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Commit 05dfb000 authored by Luke Naylor's avatar Luke Naylor
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Give examples of possible Gamma curves for Lambda stab

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    figures/Figures-notebook.ipynb
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    %% Cell type:code id:51c1d049 tags:
    ``` sage
    from pseudowalls import *
    ```
    %% Cell type:code id:cbd6285e tags:
    ``` sage
    O = Chern_Char(1,0)
    O
    ```
    %% Output
    Chern Character:\begin{array}{l} \mathrm{ch}_{0} = 1 \\ \mathrm{ch}_{1} = 0 \ell^{1} \end{array}
    <pseudowalls.chern_character.Chern_Char object at 0x7ff55f6a1c30>
    %% Cell type:code id:32366efd tags:
    ``` sage
    O1 = exponential_chern(1,1)
    O1
    ```
    %% Output
    Chern Character:\begin{array}{l} \mathrm{ch}_{0} = 1 \\ \mathrm{ch}_{1} = 1 \ell^{1} \end{array}
    <pseudowalls.chern_character.Chern_Char object at 0x7ff55f6a1840>
    %% Cell type:code id:54385028 tags:
    ``` sage
    O1inv = exponential_chern(-1,1)
    O1inv
    ```
    %% Output
    Chern Character:\begin{array}{l} \mathrm{ch}_{0} = 1 \\ \mathrm{ch}_{1} = -1 \ell^{1} \end{array}
    <pseudowalls.chern_character.Chern_Char object at 0x7ff55f6a2680>
    %% Cell type:code id:f0768fcd tags:
    ``` sage
    O2 = O1 * O1
    O2
    ```
    %% Output
    Chern Character:\begin{array}{l} \mathrm{ch}_{0} = 1 \\ \mathrm{ch}_{1} = 2 \ell^{1} \end{array}
    <pseudowalls.chern_character.Chern_Char object at 0x7ff55f6a2a10>
    %% Cell type:code id:d5c5f903 tags:
    ``` sage
    O_x = Chern_Char(0,1)
    O_x
    ```
    %% Output
    Chern Character:\begin{array}{l} \mathrm{ch}_{0} = 0 \\ \mathrm{ch}_{1} = 1 \ell^{1} \end{array}
    <pseudowalls.chern_character.Chern_Char object at 0x7ff55f6a2d10>
    %% Cell type:code id:f5e93bd3 tags:
    ``` sage
    def plot_central_charge(chern, name, argument = None, radius = None):
    Z = stability.Mumford().central_charge(chern)
    x = Z.real()
    y = Z.imag()
    if not argument:
    argument = arctan(y/x) if x != 0 else pi / 2
    if argument <= 0:
    argument += pi
    if not radius:
    radius = (pi-argument/2)/pi
    return point(
    Z,
    marker = "o",
    size = 600,
    rgbcolor = "white",
    #markeredgecolor = "purple",
    zorder = 100
    ) + point(
    Z,
    marker = name,
    size = 500,
    rgbcolor = "red",
    zorder = 101
    ) + line(
    (0, Z),
    rgbcolor = "red",
    linestyle = "dashed",
    zorder = 99
    ) + disk(
    (0,0),
    float(radius),
    (0, float(argument)),
    alpha=.2,
    fill=False,
    thickness=1,
    rgbcolor="purple"
    )
    p = sum(
    plot_central_charge(chern, name)
    for chern, name in [
    (O1, r"$\mathcal{O}(1)$"),
    (O, r"$\mathcal{O}_X$"),
    (O1inv, r"$\mathcal{O}(-1)$"),
    (O2, r"$\mathcal{O}(2)$"),
    (O_x, r"$\mathcal{O}_p$")
    ]
    )
    xmax = (2.5)
    xmin = (-2.5)
    ymin = (-0.25)
    ymax = (1.5)
    aspect_ratio = (1)
    p += polygon(
    [
    (xmax + 1,0),
    (xmin - 1,0),
    (xmin - 1,ymax + 1),
    (xmax + 1,ymax + 1)
    ],
    rgbcolor = "yellow",
    alpha = 0.2,
    zorder = 102
    )
    p.xmax(xmax)
    p.xmin(xmin)
    p.ymin(ymin)
    p.ymax(ymax)
    p.set_aspect_ratio(aspect_ratio)
    p.axes_labels([r"$\mathcal{R}$",r"$\mathcal{I}$"])
