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threefolddata.tex
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\chapter{Threefold Data}
\label{appendix1}
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
\setcounter{footnote}{0}
In this Appendix we include data needed for the results in Chapters \ref{chap:sol} and \ref{chap:R(X)}. We include the key below for ease of use. \\ \\
\textbf{Key:}
\begin{itemize}
\item ``\(\Phi\)" - the Fano divisorial polytope \(\Phi: \Box \to \wdiv_\QQ \PP^1\), specified in diagram form as values on the base polytope \(\Box\) .
\item ``\(\sigma \)" - an automorphism which permutes the vertices of \(\Box\) such that \(\deg \Phi \circ \sigma = \deg \Phi\).
\item ``\(g(\xi)\)" - the integral \(\int_\Box \langle u, v \rangle \cdot \deg \Phi (u) \cdot e^{\langle u, \xi \rangle} du\), a positive multiple of \(\DF(X \times \A^1, e_2)\).
\item ``\(R(X)\)" - the calculated greatest lower bound on Ricci curvature (Tian's beta invariant) in Chapter~\ref{chap:R(X)}.
\item ``\( \ \xi \sim \ \)" - an approximation of the K\"ahler-Ricci soliton found in Chapter~\ref{chap:sol}, representated an element of \(N_\RR\).
\item ``\(\text{Vert}(\Delta_y)\)" - the degeneration polytope \(\Delta_y\) is given by convex hull of the columns of this matrix.
\item ``\(h_y(\xi)\)" - a positive multiple of the Donaldson-Futaki invariant of the test cofiguration corresponding to \(\Delta_y\) with \(\CC^*\)-action given by \(e_3 \in N'\).
\item ``\(h_y(\xi_2) >\)" - a lower bound on \(h_y(\xi)\), found using real interval arithmetic.
\end{itemize}
{\setlength{\parindent}{0pt}
\newpage
\begin{landscape}
%
%
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%
%
%
%
%
%
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%
%
%
\begin{tabularx}{\linewidth}{clcc}
\toprule
\midrule
\textbf{Name:} & \ 2.30 \hspace{0.3\linewidth} & \textbf{Description:} & Blow up of $Q$ in a point\\
\midrule
{\small $ \sigma = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$ }, & \( g(\xi) = \frac{1}{\xi_{2}^{4}}\cdot\left({\left(2 \, \xi_{2}^{3} - 3 \, \xi_{2} - 3\right)} e^{\left(4 \, \xi_{2}\right)} + 12 \, \xi_{2} e^{\left(3 \, \xi_{2}\right)} + 3 \, \xi_{2} + 3\right) e^{\left(-3 \, \xi_{2}\right)} \), & $ R(X) = 23/29$ , & $\xi \sim (0,0.51489)$
\end{tabularx}
\begin{figure}[H]
\centering
\label{fig:data230b}
\resizebox{!}{0.25\textheight}{
\begin{subfigure}[b]{0.25\linewidth}
\centering
\begin{tikzpicture}[scale=\tfl]
% \draw[dotted,step=1,gray,] (-3,-3) grid (3,2);
\draw[line width = 1pt] (0,-3) --
(-3,0) -- (-2,1)--(2,1)--
(3,0)--(0,-3); \draw[densely dashdotted, gray, line width = 1.2pt] (0,-3) -- (0,1);
\node at (0,-3) [below] {\large{$0$}};
\node at (-3,0) [right] {\large{$0$}};
\node at (-2,1) [left] {\large{$0$}};
\node at (2,1) [right] {\large{$-2$}};
\node at (3,0) [below] {\large{$-3$}};
\node at (0,1) [above] {\large{$0$}};
\foreach \i in {-3,...,3}{
\foreach \j in {-3,...,2}{
\draw[gray] (\i,\j) node {\tiny{$\circ$}};
}
}
\draw (0,0) node {\textbullet};
\end{tikzpicture}
\caption*{$\Phi_0$}
\end{subfigure}
\begin{subfigure}[b]{0.25\linewidth}
\centering
\begin{tikzpicture}[scale=\tfl]
% \draw[dotted,step=1,gray] (-3,-3) grid (3,2);
\draw[line width = 1.2pt] (0,-3) --
(-3,0) -- (-2,1)--(2,1)--
(3,0)--(0,-3); \draw[densely dashdotted, gray,line width = 1.2pt] (-3,0) -- (3,0);
\node at (0,-3) [below] {\large{$-3$}};
\node at (-3,0) [right] {\large{$0$}};
\node at (-2,1) [left] {\large{$0$}};
\node at (2,1) [right] {\large{$0$}};
\node at (3,0) [below] {\large{$0$}};
\foreach \i in {-3,...,3}{
\foreach \j in {-3,...,2}{
\draw[gray] (\i,\j) node {\tiny{$\circ$}};
}
}
\draw (0,0) node {\textbullet};
\end{tikzpicture}
\caption*{$\Phi_1$}
\end{subfigure}
\begin{subfigure}[b]{0.25\linewidth}
\centering
\begin{tikzpicture}[scale=\tfl]
% \draw[dotted,step=1,gray] (-3,-3) grid (3,2);
\draw[line width = 1.2pt] (0,-3) --
(-3,0) -- (-2,1)--(2,1)--
(3,0)--(0,-3);
\node at (0,-3) [below] {\large{$1$}};
\node at (-3,0) [right] {\large{$-2$}};
\node at (-2,1) [left] {\large{$-2$}};
\node at (2,1) [right] {\large{$0$}};
\node at (3,0) [below] {\large{$1$}};
\foreach \i in {-3,...,3}{
\foreach \j in {-3,...,2}{
\draw[gray] (\i,\j) node {\tiny{$\circ$}};
}
}
\draw (0,0) node {\textbullet};
\end{tikzpicture}
\caption*{$\Phi_\infty$}
\end{subfigure}
\begin{subfigure}[b]{0.25\linewidth}
\centering
\begin{tikzpicture}[scale=\tfl]
% \draw[dotted,step=1,gray] (-3,-3) grid (3,2);
\draw[line width = 1.2pt] (0,-3) --
(-3,0) -- (-2,1)--(2,1)--
(3,0)--(0,-3); \draw[densely dashdotted, gray,line width = 1.2pt] (-3,0) -- (3,0); \draw[densely dashdotted, gray,line width = 1.2pt] (0,-3) -- (0,1) ;
\node at (0,-3) [below] {\large{$-2$}};
\node at (-3,0) [right] {\large{$-2$}};
\node at (-2,1) [left] {\large{$-2$}};
