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Commit de19dccf authored by Luke Naylor's avatar Luke Naylor
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Extend radius condition to problem 1

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......@@ -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{
......
%% Cell type:code id:2c1f46cb tags:
``` sage
from pseudowalls import *
%display latex
```
%% Cell type:markdown id:48112244 tags:
# Initialize Cherns
%% Cell type:code id:7a52d103 tags:
``` sage
var("R C D r c d A B", domain="real")
P = A, B
```
%% Cell type:code id:cfde4b23 tags:
``` sage
v = Chern_Char(R,C,D)
v
```
%% Output
$\displaystyle \text{Chern Character:} \\ \begin{array}{l} \mathrm{ch}_{0} = R \\ \mathrm{ch}_{1} = C \ell^{1} \\ \mathrm{ch}_{2} = D \ell^{2} \end{array}$
<pseudowalls.chern_character.Chern_Char object at 0x7f7bcab93ad0>
<pseudowalls.chern_character.Chern_Char object at 0x7f56b7c82740>
%% Cell type:code id:f015f4ab tags:
``` sage
twisted_v = Twisted_Chern_Char(B,
R,
var("twisted_v1", latex_name = r"\mathrm{ch}_1^B(v)", domain="real"),
var("twisted_v2", latex_name = r"\mathrm{ch}_2^B(v)", domain="real"),
)
twisted_v
```
%% Output
$\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}$
<pseudowalls.chern_character.Twisted_Chern_Char object at 0x7f7bbf22f110>
<pseudowalls.chern_character.Twisted_Chern_Char object at 0x7f56b0027e80>
%% Cell type:code id:6fced6d0 tags:
``` sage
u = Chern_Char(r,c,d)
u
```
%% Output
$\displaystyle \text{Chern Character:} \\ \begin{array}{l} \mathrm{ch}_{0} = r \\ \mathrm{ch}_{1} = c \ell^{1} \\ \mathrm{ch}_{2} = d \ell^{2} \end{array}$
<pseudowalls.chern_character.Chern_Char object at 0x7f56afe16140>
%% Cell type:code id:711e4205 tags:
``` sage
twisted_u = Twisted_Chern_Char(B,
r,
var("twisted_u1", latex_name = r"\mathrm{ch}_1^B(u)", domain="real"),
var("twisted_u2", latex_name = r"\mathrm{ch}_2^B(u)", domain="real"),
)
twisted_u
```
%% Output
$\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}$
<pseudowalls.chern_character.Twisted_Chern_Char object at 0x7f7bbf0399d0>
<pseudowalls.chern_character.Twisted_Chern_Char object at 0x7f56b9a5fb20>
%% Cell type:markdown id:5f3e6e12 tags:
# Numerical Conditions
%% Cell type:markdown id:1206d912 tags:
Condition of $P = (A,B)$ being on $\Theta_v$ (i.e. $ch_2^{A,B}(v) = 0$) expressed in terms of twisted Chern character for $v$ at $\beta=B$:
%% Cell type:code id:17c390cd tags:
``` sage
A2_subs = solve(
stability.Tilt(*P).degree(twisted_v) == 0,
A^2)[0]
A2_subs
```
%% Output
$\displaystyle A^{2} = \frac{2 \, {\mathrm{ch}_2^B(v)}}{R}$
A^2 == 2*twisted_v2/R
%% Cell type:markdown id:11b9c67b tags:
## Condition: $ch_2^{P}(u) > 0$
%% Cell type:code id:47b34ed7 tags:
``` sage
stability.Tilt(*P).degree(twisted_u) > 0
```
%% Output
$\displaystyle -\frac{1}{2} \, A^{2} r + {\mathrm{ch}_2^B(u)} > 0$
-1/2*A^2*r + twisted_u2 > 0
%% Cell type:code id:d8abf566 tags:
``` sage
(stability.Tilt(*P).degree(twisted_u) / r > 0).expand().subs(
A2_subs
radius_condition = expand(
(stability.Tilt(*P).degree(twisted_u) / r > 0).expand().subs(
A2_subs
) * r * R
)
radius_condition
```
%% Output
$\displaystyle R {\mathrm{ch}_2^B(u)} - r {\mathrm{ch}_2^B(v)} > 0$
R*twisted_u2 - r*twisted_v2 > 0
%% Cell type:code id:194e313d tags:
``` sage
radius_condition_d_bound = (
radius_condition
.subs(twisted_u.ch[2] == u.twist(B).ch[2])
.expand()
.add_to_both_sides(B*R*c - B^2*R*r/2 + r*twisted_v.ch[2])
.divide_both_sides(R)
.expand()
)
```
%% Output
$\displaystyle d > -\frac{1}{2} \, B^{2} r + B c + \frac{r {\mathrm{ch}_2^B(v)}}{R}$
d > -1/2*B^2*r + B*c + r*twisted_v2/R
%% Cell type:code id:e4fc4758 tags:
``` sage
u.twist(B).ch[2]
```
%% Output
$\displaystyle \frac{1}{2} \, B^{2} r - B c + d$
1/2*B^2*r - B*c + d
%% Cell type:markdown id:fe9fe5b8 tags:
## Condition: $\Delta(u) \geq 0$
%% Cell type:code id:f09514cb tags:
``` sage
```
......
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