From 0f9a603dee6f96210f1c24af96760bf9af3b49a7 Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Sat, 10 Aug 2024 23:13:27 +0100
Subject: [PATCH] Implement Jul08 feedback (non-continued)
---
tex/bounds-on-semistabilisers.tex | 98 ++++++++++++++++++-------------
tex/computing-solutions.tex | 2 +-
2 files changed, 57 insertions(+), 43 deletions(-)
diff --git a/tex/bounds-on-semistabilisers.tex b/tex/bounds-on-semistabilisers.tex
index a9d4e4a..fac1c21 100644
--- a/tex/bounds-on-semistabilisers.tex
+++ b/tex/bounds-on-semistabilisers.tex
@@ -24,7 +24,7 @@ bundle $L$ such that $\ell\coloneqq c_1(L)$ generates $\neronseveri(X)$ and
take $m\coloneqq \ell^2$.
Given a Chern character $v$ with $\Delta(v) \geq 0$ and positive rank (or
-$\chern_0(v) = 0$ but $\chern_1(v) > 0$)
+$\chern_0(v) = 0$ and $\chern_1(v) > 0$)
such that
$\beta_-\coloneqq\beta_{-}(v)\in\QQ$, the rank $r$ of
any solution $u$ of Problem \ref{problem:problem-statement-2} is
@@ -155,8 +155,8 @@ we will need to consider each of the values which
$\chern_1^{\beta}(u)$ can take.
Doing this will allow us to eliminate possible values of $\chern_0(u)$ for which
each possible value of $\chern_1^{\beta}(u)$ leads to the failure of at least one
-of the inequalities.
-As opposed to only eliminating possible values of $\chern_0(u)$ for which all
+of the inequalities,
+as opposed to only eliminating possible values of $\chern_0(u)$ for which all
corresponding $\chern_1^{\beta}(u)$ fail one of the inequalities (which is what
was implicitly happening before in the proof of Theorem
\ref{thm:loose-bound-on-r}).
@@ -172,13 +172,15 @@ of travel.
\noindent
If $u$ is a solution to the Problem then $u$ satisfies:
- \begin{align}
- q\coloneqq \chern^{\beta_0}_1(u) &\in \left( 0, \chern_1^{\beta_0}(v) \right)
+ \begin{equation}
+ q\coloneqq \chern^{\beta_0}_1(u) \in \left( 0, \chern_1^{\beta_0}(v) \right)
\label{lem:eqn:cond-for-fixed-q}
- \\
- \chern_0(u) &> \frac{q}{\mu(v) - \beta_0}
+ \qquad
+ \text{and}
+ \qquad
+ \chern_0(u) > \frac{q}{\mu(v) - \beta_0}.
\nonumber
- \end{align}
+ \end{equation}
\noindent
Conversely, any $u = (r,c\ell,d\ell^2)$
@@ -221,14 +223,18 @@ of travel.
Then any solution $u$ satisfies:
\begin{align*}
\chern^{\beta_0}_1(u)
- &= \frac{b_q}{n}
- &\text{for some }
- b_q \in \left\{ 1, 2, \ldots, n\chern_1^{\beta_0}(v) - 1 \right\}
- \\
- a_v r &\equiv -b_q \pmod{n}
- \\
- r &> \frac{q}{\mu(v) - \beta_0}
+ = \frac{b_q}{n},
+ \qquad
+ a_v r &\equiv -b_q \pmod{n},
+ \quad
+ \text{and}
+ \qquad
+ r > \frac{q}{\mu(v) - \beta_0}
\end{align*}
+ \[
+ \text{for some }
+ b_q \in \left\{ 1, 2, \ldots, n\chern_1^{\beta_0}(v) - 1 \right\}.
+ \]
And any $u = (r,c\ell,d\ell^2)$
with $r,c\in \ZZ$ and $d\in \frac{1}{\lcm(m,2)}\ZZ$
satisfying these equations is a solution to the Problem if and only if, again,
@@ -275,9 +281,9 @@ $\chern_0(v)>0$, or $\chern_0(v)=0$ and $\chern_1(v)>0$,
and consider Problem
\ref{problem:problem-statement-1} or
\ref{problem:problem-statement-2}.
