diff --git a/main.tex b/main.tex
index 0065588d181069e9c5a6b3ea9861c56916ca646d..1218309e0db4710c11f18ad4762a091d26824400 100644
--- a/main.tex
+++ b/main.tex
@@ -150,31 +150,31 @@ $\chern(F) = (R,C,D)$, and consider the possible Chern characters
$\chern(E) = (r,c,d)$ of some semistabilizer $E$.
\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
+# 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 *
+from pseudowalls import *
- v = Chern_Char(*var("R C D", domain="real"))
- u = Chern_Char(*var("r c d", domain="real"))
+v = Chern_Char(*var("R C D", domain="real"))
+u = Chern_Char(*var("r c d", domain="real"))
- Δ = lambda v: v.Q_tilt()
+Δ = lambda v: v.Q_tilt()
\end{sagesilent}
Recall [ref] that $\chern_1^{\beta_{-}}$ has fixed bounds in terms of
$\chern(F)$, and so we can write:
\begin{sagesilent}
- ts = stability.Tilt
- beta_min = var("beta", domain="real")
+ts = stability.Tilt
+beta_min = var("beta", domain="real")
- c_lower_bound = -(
- ts(beta=beta_min).rank(u)
- /ts().alpha
- ).expand() + c
+c_lower_bound = -(
+ ts(beta=beta_min).rank(u)
+ /ts().alpha
+).expand() + c
- var("q", domain="real")
- c_in_terms_of_q = c_lower_bound + q
+var("q", domain="real")
+c_in_terms_of_q = c_lower_bound + q
\end{sagesilent}
\begin{equation}
@@ -195,8 +195,8 @@ and we shall be varying $\chern_0(E) = r$ to see when certain inequalities fail.
This condition expressed in terms of $R,C,D,r,c,d$ looks as follows:
\begin{sagesilent}
- # First Bogomolov-Gieseker form expression that must be non-negative:
- bgmlv1 = Δ(v) - Δ(u) - Δ(v-u)
+# First Bogomolov-Gieseker form expression that must be non-negative:
+bgmlv1 = Δ(v) - Δ(u) - Δ(v-u)
\end{sagesilent}
\begin{equation}
@@ -209,11 +209,11 @@ Expressing $c$ in terms of $q$ as defined in (eqn \ref{eqn-cintermsofm})
we get the following:
\begin{sagesilent}
- bgmlv1_with_q = (
- bgmlv1
- .expand()
- .subs(c == c_in_terms_of_q)
- )
+bgmlv1_with_q = (
+ bgmlv1
+ .expand()
+ .subs(c == c_in_terms_of_q)
+)
\end{sagesilent}
\begin{equation}
@@ -225,17 +225,17 @@ we get the following:
This can be rearranged to express a bound on $d$ as follows:
\begin{sagesilent}
- var("r_alt",domain="real") # r_alt = r - R/2 temporary substitution
+var("r_alt",domain="real") # r_alt = r - R/2 temporary substitution
- bgmlv1_with_q_reparam = (bgmlv1_with_q.subs(r == r_alt + R/2)/r_alt).expand()
+bgmlv1_with_q_reparam = (bgmlv1_with_q.subs(r == r_alt + R/2)/r_alt).expand()
- bgmlv1_d_ineq = (
- ((0 >= -bgmlv1_with_q_reparam)/4 + d) # Rearrange for d
- .subs(r_alt == r - R/2) # Resubstitute r back in
- .expand()
- )
+bgmlv1_d_ineq = (
+ ((0 >= -bgmlv1_with_q_reparam)/4 + d) # Rearrange for d
+ .subs(r_alt == r - R/2) # Resubstitute r back in
+ .expand()
+)
- bgmlv1_d_lowerbound = bgmlv1_d_ineq.rhs() # Keep hold of lower bound for d
+bgmlv1_d_lowerbound = bgmlv1_d_ineq.rhs() # Keep hold of lower bound for d
\end{sagesilent}
\begin{dmath}
@@ -244,35 +244,35 @@ This can be rearranged to express a bound on $d$ as follows:
\end{dmath}
\begin{sagesilent}
- # Seperate out the terms of the lower bound for d
- bgmlv1_d_lowerbound_without_hyp = bgmlv1_d_lowerbound.subs(1/(R-2*r) == 0)
-
- bgmlv1_d_lowerbound_exp_term = (
- bgmlv1_d_lowerbound
- - bgmlv1_d_lowerbound_without_hyp
- ).expand()
-
- bgmlv1_d_lowerbound_const_term = bgmlv1_d_lowerbound_without_hyp.subs(r==0)
-
- bgmlv1_d_lowerbound_linear_term = (
- bgmlv1_d_lowerbound_without_hyp
- - bgmlv1_d_lowerbound_const_term
- ).expand()
-
- # Verify the simplified forms of the terms that will be mentioned in text
- assert bgmlv1_d_lowerbound_const_term == (
- v.twist(beta_min).ch[2]/2
- + beta_min*q
- )
-
- assert bgmlv1_d_lowerbound_exp_term == (
- (
- - R*v.twist(beta_min).ch[2]/2
- - R*beta_min*q
- + C*q
- - q^2
- )/(R-2*r)
- ).expand()
+# Separate out the terms of the lower bound for d
+bgmlv1_d_lowerbound_without_hyp = bgmlv1_d_lowerbound.subs(1/(R-2*r) == 0)
+
+bgmlv1_d_lowerbound_exp_term = (
+ bgmlv1_d_lowerbound
+ - bgmlv1_d_lowerbound_without_hyp
+).expand()
+
+bgmlv1_d_lowerbound_const_term = bgmlv1_d_lowerbound_without_hyp.subs(r==0)
+
+bgmlv1_d_lowerbound_linear_term = (
+ bgmlv1_d_lowerbound_without_hyp
+ - bgmlv1_d_lowerbound_const_term
+).expand()
+
+# Verify the simplified forms of the terms that will be mentioned in text
+assert bgmlv1_d_lowerbound_const_term == (
+ v.twist(beta_min).ch[2]/2
+ + beta_min*q
+)
+
+assert bgmlv1_d_lowerbound_exp_term == (
+ (
+ - R*v.twist(beta_min).ch[2]/2
+ - R*beta_min*q
+ + C*q
+ - q^2
+ )/(R-2*r)
+).expand()
\end{sagesilent}
@@ -295,8 +295,8 @@ so some of these expressions simplify.
