diff --git a/main.tex b/main.tex
index 839c5d4d90de84735482d25a5a1b6acc2a3a46be..efd077090495400b841f90ae9dc24a2833979845 100644
--- a/main.tex
+++ b/main.tex
@@ -45,25 +45,7 @@ sorting=ynt
\begin{document}
-\begin{sagesilent}
-# Requires extra package:
-#! sage -pip install "pseudowalls==0.0.3" --extra-index-url https://gitlab.com/api/v4/projects/43962374/packages/pypi/simple
-
-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{Tighter Bounds for Ranks of Tilt Semistabilizers on Picard Rank 1 Surfaces
\\[1em] \large
@@ -330,33 +312,7 @@ Because of this, when using these characteristic curves, only positive ranks are
considered, as negative rank objects are implicitly considered on the right hand
side of $V_v$.
-\begin{sagesilent}
-def charact_curves(v):
- alpha = stability.Tilt().alpha
- beta = stability.Tilt().beta
- coords_range = (beta, -4, 5), (alpha, 0, 4)
- text_args = {"fontsize":"xx-large", "clip":True}
- black_text_args = {"rgbcolor": "black", **text_args}
- p = (
- implicit_plot(stability.Tilt().degree(v), *coords_range )
- + line([(mu(v),0),(mu(v),5)], linestyle = "dashed")
- + text(r"$\Theta_v^+$",[3.5, 2], rotation=45, **text_args)
- + text(r"$V_v$", [0.43, 1.5], rotation=90, **text_args)
- + text(r"$\Theta_v^-$", [-2.2, 2], rotation=-45, **text_args)
- + text(r"$\nu_{\alpha, \beta}(v)>0$", [-3, 1], **black_text_args)
- + text(r"$\nu_{\alpha, \beta}(v)<0$", [-1, 3], **black_text_args)
- + text(r"$\nu_{\alpha, \beta}(-v)>0$", [2, 3], **black_text_args)
- + text(r"$\nu_{\alpha, \beta}(-v)<0$", [4, 1], **black_text_args)
- )
- p.xmax(5)
- p.xmin(-4)
- p.ymax(4)
- p.axes_labels([r"$\beta$", r"$\alpha$"])
- return p
-
-v1 = Chern_Char(3, 2, -2)
-v2 = Chern_Char(3, 2, 2/3)
-\end{sagesilent}
+
\begin{sagesilent}
from plots_and_expressions import \
@@ -728,9 +684,7 @@ The restrictions on $\chern^{\beta_-}_0(E)$ and $\chern^{\beta_-}_2(E)$
is best seen with the following graph:
% TODO: hyperbola restriction graph (shaded)
-\begin{sagesilent}
-var("m") # Initialize symbol for variety parameter
-\end{sagesilent}
+
This is where the rationality of $\beta_{-}$ comes in. If
$\beta_{-} = \frac{a_v}{n}$ for some $a_v,n \in \ZZ$. Then
@@ -888,17 +842,7 @@ Take $\beta = \beta(P)$ where $P\in\Theta_v^-$ is the choice made in problem
&& \text{where $r,c,2d\in \ZZ$}
\end{align}
-\begin{sagesilent}
-# Requires extra package:
-#! sage -pip install "pseudowalls==0.0.3" --extra-index-url https://gitlab.com/api/v4/projects/43962374/packages/pypi/simple
-
-from pseudowalls import *
-
-v = Chern_Char(*var("R C D", domain="real"))
-u = Chern_Char(*var("r c d", domain="real"))
-Δ = lambda v: v.Q_tilt()
-\end{sagesilent}
Recall from condition \ref{item:chern1bound:lem:num_test_prob1} in
lemma \ref{lem:num_test_prob1}
@@ -906,18 +850,7 @@ lemma \ref{lem:num_test_prob1}
that $\chern_1^{\beta}(u)$ has fixed bounds in terms of $\chern_1^{\beta}(v)$,
and so we can write:
-\begin{sagesilent}
-ts = stability.Tilt
-var("beta", domain="real")
-
-c_lower_bound = -(
- ts(beta=beta).rank(u)
- /ts().alpha
-).expand() + c
-var("q", domain="real")
-c_in_terms_of_q = c_lower_bound + q
-\end{sagesilent}
\begin{sagesilent}
from plots_and_expressions import c_in_terms_of_q
@@ -974,22 +907,10 @@ from lemma \ref{lem:num_test_prob1}
(or corollary \ref{cor:num_test_prob2}).