\node at (2,1) [right] {\large{$-2$}};
\node at (3,0) [below] {\large{$-2$}};
\node at (0,0) [below left] {\large{-$\frac{1}{2}$}};
\node at (0,1) [above] {\large{$-1$}};
\foreach \i in {-3,...,3}{
\foreach \j in {-3,...,2}{
\draw[gray] (\i,\j) node {\tiny{$\circ$}};
}
}
\draw (0,0) node {\textbullet};
\end{tikzpicture}
\caption*{$\deg \Phi $}
\end{subfigure}
}
\end{figure}
\begin{center}
\begin{tabularx}{\linewidth}{cc@{\hspace{5ex}}c@{\hspace{5ex}}c}
\toprule
\(y\) & \(\text{Vert}(\Delta_y)\) & \( h_y(\xi_2)\) & \( h_y(\xi_2) >\) \\
\midrule
\(0\) & \( \scalemath{0.8}{ \begin{pmatrix} 0 & 3 & -3 & 2 & -2 & 0 \\
-3 & 0 & 0 & 1 & 1 & 1 \\
1 & -2 & 1 & -1 & 1 & 1 \end{pmatrix}} \)& \(\frac{{\left({\left(2 \, \xi_{2}^{3} - 3 \, \xi_{2} - 3\right)} e^{\left(4 \, \xi_{2}\right)} + 3 \, {\left(3 \, \xi_{2}^{2} + 2\right)} e^{\left(3 \, \xi_{2}\right)} - 3 \, \xi_{2} - 3\right)} e^{\left(-3 \, \xi_{2}\right)}}{3 \, \xi_{2}^{4}}\) & \(1.087\) \\ \midrule
\(1\) & \( \scalemath{0.8}{ \begin{pmatrix} -3 & 0 & 3 & 2 & 0 & -2 \\
0 & -3 & 0 & 1 & 1 & 1 \\
1 & -2 & 1 & 1 & 0 & 1 \end{pmatrix}} \)& \(\frac{{\left({\left(8 \, \xi_{2}^{3} + 6 \, \xi_{2}^{2} - 3\right)} e^{\left(4 \, \xi_{2}\right)} - 12 \, {\left(3 \, \xi_{2}^{2} - 3 \, \xi_{2} + 1\right)} e^{\left(3 \, \xi_{2}\right)} + 12 \, \xi_{2} + 15\right)} e^{\left(-3 \, \xi_{2}\right)}}{6 \, \xi_{2}^{4}}\) & \(2.178\) \\ \midrule
\(\infty\) & \( \scalemath{0.8}{ \begin{pmatrix} -3 & 0 & 2 & 0 & 0 & -2 & 3 \\
0 & -3 & 1 & 0 & 1 & 1 & 0 \\
-1 & 2 & 1 & -1 & -1 & -1 & 2 \end{pmatrix}} \)& \(-\frac{{\left(2 \, {\left(2 \, \xi_{2}^{3} - 3 \, \xi_{2} - 3\right)} e^{\left(4 \, \xi_{2}\right)} - 3 \, {\left(3 \, \xi_{2}^{2} - 12 \, \xi_{2} + 2\right)} e^{\left(3 \, \xi_{2}\right)} + 12 \, \xi_{2} + 12\right)} e^{\left(-3 \, \xi_{2}\right)}}{6 \, \xi_{2}^{4}}\) & \(0.4465\) \\ \midrule
\(\notin \{0,1,\infty\} \) & \( \scalemath{0.8}{ \begin{pmatrix} 0 & 0 & -3 & 2 & 0 & -2 & 3 \\
0 & -3 & 0 & 1 & 1 & 1 & 0 \\
-\frac{1}{2} & 1 & 1 & 1 & 0 & 1 & 1 \end{pmatrix}} \)& \(\frac{{\left({\left(8 \, \xi_{2}^{3} + 6 \, \xi_{2}^{2} - 3\right)} e^{\left(4 \, \xi_{2}\right)} - 3 \, {\left(3 \, \xi_{2}^{2} - 2\right)} e^{\left(3 \, \xi_{2}\right)} - 6 \, \xi_{2} - 3\right)} e^{\left(-3 \, \xi_{2}\right)}}{6 \, \xi_{2}^{4}}\) & \(4.151\) \\ \midrule
\midrule
\end{tabularx}
\end{center}
\newpage
%
%
%
%
%
%
%
\begin{tabularx}{\linewidth}{clcc}
\toprule
\midrule
\textbf{Name:} & \ 2.31 \hspace{0.3\linewidth} & \textbf{Description:} & Blow up of $Q$ in a line\\
\midrule
{\small $ \sigma = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ }, & \( g(\xi_1,\xi_1)= \ {\mapsto}\ -\frac{{\left(9 \, \xi_{1}^{2} - 8 \, {\left(\xi_{1}^{2} - \xi_{1}\right)} e^{\left(5 \, \xi_{1}\right)} + {\left(3 \, \xi_{1}^{2} - 5 \, \xi_{1} - 3\right)} e^{\left(4 \, \xi_{1}\right)} + 9 \, \xi_{1} + 3\right)} e^{\left(-3 \, \xi_{1}\right)}}{2 \, \xi_{1}^{4}} \), & $ R(X) = 23/27$ , & $\xi \sim (0.28550,0.28550)$
\end{tabularx}
\begin{figure}[H]
\centering
\label{fig:data231}
\resizebox{!}{0.25\textheight}{
\begin{subfigure}[b]{0.25\linewidth}
\centering
\begin{tikzpicture}[scale=\tfl]
% \draw[dotted,step=1,gray,] (-3,-3) grid (3,2);
\draw[line width = 1pt] (0,-3) --
(-3,0) -- (-1,2)--(0,2)--
(2,0)--(2,-1) -- (0,-3); \draw[densely dashdotted, gray, line width = 1.2pt] (0,-3) -- (0,2);
\node at (0,-3) [below] {\large{$0$}};
\node at (-3,0) [left] {\large{$0$}};
\node at (-1,2) [above] {\large{$0$}};
\node at (0,2) [above] {\large{$0$}};
\node at (2,0) [right] {\large{$-2$}};
\node at (2,-1) [right] {\large{$-2$}};
\foreach \i in {-3,...,3}{
\foreach \j in {-3,...,2}{
\draw[gray] (\i,\j) node {\tiny{$\circ$}};
}
}
\draw (0,0) node {\textbullet};
\end{tikzpicture}
\caption*{$\Phi_0$}
\end{subfigure}
\begin{subfigure}[b]{0.25\linewidth}
\centering
\begin{tikzpicture}[scale=\tfl]
% \draw[dotted,step=1,gray] (-3,-3) grid (3,2);
\draw[line width = 1pt] (0,-3) --
(-3,0) -- (-1,2)--(0,2)--
(2,0)--(2,-1) -- (0,-3); \draw[densely dashdotted, gray,line width = 1.2pt] (-3,0) -- (2,0);
\node at (0,-3) [below] {\large{$-3$}};
\node at (-3,0) [left] {\large{$0$}};
\node at (-1,2) [above] {\large{$0$}};
\node at (0,2) [above] {\large{$0$}};
\node at (2,0) [right] {\large{$0$}};
\node at (2,-1) [right] {\large{$-1$}};
\foreach \i in {-3,...,3}{
\foreach \j in {-3,...,2}{
\draw[gray] (\i,\j) node {\tiny{$\circ$}};
}
}
\draw (0,0) node {\textbullet};
\end{tikzpicture}
\caption*{$\Phi_1$}
\end{subfigure}
\begin{subfigure}[b]{0.25\linewidth}
\centering
\begin{tikzpicture}[scale=\tfl]
% \draw[dotted,step=1,gray] (-3,-3) grid (3,2);
\draw[line width = 1pt] (0,-3) --
(-3,0) -- (-1,2)--(0,2)--
(2,0)--(2,-1) -- (0,-3); \draw[densely dashdotted, gray,line width = 1.2pt] (-1,2) -- (2,-1);