-Take $\beta_0 = \beta(P)$ where $P\in\Theta_v^-$ is the choice made in problem
+Take $\beta_0 = \beta(P)$ where $P\in\Theta_v^-$ is the choice made in Problem
\ref{problem:problem-statement-1}
-(or $\beta_0 = \beta_{-}$ for problem \ref{problem:problem-statement-2}).
+(or $\beta_0 = \beta_{-}$ for Problem \ref{problem:problem-statement-2}).
Lemma \ref{lem:fixed-q-semistabs-criterion} states that any solution
\begin{equation}
u \coloneqq \:(r,c\ell,d\ell^2)
@@ -295,7 +301,8 @@ to the Problem satisfies
\]
and also gives a lower bound for $r$ when considering $u$ with a fixed $q$.
This Section studies the extra numerical conditions that such $u$ must satisfy
-as per that Lemma, and will express them as bounds on $d$ in terms of $r$ (for a
+as given by Lemma \ref{lem:fixed-q-semistabs-criterion},
+and will express them as bounds on $d$ in terms of $r$ (for a
fixed $q$).
These bounds will later be used in Subsections
\ref{subsec:bounds-on-semistab-rank-prob-1} and
@@ -305,18 +312,21 @@ to construct upper bounds on $r$
Theorem \ref{thm:loose-bound-on-r} by considering bounds on
$\chern^{\beta_0}_0(u)$ in terms of the former).
-\subsubsection{Size of pseudo-wall\texorpdfstring{: $\chern_2^P(u)>0$}{}}
+\subsubsection{Radius of the pseudo-wall\texorpdfstring{: $\chern_2^P(u)>0$}{}}
\label{subsect-d-bound-radiuscond}
In the context of Problem \ref{problem:problem-statement-2}, this condition,
when rearranged to a bound on $d$, amounts to:
-\begin{align}
+\begin{equation}
\label{eqn:radius-cond-betamin}
- \chern_2^{\beta_{-}}(u) &> 0 \\
- d &> \frac{1}{2} {\beta_{-}}^2r + \beta_{-}q
+ \chern_2^{\beta_{-}}(u) &> 0
+ \qquad
+ \text{and}
+ \qquad
+ d > \frac{1}{2} {\beta_{-}}^2r + \beta_{-}q.
\nonumber
-\end{align}
+\end{equation}
\begin{sagesilent}
import other_P_choice as problem1
@@ -395,9 +405,9 @@ from plots_and_expressions import bgmlv3_d_upperbound_terms
d \leq
\sage{bgmlv3_d_upperbound_terms.linear}
+ \sage{bgmlv3_d_upperbound_terms.const}
- \sage{bgmlv3_d_upperbound_terms.hyperbolic}
+ \sage{bgmlv3_d_upperbound_terms.hyperbolic},
\qquad
- \text{when }r>R
+ \text{when }r>R.
\end{equation}
\noindent
@@ -411,7 +421,7 @@ given by $\chern^P_2(u)>0$ if $u$ already satisfies Equations
Since $r, R-r>0$, we have:
\begin{equation}
\label{lem:proof:slope-order-rltR}
- \beta_0, \mu(u) < \mu(v) < \mu(v-u)
+ \max(\beta_0, \mu(u)) < \mu(v) < \mu(v-u)
\end{equation}
\noindent
The first inequality coming from $P \in \Theta_v^{-}$ and Equation
@@ -443,6 +453,8 @@ see-saw principle.
&\text{since }\Delta(v) \geq 0
\:\text{and }\chern_0(v) > 0
\\
+ \text{So}
+ \quad
\frac{
\left(
q-\chern^{\beta_0}_1(v)
@@ -454,7 +466,9 @@ see-saw principle.