    p.show()
    ```
    %% Output
    %% Cell type:code id:50d8265f tags:
    ``` sage
    plot_central_charge(O, r"$\mathcal{O}_X$", radius = 0.9)
    ```
    %% Output
    Graphics object consisting of 4 graphics primitives
    %% Cell type:code id:1cfa37b3 tags:
    ``` sage
    plot_central_charge(-O, r"$\mathcal{O}_X[1]$", argument=3*pi/2, radius = 0.9)
    ```
    %% Output
    Graphics object consisting of 4 graphics primitives
    %% Cell type:code id:8149044b tags:
    ``` sage
    plot_central_charge(O, r"$\mathcal{O}_X[2]$", argument=2*pi, radius = 0.9) + \
    disk(
    (0,0),
    0.9,
    (0, float(pi/2)),
    alpha=.4,
    fill=False,
    thickness=1,
    rgbcolor="purple"
    )
    ```
    %% Output
    Graphics object consisting of 5 graphics primitives
    %% Cell type:code id:4af73988 tags:
    ``` sage
    ymin = 0
    ymax = 2
    xmin = -3
    xmax = 1
    mu = -1
    p = line(
    [(mu,0), (mu,ymax)],
    linestyle = "dashed",
    ) + text(
    r"""$\beta = \mu(E)$
    $\mathfrak{Im}(\mathcal{Z}_{\alpha,\beta}(E)) = 0$""",
    (mu, -0.25),
    fontsize = "large",
    rgbcolor = "black"
    ) + text(
    r"""$\longleftarrow$
    $\mathrm{Coh}(X) \leftarrow \mathcal{B}^\beta$""",
    (-2.5,1),
    fontsize = "large"
    ) + text(
    r"""$E \in \mathcal{B}^\beta$
    $\mathfrak{Im}(\mathcal{Z}_{\alpha,\beta}(E)) \geq 0$""",
    (-1.5,1),
    fontsize = "large"
    ) + text(
    r"""$E[1] \in \mathcal{B}^\beta$
    $\mathfrak{Im}(\mathcal{Z}_{\alpha,\beta}(E)) < 0$""",
    (-0.5,1),
    fontsize = "large",
    rgbcolor = "red"
    ) + polygon(
    [
    (mu,0),
    (xmin,0),
    (xmin,ymax),
    (mu,ymax)
    ],
    rgbcolor = "blue",
    alpha = 0.1,
    zorder = 102
    ) + polygon(
    [
    (xmax,0),
    (mu,0),
    (mu,ymax),
    (xmax,ymax)
    ],
    rgbcolor = "orange",
    alpha = 0.1,
    zorder = 102
    )
    p.axes_labels([r"$\beta$",r"$\alpha$"])
    p.tick_label_color("white")
    p.show()
    ```
    %% Output
    %% Cell type:code id:8a52b3a8 tags:
    ``` sage
    v = Chern_Char(1,0,-1)
    u = Chern_Char(1,1,-3)
    ts = stability.Tilt()
    mu = stability.Mumford().slope
    p = implicit_plot(
    ts.wall_eqn(u,v)/ts.alpha == 0,
    (ts.beta, -5, 5),
    (ts.alpha, 0, 5),
    rgbcolor = "purple",
    ) + implicit_plot(
    ts.degree(v) == 0,
    (ts.beta, -5, 5),
    (ts.alpha, 0, 5),
    rgbcolor = "red",
    linestyle = "dotted",
    ) + implicit_plot(
    ts.degree(u) == 0,
    (ts.beta, -5, 5),
    (ts.alpha, 0, 5),
    linestyle = "dotted",
    ) + line(
    [(mu(v),0),
    (mu(v),5)],
    rgbcolor = "red",
    legend_label = "Potential destabilizer",
    ) + line(
    [(mu(u),0),
    (mu(u),5)],
    legend_label = "Fixed $v$ who's repr. obj. destabilized",
    )
    p.tick_label_color("white")
    p.show()
    ```
    %% Output
    %% Cell type:code id:8f0ee2d7 tags:
    ``` sage
    plot_range = 4
    v = Chern_Char(2,0,-1,1)
    lam = stability.Lambda(s=1/3)
    Theta = lam.rank(v)
    Gamma = lam.degree(v)
    mu = stability.Mumford().slope(v)
    p = (
    implicit_plot(Theta, (lam.beta, -plot_range,plot_range), (lam.alpha, 0, plot_range))
    + implicit_plot(Gamma, (lam.beta, -plot_range,plot_range), (lam.alpha, 0, plot_range), rgbcolor="green")
    + line([(mu,0),(mu,plot_range)], rgbcolor = "red")
