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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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    main.tex
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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{
    ......
    other_P_choice.ipynb
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    View file @ de19dccf
    %% 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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