\node at (0,-3) [below] {\large{$1$}};
\node at (-3,0) [left] {\large{$-2$}};
\node at (-1,2) [left] {\large{$-2$}};
\node at (0,2) [right] {\large{$-2$}};
\node at (2,0) [right] {\large{$0$}};
\node at (2,-1) [right] {\large{$1$}};
\foreach \i in {-3,...,3}{
\foreach \j in {-3,...,2}{
\draw[gray] (\i,\j) node {\tiny{$\circ$}};
}
}
\draw (0,0) node {\textbullet};
\end{tikzpicture}
\caption*{$\Phi_\infty$}
\end{subfigure}
\begin{subfigure}[b]{0.25\linewidth}
\centering
\begin{tikzpicture}[scale=\tfl]
% \draw[dotted,step=1,gray] (-3,-3) grid (3,2);
\draw[line width = 1pt] (0,-3) --
(-3,0) -- (-1,2)--(0,2)--
(2,0)--(2,-1) -- (0,-3); \draw[densely dashdotted, gray, line width = 1.2pt] (0,-3) -- (0,2); \draw[densely dashdotted, gray,line width = 1.2pt] (-3,0) -- (2,0); \draw[densely dashdotted, gray,line width = 1.2pt] (-1,2) -- (2,-1);
\node at (0,-3) [below] {\large{$-2$}};
\node at (-3,0) [left] {\large{$-2$}};
\node at (-1,2) [left] {\large{$-2$}};
\node at (0,2) [right] {\large{$-2$}};
\node at (2,0) [right] {\large{$-2$}};
\node at (2,-1) [right] {\large{$-2$}};
\node at (1,0) {\large{$-1$}};
\node at (0,1) {\large{$-1$}};
\node at (0,0) [left] {\large{$-\frac{1}{2}$}};
\foreach \i in {-3,...,3}{
\foreach \j in {-3,...,2}{
\draw[gray] (\i,\j) node {\tiny{$\circ$}};
}
}
\draw (0,0) node {\textbullet};
\end{tikzpicture}
\caption*{$\deg \Phi $}
\end{subfigure}
}
\end{figure}
\begin{center}
\begin{tabularx}{\linewidth}{cc@{\hspace{5ex}}c@{\hspace{5ex}}c}
\toprule
\(y\) & \(\text{Vert}(\Delta_y)\) & \( h_y(\xi_2)\) & \( h_y(\xi_2) >\) \\
\midrule
\(0\) & \( \scalemath{0.8}{ \begin{pmatrix} 0 & 1 & 2 & -3 & 0 & 2 & -1 \\
-3 & 0 & -1 & 0 & 2 & 0 & 2 \\
1 & -1 & -1 & 1 & 1 & -1 & 1 \end{pmatrix}} \)& \(\frac{{\left(12 \, \xi_{2}^{2} + {\left(4 \, \xi_{2}^{2} + 2 \, \xi_{2} - 3\right)} e^{\left(4 \, \xi_{2}\right)} + 10 \, \xi_{2} + 3\right)} e^{\left(-3 \, \xi_{2}\right)}}{8 \, \xi_{2}^{4}}\) & \(1.9509\) \\ \midrule
\(1\) & \( \scalemath{0.8}{ \begin{pmatrix} -3 & 0 & -1 & 2 & 0 & 2 & 0 \\
0 & -3 & 2 & -1 & 2 & 0 & 1 \\
1 & -2 & 1 & 0 & 1 & 1 & 0 \end{pmatrix}} \)& \(-\frac{{(6 \xi_2^2 - 16 {(\xi_2^2 - \xi_2)} e^{(5 \xi_2)} + {(2 \xi_2^2 - 12 \xi_2 - 3)} e^{(4 \xi_2)} + 8 \xi_2 + 3)} e^{(-3 \xi_2)}}{8 \xi_2^{4}}\) & \(1.9410\) \\ \midrule
\(\infty\) & \( \scalemath{0.8}{ \begin{pmatrix} -3 & 0 & 0 & 2 & 0 & -1 & 2 \\
0 & -3 & 2 & 0 & 0 & 2 & -1 \\
-1 & 2 & -1 & 1 & -1 & -1 & 2 \end{pmatrix}} \)& \(\frac{{(3 \xi_2^2 - {(3 \xi_2^2 + 4 \xi_2 + 2)} e^{(4 \xi_2)} + 4 \xi_2 + 4 e^{(5 \xi_2)} - 4 e^{(3 \xi_2)} + 2)} e^{(-3 \xi_2)}}{4 \xi_2^4}\) & \(0.4632\) \\ \midrule
\(\notin \{0,1,\infty\} \) & \( \scalemath{0.8}{ \begin{pmatrix} 0 & 0 & -3 & 2 & 2 & 1 & 0 & -1 & 0 \\
0 & -3 & 0 & -1 & 0 & 0 & 2 & 2 & 1 \\
-\frac{1}{2} & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 0 \end{pmatrix}} \) & \(\frac{{(6 \xi_2^2 + 4 {(2 \xi_2^2 - 2 \xi_2 + 1)} e^{(5 \xi_2)} - {(2 \xi_2^2 - 3 \xi_2 + 2)} e^{(4 \xi_2)} + 5 \xi_2 - 4 \, e^{(3 \, \xi_2)} + 2)} e^{(-3 \, \xi_2)}}{4 \, \xi_2^4}\) & \(4.370\) \\ \midrule
\midrule
\end{tabularx}
\end{center}
\newpage
%
%
%
%
%
%
%
\begin{tabularx}{\linewidth}{clcc}
\toprule
\midrule
\textbf{Name:} & \ 3.18 \hspace{0.3\linewidth} & \textbf{Description:} & Blow up of $Q$ in a point and a conic\\
\midrule
{\small $ \sigma = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$ }, & \( g(0,\xi_2) = -\frac{{\left(4 \, \xi_{2}^{2} - {\left(3 \, \xi_{2}^{3} + \xi_{2}^{2} - 6 \, \xi_{2} - 6\right)} e^{\left(3 \, \xi_{2}\right)} - 24 \, \xi_{2} e^{\left(2 \, \xi_{2}\right)} - 6\right)} e^{\left(-2 \, \xi_{2}\right)}}{2 \, \xi_{2}^{4}} \), & $ R(X) = 48/55$ , & $\xi \sim (0,0.37970)$
\end{tabularx}
\begin{figure}[H]
\centering
\label{fig:data318}
\resizebox{!}{0.25\textheight}{
\begin{subfigure}[b]{0.25\linewidth}
\centering
\begin{tikzpicture}[scale=\tfl]
% \draw[dotted,step=1,gray,] (-3,-3) grid (3,2);
\draw[line width = 1pt] (-1,-2) --
(-3,0) -- (-2,1)--(2,1)--
(3,0)--(1,-2) -- (-1,-2);
\draw[densely dashdotted, gray, line width = 1.2pt] (1,-2) -- (1,1);
\draw[densely dashdotted, gray, line width = 1.2pt] (-1,-2) -- (-1,1);
\node at (1,-2) [below] {\large{$-1$}};
\node at (-1,-2) [below] {\large{$0$}};
\node at (-3,0) [below] {\large{$0$}};
\node at (-2,1) [above] {\large{$0$}};
\node at (-1,1) [above] {\large{$0$}};
\node at (1,1) [above] {\large{$-1$}};
\node at (2,1) [right] {\large{$-2$}};
\node at (3,0) [below] {\large{$-3$}};
\foreach \i in {-3,...,3}{
\foreach \j in {-3,...,2}{
\draw[gray] (\i,\j) node {\tiny{$\circ$}};
}
}
\draw (0,0) node {\textbullet};
\end{tikzpicture}
\caption*{$\Phi_0$}
\end{subfigure}
\begin{subfigure}[b]{0.25\linewidth}
\centering
\begin{tikzpicture}[scale=\tfl]
% \draw[dotted,step=1,gray] (-3,-3) grid (3,2);
\draw[line width = 1pt] (-1,-2) --