}
&>
2 \frac{\chern^{\beta_0}_2(v)}{R}
-\\
+ &
+ \text{and}
+ \quad
\chern_2^{\beta_0}(v)
- \frac{
\left(
@@ -491,25 +505,25 @@ for a potential solution to the problem of the form in Equation
from plots_and_expressions import bgmlv2_d_upperbound_terms
\end{sagesilent}
\begin{align}
- d &>&
+ d &>
\frac{1}{2}{\beta_0}^2 r
- &+ {\beta_0} q,
- \phantom{+}& % to keep terms aligned
+ + {\beta_0} q,
+ \phantom{+} % to keep terms aligned
&\qquad\text{when\:} r > 0
\label{eqn:radiuscond_d_bound_betamin}
\\
- d &\leq&
+ d &\leq
\sage{bgmlv2_d_upperbound_terms.problem2.linear}
- &+ \sage{bgmlv2_d_upperbound_terms.problem2.const}
- & +\sage{bgmlv2_d_upperbound_terms.problem2.hyperbolic},
+ + \sage{bgmlv2_d_upperbound_terms.problem2.const}
+ +\sage{bgmlv2_d_upperbound_terms.problem2.hyperbolic},
&\qquad\text{when\:} r > 0
\label{eqn:bgmlv2_d_bound_betamin}
\\
- d &\leq&
+ d &\leq
\sage{bgmlv3_d_upperbound_terms.problem2.linear}
- &+ \sage{bgmlv3_d_upperbound_terms.problem2.const}
+ + \sage{bgmlv3_d_upperbound_terms.problem2.const}
% ^ ch_2^\beta(F)=0 for beta_{-}
- & \sage{bgmlv3_d_upperbound_terms.problem2.hyperbolic},
+ \sage{bgmlv3_d_upperbound_terms.problem2.hyperbolic},
&\qquad\text{when\:} r > R
\label{eqn:bgmlv3_d_bound_betamin}
\end{align}
@@ -755,9 +769,9 @@ beta_value_expr
\label{eqn:positive_rad_condition_in_terms_of_q_beta}
\frac{1}{\lcm(m,2)}\ZZ
\ni
- \qquad
+ \:\:
\sage{positive_radius_condition_with_q.subs([q_value_expr,beta_value_expr]).factor()}
- \qquad
+ \:\:
\in
\frac{1}{2n^2}\ZZ
\end{equation}
@@ -915,12 +929,12 @@ we get the bounds as stated in the statement of the Corollary.
\begin{example}[$v=(3, 2\ell, -2)$ on $\PP^2$]
\label{exmpl:recurring-second}
-Just like in example \ref{exmpl:recurring-first}, take
+Just like in Example \ref{exmpl:recurring-first}, take
$\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so
that $m=2$, $\beta=\sage{recurring.betaminus}$,
giving $n=\sage{recurring.n}$.
-Using the above corollary \ref{cor:direct_rmax_with_uniform_eps}, we get that
+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
$\sage{recurring.corrolary_bound} \approx
\sage{round(float(recurring.corrolary_bound), 1)}$,
@@ -929,12 +943,12 @@ which is much closer to real maximum 25 than the original bound 144.
\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
+Just like in Example \ref{exmpl:extravagant-first}, take
$\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so
that $m=2$, $\beta=\sage{extravagant.betaminus}$,
giving $n=\sage{extravagant.n}$.
-Using the above corollary \ref{cor:direct_rmax_with_uniform_eps}, we get that
+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
$\sage{extravagant.corrolary_bound} \approx
\sage{round(float(extravagant.corrolary_bound), 1)}$,
diff --git a/tex/computing-solutions.tex b/tex/computing-solutions.tex
index ba6d1d1..8f6a641 100644
--- a/tex/computing-solutions.tex
+++ b/tex/computing-solutions.tex
@@ -58,7 +58,7 @@ In particular, it is the $\chern_1^{\beta_{-}}(v-u) \geq 0$ condition which
fails for $r$,$c$ pairs with large $r$ and $\frac{c}{r}$ too far from $\beta_{-}$.
This condition is only checked within the internal loop.
This, along with a conservative estimate for a bound on the $r$ values (as
-illustrated in example \ref{exmpl:recurring-first}) occasionally leads to slow
+illustrated in Example \ref{exmpl:recurring-first}) occasionally leads to slow
computations.
Here are some benchmarks to illustrate the performance benefits of the
--
GitLab