From de19dccfcf0cd71d6d15af35d47d941b1c95ba8a Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Thu, 14 Sep 2023 14:04:34 +0100
Subject: [PATCH] Extend radius condition to problem 1
---
main.tex | 26 +++++++-
other_P_choice.ipynb | 154 +++++++++++++++++++++++++++++++++++++++----
2 files changed, 165 insertions(+), 15 deletions(-)
diff --git a/main.tex b/main.tex
index e9975ab..2c14f55 100644
--- a/main.tex
+++ b/main.tex
@@ -1014,10 +1014,32 @@ amounts to:
\begin{align}
\label{eqn:radius-cond-betamin}
- \chern_2^{\beta_{-}}(u) &\geq 0 \\
- d &\geq \beta_{-}q + \frac{1}{2} \beta_{-}^2r
+ \chern_2^{\beta_{-}}(u) &> 0 \\
+ d &> \beta_{-}q + \frac{1}{2} \beta_{-}^2r
\end{align}
+\begin{sagesilent}
+import other_P_choice as problem1
+\end{sagesilent}
+
+In the case where we are tackling problem \ref{problem:problem-statement-1},
+with some Chern character $v$ with positive rank, and some choice of point
+$P=(A,B) \in \Theta_v^-$.
+Then $\sage{problem1.A2_subs}$ follows from $\chern_2^P(v)=0$. Using this substitution into the
+condition $\chern_2^P(u)>0$ yields:
+
+\begin{equation}
+ \sage{problem1.radius_condition}
+\end{equation}
+
+\noindent
+Expressing this as a bound on $d$, then yields:
+
+\begin{equation}
+ \sage{problem1.radius_condition_d_bound}
+\end{equation}
+
+
\subsubsection{
Semistability of the Semistabilizer:
\texorpdfstring{
diff --git a/other_P_choice.ipynb b/other_P_choice.ipynb
index cc0012d..acb0c98 100644
--- a/other_P_choice.ipynb
+++ b/other_P_choice.ipynb
@@ -45,7 +45,7 @@
"$\\displaystyle \\text{Chern Character:} \\\\ \\begin{array}{l} \\mathrm{ch}_{0} = R \\\\ \\mathrm{ch}_{1} = C \\ell^{1} \\\\ \\mathrm{ch}_{2} = D \\ell^{2} \\end{array}$"
],
"text/plain": [
- "<pseudowalls.chern_character.Chern_Char object at 0x7f7bcab93ad0>"
+ "<pseudowalls.chern_character.Chern_Char object at 0x7f56b7c82740>"
]
},
"execution_count": 3,
@@ -73,7 +73,7 @@
"$\\displaystyle \\text{ Twisted Chern Character for $\\beta={ B }$ } \\\\ \\begin{array}{l} \\mathrm{ch}_{0} = R \\\\ \\mathrm{ch}_{1} = {\\mathrm{ch}_1^B(v)} \\ell^{1} \\\\ \\mathrm{ch}_{2} = {\\mathrm{ch}_2^B(v)} \\ell^{2} \\end{array}$"
],
"text/plain": [
- "<pseudowalls.chern_character.Twisted_Chern_Char object at 0x7f7bbf22f110>"
+ "<pseudowalls.chern_character.Twisted_Chern_Char object at 0x7f56b0027e80>"
]
},
"execution_count": 4,
@@ -90,6 +90,34 @@
"twisted_v"
]
},
+ {
+ "cell_type": "code",
+ "execution_count": 12,
+ "id": "6fced6d0",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<html>\\(\\displaystyle \\text{Chern Character:} \\\\ \\begin{array}{l} \\mathrm{ch}_{0} = r \\\\ \\mathrm{ch}_{1} = c \\ell^{1} \\\\ \\mathrm{ch}_{2} = d \\ell^{2} \\end{array}\\)</html>"
+ ],
+ "text/latex": [
+ "$\\displaystyle \\text{Chern Character:} \\\\ \\begin{array}{l} \\mathrm{ch}_{0} = r \\\\ \\mathrm{ch}_{1} = c \\ell^{1} \\\\ \\mathrm{ch}_{2} = d \\ell^{2} \\end{array}$"
+ ],
+ "text/plain": [
+ "<pseudowalls.chern_character.Chern_Char object at 0x7f56afe16140>"
+ ]
+ },
+ "execution_count": 12,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "u = Chern_Char(r,c,d)\n",
+ "u"
+ ]
+ },
{
"cell_type": "code",
"execution_count": 5,
@@ -105,7 +133,7 @@
"$\\displaystyle \\text{ Twisted Chern Character for $\\beta={ B }$ } \\\\ \\begin{array}{l} \\mathrm{ch}_{0} = r \\\\ \\mathrm{ch}_{1} = {\\mathrm{ch}_1^B(u)} \\ell^{1} \\\\ \\mathrm{ch}_{2} = {\\mathrm{ch}_2^B(u)} \\ell^{2} \\end{array}$"