(-3,0) -- (-2,1)--(2,1)--
(3,0)--(1,-2) -- (-1,-2);
\draw[densely dashdotted, gray, line width = 1.2pt] (-3,0) -- (3,0);
\node at (1,-2) [below] {\large{$-2$}};
\node at (-1,-2) [below] {\large{$-2$}};
\node at (-3,0) [below] {\large{$0$}};
\node at (-2,1) [left] {\large{$0$}};
\node at (2,1) [right] {\large{$0$}};
\node at (3,0) [below] {\large{$0$}};
\foreach \i in {-3,...,3}{
\foreach \j in {-3,...,2}{
\draw[gray] (\i,\j) node {\tiny{$\circ$}};
}
}
\draw (0,0) node {\textbullet};
\end{tikzpicture}
\caption*{$\Phi_1$}
\end{subfigure}
\begin{subfigure}[b]{0.25\linewidth}
\centering
\begin{tikzpicture}[scale=\tfl]
% \draw[dotted,step=1,gray] (-3,-3) grid (3,2);
\draw[line width = 1pt] (-1,-2) --
(-3,0) -- (-2,1)--(2,1)--
(3,0)--(1,-2) -- (-1,-2);
\node at (1,-2) [below] {\large{$1$}};
\node at (-1,-2) [below] {\large{$0$}};
\node at (-3,0) [below] {\large{$-2$}};
\node at (-2,1) [left] {\large{$-2$}};
\node at (2,1) [right] {\large{$0$}};
\node at (3,0) [below] {\large{$1$}};
\foreach \i in {-3,...,3}{
\foreach \j in {-3,...,2}{
\draw[gray] (\i,\j) node {\tiny{$\circ$}};
}
}
\draw (0,0) node {\textbullet};
\end{tikzpicture}
\caption*{$\Phi_\infty$}
\end{subfigure}
\begin{subfigure}[b]{0.25\linewidth}
\centering
\begin{tikzpicture}[scale=\tfl]
% \draw[dotted,step=1,gray] (-3,-3) grid (3,2);
\draw[line width = 1pt] (-1,-2) --
(-3,0) -- (-2,1)--(2,1)--
(3,0)--(1,-2) -- (-1,-2); \draw[densely dashdotted, gray,line width = 1.2pt] (-3,0) -- (3,0);
\draw[densely dashdotted, gray, line width = 1.2pt] (1,-2) -- (1,1);
\draw[densely dashdotted, gray, line width = 1.2pt] (-1,-2) -- (-1,1);
\node at (1,-2) [below] {\large{$0$}};
\node at (-1,-2) [below] {\large{$0$}};
\node at (-3,0) [below] {\large{$0$}};
\node at (-2,1) [left] {\large{$0$}};
\node at (-1,1) [above] {\large{$1/2$}};
\node at (1,1) [above] {\large{$1/2$}};
\node at (2,1) [right] {\large{$0$}};
\node at (3,0) [below] {\large{$0$}};
\node at (1,0) {\large{$1$}};
\node at (-1,0) {\large{$1$}};
\foreach \i in {-3,...,3}{
\foreach \j in {-3,...,2}{
\draw[gray] (\i,\j) node {\tiny{$\circ$}};
}
}
\draw (0,0) node {\textbullet};
\end{tikzpicture}
\caption*{$\deg \Phi $}
\end{subfigure}
}
\end{figure}
\begin{center}
\begin{tabularx}{\linewidth}{cc@{\hspace{3ex}}c@{\hspace{2ex}}c}
\toprule
\(y\) & \(\text{Vert}(\Delta_y)\) & \( h_y(\xi_2)\) & \( h_y(\xi_2) >\) \\
\midrule
\(0\) & \( \scalemath{0.8}{ \begin{pmatrix} -1 & 3 & -3 & 1 & 1 & -1 & 2 & -2 \\
-2 & 0 & 0 & -2 & 1 & 1 & 1 & 1 \\
1 & -2 & 1 & 0 & 0 & 1 & -1 & 1 \end{pmatrix}} \) & \( \frac{{\left(3 \, \xi_{2}^{2} + 2 \, {\left(\xi_{2}^{3} - 3 \, \xi_{2} - 3\right)} e^{\left(3 \, \xi_{2}\right)} + 6 \, {\left(3 \, \xi_{2}^{2} + 2\right)} e^{\left(2 \, \xi_{2}\right)} - 6\right)} e^{\left(-2 \, \xi_{2}\right)}}{6 \, \xi_{2}^{4}}\) & \(0.3681\) \\ \midrule
\(1\) & \( \scalemath{0.8}{ \begin{pmatrix} -3 & -1 & 3 & 2 & 1 & 1 & -2 & -1 \\
0 & -2 & 0 & 1 & 1 & -2 & 1 & 1 \\
1 & -1 & 1 & 1 & \frac{1}{2} & -1 & 1 & \frac{1}{2} \end{pmatrix}} \) & \( \frac{{\left(12 \, \xi_{2}^{2} - {\left(14 \, \xi_{2}^{3} + 15 \, \xi_{2}^{2} - 6\right)} e^{\left(3 \, \xi_{2}\right)} + 24 \, {\left(3 \, \xi_{2}^{2} - 3 \, \xi_{2} + 1\right)} e^{\left(2 \, \xi_{2}\right)} + 6 \, \xi_{2} - 30\right)} e^{\left(-2 \, \xi_{2}\right)}}{12 \, \xi_{2}^{4}}\) & \(2.516\) \\
\midrule
\(\infty\) & \( \scalemath{0.8}{ \begin{pmatrix} -3 & -1 & 1 & 2 & -1 & 1 & 1 & -1 & -2 & 3 \\
0 & -2 & -2 & 1 & 0 & 0 & 1 & 1 & 1 & 0 \\
-1 & 1 & 2 & 1 & -1 & 0 & 0 & -1 & -1 & 2 \end{pmatrix}} \) & \(\frac{{\left(9 \, \xi_{2}^{2} - 2 \, {\left(\xi_{2}^{3} - 3 \, \xi_{2} - 3\right)} e^{\left(3 \, \xi_{2}\right)} + 6 \, {\left(\xi_{2}^{2} - 6 \, \xi_{2} + 1\right)} e^{\left(2 \, \xi_{2}\right)} - 12\right)} e^{\left(-2 \, \xi_{2}\right)}}{6 \, \xi_{2}^{4}}\) & \(1.013\) \\ \midrule
\(\notin \{0,1,\infty\} \) & \( \scalemath{0.8}{ \begin{pmatrix} -1 & -1 & -3 & 2 & 1 & 3 & 1 & -2 & -1 & 1 \\
0 & -2 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & -2 \\
0 & 1 & 1 & 1 & \frac{1}{2} & 1 & 0 & 1 & \frac{1}{2} & 1 \end{pmatrix}} \) & \(\frac{{\left(12 \, \xi_{2}^{2} + {\left(14 \, \xi_{2}^{3} + 15 \, \xi_{2}^{2} - 6\right)} e^{\left(3 \, \xi_{2}\right)} - 12 \, {\left(2 \, \xi_{2}^{2} - 1\right)} e^{\left(2 \, \xi_{2}\right)} - 6 \, \xi_{2} - 6\right)} e^{\left(-2 \, \xi_{2}\right)}}{12 \, \xi_{2}^{4}}\) & \(3.946\) \\ \midrule
\midrule
\end{tabularx}
\end{center}
\newpage
%
%
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%
%
%
%
\begin{tabularx}{\linewidth}{clcc}
\toprule
\midrule
\textbf{Name:} & \ 3.21 \hspace{0.3\linewidth} & \textbf{Description:} & Blow up of \(\PP^1 \times \PP^2\) in curve of bidegree \((2,1)\).\\
\midrule