-\begin{sagesilent}
-# First Bogomolov-Gieseker form expression that must be non-negative:
-bgmlv2 = Δ(u)
-\end{sagesilent}
-
\noindent
Expressing $\Delta(u)\geq 0$ in terms of $q$ as defined in eqn \ref{eqn-cintermsofm}
we get the following:
-\begin{sagesilent}
-bgmlv2_with_q = (
- bgmlv2
- .expand()
- .subs(c == c_in_terms_of_q)
-)
-\end{sagesilent}
\begin{sagesilent}
from plots_and_expressions import bgmlv2_with_q
@@ -1006,15 +927,6 @@ This can be rearranged to express a bound on $d$ as follows
in lemma \ref{lem:num_test_prob1} or corollary
\ref{cor:num_test_prob2} that $r>0$):
-\begin{sagesilent}
-bgmlv2_d_ineq = (
- (0 <= bgmlv2_with_q)/2/r # rescale assuming r > 0
- + d # Rearrange for d
-).expand()
-
-# Keep hold of lower bound for d
-bgmlv2_d_upperbound = bgmlv2_d_ineq.rhs()
-\end{sagesilent}
\begin{sagesilent}
from plots_and_expressions import bgmlv2_d_ineq
@@ -1024,30 +936,6 @@ from plots_and_expressions import bgmlv2_d_ineq
\sage{bgmlv2_d_ineq}
\end{equation}
-\begin{sagesilent}
-# Seperate out the terms of the lower bound for d
-
-bgmlv2_d_upperbound_without_hyp = (
- bgmlv2_d_upperbound
- .subs(1/r == 0)
-)
-
-bgmlv2_d_upperbound_const_term = (
- bgmlv2_d_upperbound_without_hyp
- .subs(r==0)
-)
-
-bgmlv2_d_upperbound_linear_term = (
- bgmlv2_d_upperbound_without_hyp
- - bgmlv2_d_upperbound_const_term
-).expand()
-
-bgmlv2_d_upperbound_exp_term = (
- bgmlv2_d_upperbound
- - bgmlv2_d_upperbound_without_hyp
-).expand()
-\end{sagesilent}
-
\begin{sagesilent}
from plots_and_expressions import \
bgmlv2_d_upperbound_const_term, \
@@ -1084,100 +972,6 @@ from lemma \ref{lem:num_test_prob1}
Expressing $\Delta(v-u)\geq 0$ in term of $q$ and rearranging as a bound on
$d$ yields:
-\begin{sagesilent}
-# Third Bogomolov-Gieseker form expression that must be non-negative:
-bgmlv3 = Δ(v-u)
-bgmlv3_with_q = (
- bgmlv3
- .expand()
- .subs(c == c_in_terms_of_q)
-)
-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
-assert bgmlv3_d_ineq.lhs() == d
-
-bgmlv3_d_upperbound = bgmlv3_d_ineq.rhs() # Keep hold of lower bound for d
-
-# 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)
-
-assert bgmlv3_d_upperbound_const_term == (
- (
- # keep hold of this alternative expression:
- bgmlv3_d_upperbound_const_term_alt := (
- phi
- + beta*q
- )
- )
- .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_alt :=
- (
- R*phi
- + (C - q)^2/2
- + R*beta*q
- - 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()
-)
-\end{sagesilent}
\begin{sagesilent}
from plots_and_expressions import \
@@ -1271,98 +1065,6 @@ from plots_and_expressions import phi
\end{align}
\egroup
-\begin{sagesilent}
-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()
-)
-
-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:
- 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: $\mathrm{ch}_2^{\beta_{-}}(u)>0$"
- )
- )
- 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{sagesilent}
from plots_and_expressions import \
@@ -1451,11 +1153,6 @@ $u=(r,c\ell,d\ell^2)$ to problem \ref{problem:problem-statement-2}.
The strategy here is similar to what was shown in theorem
\ref{thm:loose-bound-on-r}.
-\begin{sagesilent}
-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}}
@@ -1493,27 +1190,6 @@ beta_value_expr
\frac{1}{2n^2}\ZZ
\end{equation}
-\begin{sagesilent}
-# 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)
-
-r_upper_bound1 = (
- assymptote_gap_condition1
- * r * 2*n^2 / kappa
-)
-
-assert r_upper_bound1.lhs() == r
-
-r_upper_bound2 = (
- assymptote_gap_condition2
- * (r-R) * 2*n^2 / kappa + R
-)
-
-assert r_upper_bound2.lhs() == r
-\end{sagesilent}
\begin{sagesilent}
from plots_and_expressions import r_upper_bound1, r_upper_bound2, kappa
@@ -1566,22 +1242,6 @@ considering equations
\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}
\begin{sagesilent}
from plots_and_expressions import bounds_too_tight_condition1, bounds_too_tight_condition2
@@ -1621,24 +1281,9 @@ This is equivalent to:
\end{proof}
-\begin{sagesilent}
-var("Delta nu", domain="real")
-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(kappa==1)
- .subs(psi**2 == Delta/nu^2)
- .subs(1/psi**2 == nu^2/Delta)
-)
-\end{sagesilent}
\begin{sagesilent}
-from plots_and_expressions import r_upper_bound_all_q, q_sol
+from plots_and_expressions import r_upper_bound_all_q, q_sol, nu, Delta, psi
\end{sagesilent}
\begin{corollary}[Bound on $r$ \#2]
@@ -1749,15 +1394,6 @@ integral:
That is, $r \equiv -\aa^{-1}\bb$ mod $n$ ($\aa$ is coprime to
$n$, and so invertible mod $n$).
-\begin{sagesilent}
- rhs_numerator = (
- 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 $\ZZ/n\ZZ$.
@@ -1856,10 +1492,6 @@ $\epsilon_{v,q}\geq\epsilon_v$, with equality when $k_{v,q}=1$.
$\chern_1^\beta(u) = q = \frac{b_q}{n}$
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_{v,q}}
\def\psi{\chern_1^{\beta}(F)}