This condition expressed in terms of $R,C,D,r,c,d$ looks as follows:
\begin{sagesilent}
- # First Bogomolov-Gieseker form expression that must be non-negative:
- bgmlv2 = Δ(u)
+# First Bogomolov-Gieseker form expression that must be non-negative:
+bgmlv2 = Δ(u)
\end{sagesilent}
\begin{equation}
@@ -309,11 +309,11 @@ Expressing $c$ 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)
- )
+bgmlv2_with_q = (
+ bgmlv2
+ .expand()
+ .subs(c == c_in_terms_of_q)
+)
\end{sagesilent}
\begin{equation}
@@ -325,13 +325,13 @@ we get the following:
This can be rearranged to express a bound on $d$ as follows:
\begin{sagesilent}
- bgmlv2_d_ineq = (
- (0 <= bgmlv2_with_q)/2/r # rescale assuming r > 0
- + d # Rearrange for d
- ).expand()
+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_lowerbound = bgmlv2_d_ineq.rhs()
+# Keep hold of lower bound for d
+bgmlv2_d_lowerbound = bgmlv2_d_ineq.rhs()
\end{sagesilent}
\begin{equation}
@@ -340,17 +340,17 @@ This can be rearranged to express a bound on $d$ as follows:
\end{equation}
\begin{sagesilent}
- # Seperate out the terms of the lower bound for d
- bgmlv2_d_lowerbound_without_hyp = bgmlv2_d_lowerbound.subs(1/r == 0)
- bgmlv2_d_lowerbound_const_term = bgmlv2_d_lowerbound_without_hyp.subs(r==0)
- bgmlv2_d_lowerbound_linear_term = (
- bgmlv2_d_lowerbound_without_hyp
- - bgmlv2_d_lowerbound_const_term
- ).expand()
- bgmlv2_d_lowerbound_exp_term = (
- bgmlv2_d_lowerbound
- - bgmlv2_d_lowerbound_without_hyp
- ).expand()
+# Seperate out the terms of the lower bound for d
+bgmlv2_d_lowerbound_without_hyp = bgmlv2_d_lowerbound.subs(1/r == 0)
+bgmlv2_d_lowerbound_const_term = bgmlv2_d_lowerbound_without_hyp.subs(r==0)
+bgmlv2_d_lowerbound_linear_term = (
+ bgmlv2_d_lowerbound_without_hyp
+ - bgmlv2_d_lowerbound_const_term
+).expand()
+bgmlv2_d_lowerbound_exp_term = (
+ bgmlv2_d_lowerbound
+ - bgmlv2_d_lowerbound_without_hyp
+).expand()
\end{sagesilent}
Viewing equation \ref{eqn-bgmlv2_d_lowerbound} as a lower bound for $d$ in term
@@ -370,8 +370,8 @@ for the bound found for $d$ in subsection \ref{subsect-d-bound-bgmlv1}.
This condition expressed in terms of $R,C,D,r,c,d$ looks as follows:
\begin{sagesilent}
- # Third Bogomolov-Gieseker form expression that must be non-negative:
- bgmlv3 = Δ(v-u)
+# Third Bogomolov-Gieseker form expression that must be non-negative:
+bgmlv3 = Δ(v-u)
\end{sagesilent}
\begin{equation}
@@ -384,11 +384,11 @@ 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)
- )
+bgmlv3_with_q = (
+ bgmlv3
+ .expand()
+ .subs(c == c_in_terms_of_q)
+)
\end{sagesilent}
\begin{equation}
@@ -400,23 +400,23 @@ we get the following:
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
- assert bgmlv3_d_ineq.lhs() == d, f"Inequality is of the form: {bgmlv3_d_ineq}"
- bgmlv3_d_upperbound = bgmlv3_d_ineq.rhs() # Keep hold of lower bound for d
+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, f"Inequality is of the form: {bgmlv3_d_ineq}"
+bgmlv3_d_upperbound = bgmlv3_d_ineq.rhs() # Keep hold of lower bound for d
\end{sagesilent}
\begin{dmath}
@@ -425,35 +425,35 @@ This can be rearranged to express a bound on $d$ as follows:
\end{dmath}
\begin{sagesilent}
- # Seperate out the terms of the 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_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_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_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()
+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
+# Verify the simplified forms of the terms that will be mentioned in text
- assert bgmlv3_d_upperbound_const_term == (
- v.twist(beta_min).ch[2]
- + beta_min*q
- ).expand()
+assert bgmlv3_d_upperbound_const_term == (
+ v.twist(beta_min).ch[2]
+ + beta_min*q
+).expand()
- assert bgmlv3_d_upperbound_exp_term == (
- R*v.twist(beta_min).ch[2]
- + (C - q)^2/2
- + R*beta_min*q
- - D*R
- )/(r-R)
+assert bgmlv3_d_upperbound_exp_term == (
+ R*v.twist(beta_min).ch[2]
+ + (C - q)^2/2
+ + R*beta_min*q
+ - D*R
+ )/(r-R)
\end{sagesilent}