{\small $ \sigma = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ }, & \( g(\xi_1,\xi_1) = \frac{{\left(\xi_{1}^{3} - \xi_{1}^{2} + 4 \, {\left(\xi_{1}^{2} - \xi_{1}\right)} e^{\left(3 \, \xi_{1}\right)} + 6 \, {\left(2 \, \xi_{1} - 1\right)} e^{\xi_{1}} - 2 \, \xi_{1} + 6\right)} e^{\left(-\xi_{1}\right)}}{\xi_{1}^{4}} \), & $ R(X) = 76/97$ , & $\xi \sim (-0.69622,-0.69622)$
\end{tabularx}
\begin{figure}[H]
\centering
\label{fig:data321}
\resizebox{!}{0.25\textheight}{
\begin{subfigure}[b]{0.25\linewidth}
\centering
\begin{tikzpicture}[scale=\tfl]
% \draw[dotted,step=1,gray,] (-3,-3) grid (3,3);
\draw[line width = 1pt] (3,-3) --
(1,-2) -- (-2,1)--(-3,3)--
(-1,3)--(3,-1) -- (3,-3);
\draw[densely dashdotted, gray, line width = 1.2pt] (-1,0) -- (-1,3);
\node at (-1,0) [above] {\large{$0$}};
\node at (-2,1) [below] {\large{$0$}};
\node at (-3,3) [left] {\large{$0$}};
\node at (-1,3) [right] {\large{$0$}};
\node at (3,-1) [right] {\large{$-2$}};
\node at (3,-3) [right] {\large{$-2$}};
\node at (1,-2) [left] {\large{$-1$}};
\foreach \i in {-3,...,3}{
\foreach \j in {-3,...,3}{
\draw[gray] (\i,\j) node {\tiny{$\circ$}};
}
}
\draw (0,0) node {\textbullet};
\end{tikzpicture}
\caption*{$\Phi_0$}
\end{subfigure}
\begin{subfigure}[b]{0.25\linewidth}
\centering
\begin{tikzpicture}[scale=\tfl]
% \draw[dotted,step=1,gray,] (-3,-3) grid (3,3);
\draw[line width = 1pt] (3,-3) --
(1,-2) -- (-2,1)--(-3,3)--
(-1,3)--(3,-1) -- (3,-3);
\draw[densely dashdotted, gray, line width = 1.2pt] (0,-1) -- (3,-1);
\node at (0,-1) [left] {\large{$0$}};
\node at (1,-2) [left] {\large{$0$}};
\node at (3,-3) [right] {\large{$0$}};
\node at (3,-1) [right] {\large{$0$}};
\node at (-1,3) [right] {\large{$-2$}};
\node at (-3,3) [left] {\large{$-2$}};
\node at (-2,1) [left] {\large{$-1$}};
\foreach \i in {-3,...,3}{
\foreach \j in {-3,...,3}{
\draw[gray] (\i,\j) node {\tiny{$\circ$}};
}
}
\draw (0,0) node {\textbullet};
\end{tikzpicture}
\caption*{$\Phi_1$}
\end{subfigure}
\begin{subfigure}[b]{0.25\linewidth}
\centering
\begin{tikzpicture}[scale=\tfl]
% \draw[dotted,step=1,gray,] (-3,-3) grid (3,3);
\draw[line width = 1pt] (3,-3) --
(1,-2) -- (-2,1)--(-3,3)--
(-1,3)--(3,-1) -- (3,-3);
\draw[densely dashdotted, gray, line width = 1.2pt] (3,-3) -- (-3,3);
\node at (-2,1) [left] {\large{$-1$}};
\node at (-3,3) [left] {\large{$0$}};
\node at (-1,3) [right] {\large{$0$}};
\node at (3,-1) [right] {\large{$0$}};
\node at (3,-3) [right] {\large{$0$}};
\node at (1,-2) [below] {\large{$-1$}};
\foreach \i in {-3,...,3}{
\foreach \j in {-3,...,3}{
\draw[gray] (\i,\j) node {\tiny{$\circ$}};
}
}
\draw (0,0) node {\textbullet};
\end{tikzpicture}
\caption*{$\Phi_\infty$}
\end{subfigure}
\begin{subfigure}[b]{0.25\linewidth}
\centering
\begin{tikzpicture}[scale=\tfl]
% \draw[dotted,step=1,gray,] (-3,-3) grid (3,3);
\draw[line width = 1pt] (3,-3) --
(1,-2) -- (-2,1)--(-3,3)--
(-1,3)--(3,-1) -- (3,-3);
\draw[densely dashdotted, gray, line width = 1.2pt] (3,-3) -- (-3,3);
\draw[densely dashdotted, gray, line width = 1.2pt] (-1,0) -- (-1,3);
\draw[densely dashdotted, gray, line width = 1.2pt] (0,-1) -- (3,-1);
\node at (-2,1) [left] {\large{$0$}};
\node at (-3,3) [left] {\large{$0$}};
\node at (-1,3) [right] {\large{$0$}};
\node at (3,-1) [right] {\large{$-2$}};
\node at (3,-3) [right] {\large{$-3$}};
\node at (1,-2) [below] {\large{$0$}};
\node at (0,-1) [below] {\large{$1/2$}};
\node at (-1,0) [left] {\large{$1/2$}};
\node at (1,-1) {\large{$1$}};
\node at (-1,1) {\large{$1$}};
\foreach \i in {-3,...,3}{
\foreach \j in {-3,...,3}{
\draw[gray] (\i,\j) node {\tiny{$\circ$}};
}
}
\draw (0,0) node {\textbullet};
\end{tikzpicture}
\caption*{$\deg \Phi $}
\end{subfigure}
}
\end{figure}
\begin{center}
\begin{tabularx}{\linewidth}{cc@{\hspace{3ex}}c@{\hspace{2ex}}c}
\toprule
\(y\) & \(\text{Vert}(\Delta_y)\) & \( h_y(\xi_2)\) & \( h_y(\xi_2) \in\) \\
\midrule
\(0\) & \( \scalemath{0.8}{ \begin{pmatrix} -2 & 1 & -3 & 3 & 1 & -1 & 3 & -1 & 0 \\
1 & -1 & 3 & -1 & -2 & 3 & -3 & 0 & -1 \\
1 & -1 & 1 & -1 & 0 & 1 & -1 & 1 & 0 \end{pmatrix}} \) & \(-\frac{{\left(\xi_{2}^{3} - \xi_{2}^{2} - 2 \, {\left(2 \, \xi_{2}^{2} - 4 \, \xi_{2} + 3\right)} e^{\xi_{2}} - 2 \, \xi_{2} + 6\right)} e^{\left(-\xi_{2}\right)}}{2 \, \xi_{2}^{4}}\) & \(0.6910\) \\ \midrule
\(1\) & \( \scalemath{0.8}{ \begin{pmatrix} -3 & -2 & 3 & 1 & 3 & -1 & -1 & 0 & -1 \\
3 & 1 & -3 & -2 & -1 & 1 & 3 & -1 & 0 \\
-1 & 0 & 1 & 1 & 1 & -1 & -1 & 1 & 0 \end{pmatrix}} \) & \(-\frac{{\left(\xi_{2}^{3} - \xi_{2}^{2} - 2 \, {\left(2 \, \xi_{2}^{2} - 4 \, \xi_{2} + 3\right)} e^{\xi_{2}} - 2 \, \xi_{2} + 6\right)} e^{\left(-\xi_{2}\right)}}{2 \, \xi_{2}^{4}}\) & \(0.6910\) \\ \midrule
\(\infty\) & \( \scalemath{0.8}{ \begin{pmatrix} -3 & -2 & -1 & 3 & 1 & 3 & 0 & -1 \\
3 & 1 & 3 & -3 & -2 & -1 & -1 & 0 \\
