b}$, giving $n=\sage{recurring.b.denominator()}$
and $\chern_1^{\sage{recurring.b}}(F) = \sage{recurring.twisted.ch[1]}$.
The (non-exclusive) upper bounds for $r\coloneqq\chern_0(u)$ of a tilt semistabilizer $u$ of $v$
in terms of the possible values for $q\coloneqq\chern_1^{\beta}(u)$ are as follows:
\begin{sagesilent}
import numpy as np
def bound_comparisons(example):
n = example.b.denominator()
a_v = example.b.numerator()
def theorem_bound(v_twisted, q_val, k):
return int(min(
n^2*q_val^2/k
,
v_twisted.ch[0]
+ n^2*(v_twisted.ch[1] - q_val)^2/k
))
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
b_qs = list(range(example.twisted.ch[1]*n+1))
qs = list(map(lambda x: x/n,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
]
theorem3_bounds = [
theorem_bound(example.twisted, q_val, k)
for q_val, k in zip(qs,ks)
return qs, theorem2_bounds, theorem3_bounds
qs, theorem2_bounds, theorem3_bounds = bound_comparisons(recurring)
\directlua{ table_width = 3*4+1 }
\begin{tabular}{l\directlua{for i=0,table_width-1 do tex.sprint([[|c]]) end}}
$q=\chern_1^\beta(u)$
\directlua{for i=0,table_width-1 do
local cell = [[&$\noexpand\sage{qs[]] .. i .. "]}$"
tex.sprint(cell)
end}
\\ \hline
Thm \ref{thm:rmax_with_uniform_eps}
\directlua{for i=0,table_width-1 do
local cell = [[&$\noexpand\sage{theorem2_bounds[]] .. i .. "]}$"
tex.sprint(cell)
end}
\\
Thm \ref{thm:rmax_with_eps1}
\directlua{for i=0,table_width-1 do
local cell = [[&$\noexpand\sage{theorem3_bounds[]] .. i .. "]}$"
tex.sprint(cell)
end}
\end{tabular}
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}
\begin{example}[extravagant example: $v=(29, 13\ell, -3/2)$ on $\PP^2$]
\label{exmpl:extravagant-third}
Just like in examples \ref{exmpl:extravagant-first} and
\ref{exmpl:extravagant-second},
take $\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so that
$\beta=\sage{extravagant.b}$, giving $n=\sage{n:=extravagant.b.denominator()}$
and $\chern_1^{\sage{extravagant.b}}(F) = \sage{extravagant.twisted.ch[1]}$.
This example was chosen because the $n$ value is moderatly large, giving more
possible values for $k_{v,q}$, in dfn/lemma \ref{lemdfn:epsilon_q}. This allows
for a larger possible difference between the bounds given by theorems
\ref{thm:rmax_with_uniform_eps} and \ref{thm:rmax_with_eps1}, with the bound
from the second being up to $\sage{n}$ smaller, for any given $q$ value.
The (non-exclusive) upper bounds for $r\coloneqq\chern_0(u)$ of a tilt semistabilizer $u$ of $v$
in terms of the first few smallest possible values for $q\coloneqq\chern_1^{\beta}(u)$ are as follows:
\begin{sagesilent}
qs, theorem2_bounds, theorem3_bounds = bound_comparisons(extravagant)
\end{sagesilent}
1938
1939
1940
1941
1942
1943
1944
1945
1946
1947
1948
1949
1950
1951
1952
1953
1954
1955
1956
1957
1958
1959
1960
1961
\noindent
\directlua{ table_width = 12 }
\begin{tabular}{l\directlua{for i=0,table_width do tex.sprint([[|c]]) end}}
$q=\chern_1^\beta(u)$
\directlua{for i=0,table_width-1 do
local cell = [[&$\noexpand\sage{qs[]] .. i .. "]}$"
tex.sprint(cell)
end}
&$\cdots$
\\ \hline
Thm \ref{thm:rmax_with_uniform_eps}
\directlua{for i=0,table_width-1 do
local cell = [[&$\noexpand\sage{theorem2_bounds[]] .. i .. "]}$"
tex.sprint(cell)
end}
&$\cdots$
\\
Thm \ref{thm:rmax_with_eps1}
\directlua{for i=0,table_width-1 do
local cell = [[&$\noexpand\sage{theorem3_bounds[]] .. i .. "]}$"
tex.sprint(cell)
end}
&$\cdots$
\end{tabular}
\noindent
However the reduction in the overall bound on $r$ is not as drastic, since all
possible values for $k_{v,q}$ in $\{1,2,\ldots,\sage{n}\}$ are iterated through
cyclically as we consider successive possible values for $q$.
Calculating the maximums over all values of $q$ yields
$\sage{max(theorem2_bounds)}$ for theorem \ref{thm:rmax_with_uniform_eps}, and
$\sage{max(theorem3_bounds)}$ for theorem \ref{thm:rmax_with_eps1}.
\end{example}
\egroup % end scope where beta redefined to beta_{-}
\subsubsection{All Semistabilizers Giving Sufficiently Large Circular Walls Left
of Vertical Wall}
Goals:
\begin{itemize}
\item refresher on strategy
\item point out no need for rational beta
\item calculate intersection of bounds?
\end{itemize}
\subsection{Irrational $\beta_{-}$}
Goals:
\begin{itemize}
\item Point out if only looking for sufficiently large wall, look at above
subsubsection
\item Relate to Pell's equation through coordinate change?
\item Relate to numerical condition described by Yanagida/Yoshioka
\end{itemize}
\newpage
\printbibliography