    + text(r"$\beta=\mu$", (mu, -0.5), fontsize="x-large", rgbcolor="red")
    + text(r"$\nu = 0$", (mu-1, -0.5), fontsize="x-large", rgbcolor="blue")
    + text(r"$\lambda = 0$", (mu+2, -0.5), fontsize="x-large", rgbcolor="green")
    )
    p.tick_label_color("white")
    p.ymin(0)
    p
    ```
    %% Output
    Graphics object consisting of 6 graphics primitives
    %% Cell type:code id:4b66636a tags:
    ``` sage
    plot_range = 4
    v = Chern_Char(2,0,-1,1/2)
    lam = stability.Lambda(s=1/3)
    Theta = lam.rank(v)
    Gamma = lam.degree(v)
    mu = stability.Mumford().slope(v)
    p = (
    implicit_plot(Theta, (lam.beta, -plot_range,plot_range), (lam.alpha, 0, plot_range))
    + implicit_plot(Gamma, (lam.beta, -plot_range,plot_range), (lam.alpha, 0, plot_range), rgbcolor="green")
    + line([(mu,0),(mu,plot_range)], rgbcolor = "red")
    + text(r"$\beta=\mu$", (mu, -0.5), fontsize="x-large", rgbcolor="red")
    + text(r"$\nu = 0$", (mu+1, -0.5), fontsize="x-large", rgbcolor="blue")
    + text(r"$\lambda = 0$", (mu-1.5, -0.5), fontsize="x-large", rgbcolor="green")
    )
    p.tick_label_color("white")
    p.ymin(0)
    p
    ```
    %% Output
    Graphics object consisting of 6 graphics primitives
    ......
    main.tex
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    View file @ 05dfb000
    ......@@ -440,4 +440,12 @@
    \end{itemize}
    \end{frame}
    \begin{frame}
    \resizebox{\hsize}{!}{ \sageplot{fig7.plot()} }
    \end{frame}
    \begin{frame}
    \resizebox{\linewidth}{!}{ \sageplot{fig8.plot()} }
    \end{frame}
    \end{document}
    sagetexscripts/fig7.sage 0 → 100644
    + 21
    − 0
    View file @ 05dfb000
    from pseudowalls import *
    def plot():
    plot_range = 4
    v = Chern_Char(2,0,-1,1/2)
    lam = stability.Lambda(s=1/3)
    Theta = lam.rank(v)
    Gamma = lam.degree(v)
    mu = stability.Mumford().slope(v)
    p = (
    implicit_plot(Theta, (lam.beta, -plot_range,plot_range), (lam.alpha, 0, plot_range))
    + implicit_plot(Gamma, (lam.beta, -plot_range,plot_range), (lam.alpha, 0, plot_range), rgbcolor="green")
    + line([(mu,0),(mu,plot_range)], rgbcolor = "red")
    + text(r"$\beta=\mu$", (mu, -0.5), fontsize="x-large", rgbcolor="red")
    + text(r"$\nu = 0$", (mu+1, -0.5), fontsize="x-large", rgbcolor="blue")
    + text(r"$\lambda = 0$", (mu-1.5, -0.5), fontsize="x-large", rgbcolor="green")
    )
    p.tick_label_color("white")
    p.ymin(0)
    p
    return p
    sagetexscripts/fig8.sage 0 → 100644
    + 21
    − 0
    View file @ 05dfb000
    from pseudowalls import *
    def plot():
    plot_range = 4
    v = Chern_Char(2,0,-1,1)
    lam = stability.Lambda(s=1/3)
    Theta = lam.rank(v)
    Gamma = lam.degree(v)
    mu = stability.Mumford().slope(v)
    p = (
    implicit_plot(Theta, (lam.beta, -plot_range,plot_range), (lam.alpha, 0, plot_range))
    + implicit_plot(Gamma, (lam.beta, -plot_range,plot_range), (lam.alpha, 0, plot_range), rgbcolor="green")
    + line([(mu,0),(mu,plot_range)], rgbcolor = "red")
    + text(r"$\beta=\mu$", (mu, -0.5), fontsize="x-large", rgbcolor="red")
    + text(r"$\nu = 0$", (mu-1, -0.5), fontsize="x-large", rgbcolor="blue")
    + text(r"$\lambda = 0$", (mu+2, -0.5), fontsize="x-large", rgbcolor="green")
    )
    p.tick_label_color("white")
    p.ymin(0)
    p
    return p
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