1 & 0 & 1 & 1 & 0 & 1 & -\frac{1}{2} & -\frac{1}{2} \end{pmatrix}} \) & \(\frac{{\left(5 \, \xi_{2}^{3} + 9 \, \xi_{2}^{2} + 6 \, {\left(8 \, \xi_{2}^{2} - 4 \, \xi_{2} + 1\right)} e^{\left(3 \, \xi_{2}\right)} - 24 \, {\left(6 \, \xi_{2}^{2} - 8 \, \xi_{2} + 5\right)} e^{\xi_{2}} - 66 \, \xi_{2} + 114\right)} e^{\left(-\xi_{2}\right)}}{24 \, \xi_{2}^{4}}\) & \(0.6910\) \\ \midrule
\(\infty\) & \( \scalemath{0.8}{ \begin{pmatrix} -1 & -3 & -2 & 0 & 3 & 1 & -1 & 3 & 1 & -1 \\
1 & 3 & 1 & -1 & -1 & -1 & 3 & -3 & -2 & 0 \\
0 & 1 & 1 & \frac{1}{2} & 1 & 0 & 1 & 1 & 1 & \frac{1}{2} \end{pmatrix}} \) & \(-\frac{{\left(19 \, \xi_{2}^{3} - 33 \, \xi_{2}^{2} - 6 \, {\left(8 \, \xi_{2}^{2} - 4 \, \xi_{2} + 1\right)} e^{\left(3 \, \xi_{2}\right)} + 24 \, {\left(2 \, \xi_{2}^{2} - 1\right)} e^{\xi_{2}} + 18 \, \xi_{2} + 30\right)} e^{\left(-\xi_{2}\right)}}{24 \, \xi_{2}^{4}}\) & \(3.907\) \\ \midrule
\midrule
\end{tabularx}
\end{center}
\newpage
%
%
%
%
%
%
%
\begin{tabularx}{\linewidth}{clcc}
\toprule
\midrule
\textbf{Name:} & \ 3.22 \hspace{0.3\linewidth} & \textbf{Description:} & Blow up of \(\PP^1 \times \PP^2\) in \(\{0\} \times \PP^2\)\\
\midrule
{\small $ \sigma = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$ }, & \( g(0,\xi_2) = \frac{{\left(9 \, \xi_{2}^{3} + 9 \, \xi_{2}^{2} + {\left(\xi_{2}^{3} + 3 \, \xi_{2}^{2} - 24\right)} e^{\left(2 \, \xi_{2}\right)} + 24 \, {\left(\xi_{2} + 1\right)} e^{\xi_{2}}\right)} e^{\left(-\xi_{2}\right)}}{2 \, \xi_{2}^{4}} \), & $ R(X) = 40/49$ , & $\xi \sim (0,0.91479)$
\end{tabularx}
\begin{figure}[H]
\centering
\label{fig:data322}
\resizebox{!}{0.25\textheight}{
\begin{subfigure}[b]{0.25\linewidth}
\centering
\begin{tikzpicture}[scale=\tfl]
% \draw[dotted,step=1,gray,] (-3,-3) grid (3,2);
\draw[line width = 1pt] (-3,-1) --
(-3,0) -- (-1,1)--(1,1)--
(3,0)--(3,-1) -- (-3,-1);
\draw[densely dashdotted, gray, line width = 1.2pt] (0,-1) -- (0,1);
\node at (-3,-1) [below] {\large{$0$}};
\node at (-3,0) [above] {\large{$0$}};
\node at (-1,1) [above] {\large{$0$}};
\node at (0,1) [above] {\large{$0$}};
\node at (1,1) [above] {\large{$-1$}};
\node at (3,0) [above] {\large{$-3$}};
\node at (3,-1) [below] {\large{$-3$}};
\node at (0,-1) [below] {\large{$0$}};
\foreach \i in {-3,...,3}{
\foreach \j in {-2,...,2}{
\draw[gray] (\i,\j) node {\tiny{$\circ$}};
}
}
\draw (0,0) node {\textbullet};
\end{tikzpicture}
\caption*{$\Phi_0$}
\end{subfigure}
\begin{subfigure}[b]{0.25\linewidth}
\centering
\begin{tikzpicture}[scale=\tfl]
% \draw[dotted,step=1,gray] (-3,-3) grid (3,2);
\draw[line width = 1pt] (-3,-1) --
(-3,0) -- (-1,1)--(1,1)--
(3,0)--(3,-1) -- (-3,-1);
\draw[densely dashdotted, gray, line width = 1.2pt] (-3,0) -- (3,0);
\node at (-3,-1) [below] {\large{$0$}};
\node at (-3,0) [above] {\large{$0$}};
\node at (-1,1) [above] {\large{$-1$}};
\node at (1,1) [above] {\large{$-1$}};
\node at (3,0) [above] {\large{$0$}};
\node at (3,-1) [below] {\large{$0$}};
\foreach \i in {-3,...,3}{
\foreach \j in {-2,...,2}{
\draw[gray] (\i,\j) node {\tiny{$\circ$}};
}
}
\draw (0,0) node {\textbullet};
\end{tikzpicture}
\caption*{$\Phi_1$}
\end{subfigure}
\begin{subfigure}[b]{0.25\linewidth}
\centering
\begin{tikzpicture}[scale=\tfl]
% \draw[dotted,step=1,gray] (-3,-3) grid (3,2);
\draw[line width = 1pt] (-3,-1) --
(-3,0) -- (-1,1)--(1,1)--
(3,0)--(3,-1) -- (-3,-1);
\node at (-3,-1) [below] {\large{$-2$}};
\node at (-3,0) [above] {\large{$-2$}};
\node at (-1,1) [above] {\large{$-1$}};
\node at (1,1) [above] {\large{$0$}};
\node at (3,0) [above] {\large{$1$}};
\node at (3,-1) [below] {\large{$1$}};
\foreach \i in {-3,...,3}{
\foreach \j in {-2,...,2}{
\draw[gray] (\i,\j) node {\tiny{$\circ$}};
}
}
\draw (0,0) node {\textbullet};
\end{tikzpicture}
\caption*{$\Phi_\infty$}
\end{subfigure}
\begin{subfigure}[b]{0.25\linewidth}
\centering
\begin{tikzpicture}[scale=\tfl]
% \draw[dotted,step=1,gray] (-3,-3) grid (3,2);
\draw[line width = 1pt] (-3,-1) --
(-3,0) -- (-1,1)--(1,1)--
(3,0)--(3,-1) -- (-3,-1);
\draw[densely dashdotted, gray, line width = 1.2pt] (0,-1) -- (0,1);
\draw[densely dashdotted, gray, line width = 1.2pt] (-3,0) -- (3,0);
\node at (-3,-1) [below] {\large{$-2$}};
\node at (-3,0) [above] {\large{$-2$}};
\node at (-1,1) [left] {\large{$-2$}};
\node at (0,1) [above] {\large{$-\nicefrac{1}{2}$}};
\node at (1,1) [right] {\large{$-2$}};
\node at (3,0) [above] {\large{$-2$}};
\node at (3,-1) [below] {\large{$-2$}};
\node at (0,0) [below left] {\large{$-\nicefrac{1}{2}$}};
\node at (0,-1) [below] {\large{$-\nicefrac{1}{2}$}};
\foreach \i in {-3,...,3}{
\foreach \j in {-2,...,2}{
\draw[gray] (\i,\j) node {\tiny{$\circ$}};
}
}
\draw (0,0) node {\textbullet};
\end{tikzpicture}
\caption*{$\deg \Phi $}
\end{subfigure}
}
\end{figure}
\begin{center}
\begin{tabularx}{\linewidth}{cc@{\hspace{5ex}}c@{\hspace{5ex}}c}
\toprule