],
"text/plain": [
- "<pseudowalls.chern_character.Twisted_Chern_Char object at 0x7f7bbf0399d0>"
+ "<pseudowalls.chern_character.Twisted_Chern_Char object at 0x7f56b9a5fb20>"
]
},
"execution_count": 5,
@@ -140,7 +168,7 @@
},
{
"cell_type": "code",
- "execution_count": 7,
+ "execution_count": 6,
"id": "17c390cd",
"metadata": {},
"outputs": [
@@ -156,7 +184,7 @@
"A^2 == 2*twisted_v2/R"
]
},
- "execution_count": 7,
+ "execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
@@ -179,26 +207,126 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 7,
"id": "47b34ed7",
"metadata": {},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<html>\\(\\displaystyle -\\frac{1}{2} \\, A^{2} r + {\\mathrm{ch}_2^B(u)} > 0\\)</html>"
+ ],
+ "text/latex": [
+ "$\\displaystyle -\\frac{1}{2} \\, A^{2} r + {\\mathrm{ch}_2^B(u)} > 0$"
+ ],
+ "text/plain": [
+ "-1/2*A^2*r + twisted_u2 > 0"
+ ]
+ },
+ "execution_count": 7,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"stability.Tilt(*P).degree(twisted_u) > 0"
]
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 11,
"id": "d8abf566",
"metadata": {},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<html>\\(\\displaystyle R {\\mathrm{ch}_2^B(u)} - r {\\mathrm{ch}_2^B(v)} > 0\\)</html>"
+ ],
+ "text/latex": [
+ "$\\displaystyle R {\\mathrm{ch}_2^B(u)} - r {\\mathrm{ch}_2^B(v)} > 0$"
+ ],
+ "text/plain": [
+ "R*twisted_u2 - r*twisted_v2 > 0"
+ ]
+ },
+ "execution_count": 11,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "radius_condition = expand(\n",
+ " (stability.Tilt(*P).degree(twisted_u) / r > 0).expand().subs(\n",
+ " A2_subs\n",
+ " ) * r * R\n",
+ ")\n",
+ "radius_condition"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 28,
+ "id": "194e313d",
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<html>\\(\\displaystyle d > -\\frac{1}{2} \\, B^{2} r + B c + \\frac{r {\\mathrm{ch}_2^B(v)}}{R}\\)</html>"
+ ],
+ "text/latex": [
+ "$\\displaystyle d > -\\frac{1}{2} \\, B^{2} r + B c + \\frac{r {\\mathrm{ch}_2^B(v)}}{R}$"
+ ],
+ "text/plain": [
+ "d > -1/2*B^2*r + B*c + r*twisted_v2/R"
+ ]
+ },
+ "execution_count": 28,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
- "(stability.Tilt(*P).degree(twisted_u) / r > 0).expand().subs(\n",
- " A2_subs\n",
+ "radius_condition_d_bound = (\n",
+ " radius_condition\n",
+ " .subs(twisted_u.ch[2] == u.twist(B).ch[2])\n",
+ " .expand()\n",
+ " .add_to_both_sides(B*R*c - B^2*R*r/2 + r*twisted_v.ch[2])\n",
+ " .divide_both_sides(R)\n",
+ " .expand()\n",
")"
]
},
+ {
+ "cell_type": "code",
+ "execution_count": 16,
+ "id": "e4fc4758",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<html>\\(\\displaystyle \\frac{1}{2} \\, B^{2} r - B c + d\\)</html>"
+ ],
+ "text/latex": [
+ "$\\displaystyle \\frac{1}{2} \\, B^{2} r - B c + d$"
+ ],
+ "text/plain": [
+ "1/2*B^2*r - B*c + d"
+ ]
+ },
+ "execution_count": 16,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "u.twist(B).ch[2]"
+ ]
+ },
{
"cell_type": "markdown",
"id": "fe9fe5b8",
@@ -218,7 +346,7 @@
],
"metadata": {
"kernelspec": {
- "display_name": "SageMath 9.8",
+ "display_name": "SageMath 9.7",
"language": "sage",
"name": "sagemath"
},
@@ -232,7 +360,7 @@
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
- "version": "3.11.3"
+ "version": "3.10.8"
}
},
"nbformat": 4,
--
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