\(y\) & \(\text{Vert}(\Delta_y)\) & \( h_y(\xi_2)\) & \( h_y(\xi_2) >\) \\
\midrule
\(0\) & \( \scalemath{0.8}{ \begin{pmatrix} 3 & -3 & 3 & -3 & 1 & 0 & 0 & -1 \\
0 & 0 & -1 & -1 & 1 & -1 & 1 & 1 \\
-2 & 1 & -2 & 1 & 0 & 1 & 1 & 1 \end{pmatrix}} \) & \(\frac{9 \, \xi_{2}^{2} + {\left(\xi_{2}^{3} + 3 \, \xi_{2}^{2} - 24\right)} e^{\xi_{2}} + 24 \, \xi_{2} + 24}{3 \, \xi_{2}^{4}}\) & \(0.8323\) \\ \midrule
\(1\) & \( \scalemath{0.8}{ \begin{pmatrix} -3 & -1 & -3 & 3 & 1 & 3 & 0 & 0 \\
0 & 1 & -1 & 0 & 1 & -1 & -1 & 1 \\
1 & 0 & 1 & 1 & 0 & 1 & -\frac{1}{2} & -\frac{1}{2} \end{pmatrix}} \) & \(-\frac{{\left(27 \, \xi_{2}^{3} + {\left(\xi_{2}^{3} - 24 \, \xi_{2} - 96\right)} e^{\left(2 \, \xi_{2}\right)} + 24 \, {\left(3 \, \xi_{2}^{2} + 5 \, \xi_{2} + 4\right)} e^{\xi_{2}}\right)} e^{\left(-\xi_{2}\right)}}{12 \, \xi_{2}^{4}}\) & \(2.164\) \\ \midrule
\(\infty\) & \( \scalemath{0.8}{ \begin{pmatrix} -3 & -3 & 3 & 1 & -1 & 0 & 0 & 0 & 3 \\
0 & -1 & 0 & 1 & 1 & 0 & -1 & 1 & -1 \\
-1 & -1 & 2 & 1 & 0 & -1 & -1 & 0 & 2 \end{pmatrix}} \) & \( \frac{9 \, \xi_{2}^{2} + {\left(\xi_{2}^{3} + 3 \, \xi_{2}^{2} - 24\right)} e^{\xi_{2}} + 24 \, \xi_{2} + 24}{6 \, \xi_{2}^{4}}\) & \(0.4161\) \\ \midrule
\(\notin \{0,1,\infty\} \) & \( \scalemath{0.8}{ \begin{pmatrix} 0 & -3 & -3 & 3 & 1 & 0 & -1 & 3 & 0 \\
0 & -1 & 0 & -1 & 1 & 1 & 1 & 0 & -1 \\
-\frac{1}{2} & 1 & 1 & 1 & 1 & \frac{1}{2} & 1 & 1 & -\frac{1}{2} \end{pmatrix}} \) & \(-\frac{{\left(27 \, \xi_{2}^{3} - {\left(5 \, \xi_{2}^{3} + 18 \, \xi_{2}^{2} + 24 \, \xi_{2} - 48\right)} e^{\left(2 \, \xi_{2}\right)} + 6 \, {\left(3 \, \xi_{2}^{2} - 4 \, \xi_{2} - 8\right)} e^{\xi_{2}}\right)} e^{\left(-\xi_{2}\right)}}{12 \, \xi_{2}^{4}}\) & \(3.419\) \\ \midrule
\midrule
\end{tabularx}
\end{center}
\newpage
%
%
%
%
%
%
%
\begin{tabularx}{\linewidth}{clcc}
\toprule
\midrule
\textbf{Name:} & \ 3.24 \hspace{0.3\linewidth} & \textbf{Description:} & Blow up of \(W\) in \((0:0:1:*:*:0)\).\\
\midrule
{\small $ \sigma = \begin{pmatrix} -1 & -1 \\ 1 & 0 \end{pmatrix}$ }, & \( g_1(0,\xi_2) = \xi_{2} \ {\mapsto}\ \frac{2 \, {\left({\left(\xi_{2}^{2} - 2\right)} e^{\left(3 \, \xi_{2}\right)} - 2 \, \xi_{2} + 4 \, e^{\left(2 \, \xi_{2}\right)} - 2\right)} e^{\left(-2 \, \xi_{2}\right)}}{\xi_{2}^{3}} \), & $ R(X) = 21/25$ , & $\xi \sim (0,0.43475)$
\end{tabularx}
\begin{figure}[H]
\centering
\label{fig:data324}
\resizebox{!}{0.25\textheight}{
\begin{subfigure}[b]{0.25\linewidth}
\centering
\begin{tikzpicture}[scale=\tfl]
% \draw[dotted,step=1,gray,] (-3,-3) grid (3,2);
\draw[line width = 1pt] (0,-2) --
(-2,0) -- (-2,1)--(1,1)--
(2,0)--(2,-2) -- (0,-2);
\draw[densely dashdotted, gray, line width = 1.2pt] (0,-2) -- (0,1);
\node at (0,-2) [below] {\large{$0$}};
\node at (-2,0) [left] {\large{$-2$}};
\node at (-2,1) [left] {\large{$-2$}};
\node at (0,1) [above] {\large{$0$}};
\node at (1,1) [right] {\large{$0$}};
\node at (2,0) [right] {\large{$0$}};
\node at (2,-2) [below] {\large{$0$}};
\foreach \i in {-3,...,3}{
\foreach \j in {-3,...,2}{
\draw[gray] (\i,\j) node {\tiny{$\circ$}};
}
}
\draw (0,0) node {\textbullet};
\end{tikzpicture}
\caption*{$\Phi_0$}
\end{subfigure}
\begin{subfigure}[b]{0.25\linewidth}
\centering
\begin{tikzpicture}[scale=\tfl]
% \draw[dotted,step=1,gray] (-3,-3) grid (3,2);
\draw[line width = 1pt] (0,-2) --
(-2,0) -- (-2,1)--(1,1)--
(2,0)--(2,-2) -- (0,-2);
\draw[densely dashdotted, gray, line width = 1.2pt] (-2,0) -- (2,0);
\node at (0,-2) [below] {\large{$-2$}};
\node at (-2,0) [left] {\large{$0$}};
\node at (-2,1) [left] {\large{$0$}};
\node at (1,1) [right] {\large{$0$}};
\node at (2,0) [right] {\large{$0$}};
\node at (2,-2) [below] {\large{$-2$}};
\foreach \i in {-3,...,3}{
\foreach \j in {-3,...,2}{
\draw[gray] (\i,\j) node {\tiny{$\circ$}};
}
}
\draw (0,0) node {\textbullet};
\end{tikzpicture}
\caption*{$\Phi_1$}
\end{subfigure}
\begin{subfigure}[b]{0.25\linewidth}
\centering
\begin{tikzpicture}[scale=\tfl]
% \draw[dotted,step=1,gray] (-3,-3) grid (3,2);
\draw[line width = 1pt] (0,-2) --
(-2,0) -- (-2,1)--(1,1)--
(2,0)--(2,-2) -- (0,-2);
\draw[densely dashdotted, gray, line width = 1.2pt] (2,-2) -- (-1,1);
\node at (0,-2) [below] {\large{$0$}};
\node at (-2,0) [left] {\large{$0$}};
\node at (-2,1) [left] {\large{$0$}};
\node at (-1,1) [above] {\large{$0$}};
\node at (1,1) [right] {\large{$-2$}};
\node at (2,0) [right] {\large{$-2$}};
\node at (2,-2) [below] {\large{$0$}};
\draw (0,0) node {\textbullet};
\foreach \i in {-3,...,3}{
\foreach \j in {-3,...,2}{
\draw[gray] (\i,\j) node {\tiny{$\circ$}};
}
}
\end{tikzpicture}
\caption*{$\Phi_\infty$}
\end{subfigure}
\begin{subfigure}[b]{0.25\linewidth}
\centering
\begin{tikzpicture}[scale=\tfl]
% \draw[dotted,step=1,gray] (-3,-3) grid (3,2);
\draw[line width = 1pt] (0,-2) --
(-2,0) -- (-2,1)--(1,1)--
(2,0)--(2,-2) -- (0,-2);
\draw[densely dashdotted, gray, line width = 1.2pt] (0,-2) -- (0,1);
\draw[densely dashdotted, gray, line width = 1.2pt] (-2,0) -- (2,0);
\draw[densely dashdotted, gray, line width = 1.2pt] (2,-2) -- (-1,1);
\node at (0,-2) [below] {\large{$-2$}};
\node at (-2,0) [left] {\large{$-2$}};
\node at (-2,1) [left] {\large{$-2$}};
\node at (-1,1) [above] {\large{$-1$}};
\node at (1,1) [right] {\large{$-2$}};
\node at (2,0) [right] {\large{$-2$}};
\node at (2,-2) [below] {\large{$-2$}};
\node at (0,1) [above] {\large{$-1$}};
\node at (0,0) [below left] {\large{$-\nicefrac{1}{2}$}};
\foreach \i in {-3,...,3}{
\foreach \j in {-3,...,2}{
\draw[gray] (\i,\j) node {\tiny{$\circ$}};
}
}
\draw (0,0) node {\textbullet};
\end{tikzpicture}
\caption*{$\deg \Phi $}
\end{subfigure}
}
\end{figure}
\begin{center}
\begin{tabularx}{\linewidth}{cc@{\hspace{5ex}}c@{\hspace{5ex}}c}
\toprule
\(y\) & \(\text{Vert}(\Delta_y)\) & \( h_y(\xi_2)\) & \( h_y(\xi_2) >\) \\
\midrule
\(0\) & \( \scalemath{0.8}{ \begin{pmatrix} -3 & -2 & 2 & 1 & -1 & 0 \\ 0 & 1 & 1 & 0 & -3 & 1 \\ 1 & 1 & -1 & -2 & 1 & 1\end{pmatrix}} \) & \( \frac{2 \, {\left({\left(\xi_{2} - 1\right)} e^{\left(2 \, \xi_{2}\right)} + \xi_{2} + 1\right)} e^{\left(-2 \, \xi_{2}\right)}}{\xi_{2}^{3}}\) & \(0.8793\) \\ \midrule
\(1\) & \( \scalemath{0.8}{ \begin{pmatrix} -3 & -2 & 2 & 1 & -1 & 0 \\ 0 & 1 & 1 & 0 & -3 & 1 \\ 1 & 1 & -1 & -2 & 1 & 1\end{pmatrix}} \) & \( -\frac{{\left(12 \, \xi_{2}^{2} - {\left(7 \, \xi_{2}^{3} + 3 \, \xi_{2}^{2} - 6 \, \xi_{2} + 6\right)} e^{\left(3 \, \xi_{2}\right)} + 12 \, {\left(2 \, \xi_{2}^{2} - 2 \, \xi_{2} + 1\right)} e^{\left(2 \, \xi_{2}\right)} + 12 \, \xi_{2} - 6\right)} e^{\left(-2 \, \xi_{2}\right)}}{6 \, \xi_{2}^{4}}\) & \(1.2944\) \\ \midrule
\(\infty\) & \( \scalemath{0.8}{ \begin{pmatrix} -3 & -2 & 2 & 1 & -1 & 0 \\ 0 & 1 & 1 & 0 & -3 & 1 \\ 1 & 1 & -1 & -2 & 1 & 1\end{pmatrix}} \) & \( \frac{2 \, {\left({\left(\xi_{2} - 1\right)} e^{\left(2 \, \xi_{2}\right)} + \xi_{2} + 1\right)} e^{\left(-2 \, \xi_{2}\right)}}{\xi_{2}^{3}}\) & \(0.8793\) \\ \midrule
\(\notin \{0,1,\infty\} \) & \( \scalemath{0.8}{ \begin{pmatrix} -3 & -2 & 2 & 1 & -1 & 0 \\ 0 & 1 & 1 & 0 & -3 & 1 \\ 1 & 1 & -1 & -2 & 1 & 1\end{pmatrix}} \) & \( \frac{{\left(12 \, \xi_{2}^{2} + {\left(7 \, \xi_{2}^{3} + 3 \, \xi_{2}^{2} - 6 \, \xi_{2} + 6\right)} e^{\left(3 \, \xi_{2}\right)} + 12 \, \xi_{2} - 12 \, e^{\left(2 \, \xi_{2}\right)} + 6\right)} e^{\left(-2 \, \xi_{2}\right)}}{6 \, \xi_{2}^{4}}\) & \(3.0544\) \\ \midrule
\midrule
\end{tabularx}
\end{center}
\newpage
%
%
%
%
%
%
%
\begin{tabularx}{\linewidth}{clcc}
\toprule
\midrule
\textbf{Name:} & \ 4.8 \hspace{0.3\linewidth} & \textbf{Description:} & Blow up of \((\PP^1)^3\) in a curve.\\
\midrule
{\small $ \sigma = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$ }, & \( g(0,\xi_2) = \frac{{\left(4 \, \xi_{2}^{3} + 4 \, \xi_{2}^{2} + {\left(\xi_{2}^{3} + \xi_{2}^{2} - 2 \, \xi_{2} - 6\right)} e^{\left(2 \, \xi_{2}\right)} + 2 \, {\left(4 \, \xi_{2} + 3\right)} e^{\xi_{2}}\right)} e^{\left(-\xi_{2}\right)}}{\xi_{2}^{4}}\), & $ R(X) = 76/89$ , & $\xi \sim (0,0.62431)$
\end{tabularx}
\begin{figure}[H]
\centering
\label{fig:data48}
\resizebox{!}{0.25\textheight}{
\begin{subfigure}[b]{0.25\linewidth}
\centering
\begin{tikzpicture}[scale=\tfl]
% \draw[dotted,step=1,gray,] (-3,-3) grid (3,2);
\draw[line width = 1pt] (-2,-1) --
(-2,0) -- (-1,1)--(1,1)--
(2,0)--(2,-1) -- (-2,-1); \draw[densely dashdotted, gray, line width = 1.2pt] (0,-1) -- (0,1);
\node at (-2,-1) [left] {\large{$-2$}};
\node at (-2,0) [left] {\large{$-2$}};
\node at (-1,1) [left] {\large{$-1$}};
\node at (0,1) [above] {\large{$0$}};
\node at (1,1) [right] {\large{$0$}};
\node at (2,0) [right] {\large{$0$}};
\node at (2,-1) [right] {\large{$0$}};
\node at (0,-1) [below] {\large{$0$}};
\foreach \i in {-3,...,3}{
\foreach \j in {-2,...,2}{
\draw[gray] (\i,\j) node {\tiny{$\circ$}};
}
}
\draw (0,0) node {\textbullet};
\end{tikzpicture}
\caption*{$\Phi_0$}
\end{subfigure}
\begin{subfigure}[b]{0.25\linewidth}
\centering
\begin{tikzpicture}[scale=\tfl]
% \draw[dotted,step=1,gray] (-3,-3) grid (3,2);
\draw[line width = 1pt] (-2,-1) --
(-2,0) -- (-1,1)--(1,1)--
(2,0)--(2,-1) -- (-2,-1);
\draw[densely dashdotted, gray,line width = 1.2pt] (-2,0) -- (2,0);
\node at (-2,-1) [left] {\large{$0$}};
\node at (-2,0) [left] {\large{$0$}};
\node at (-1,1) [left] {\large{$-1$}};
\node at (1,1) [right] {\large{$-1$}};
\node at (2,0) [right] {\large{$0$}};
\node at (2,-1) [right] {\large{$0$}};
\foreach \i in {-3,...,3}{
\foreach \j in {-2,...,2}{
\draw[gray] (\i,\j) node {\tiny{$\circ$}};
}
}
\draw (0,0) node {\textbullet};
\end{tikzpicture}
\caption*{$\Phi_1$}
\end{subfigure}
\begin{subfigure}[b]{0.25\linewidth}
\centering
\begin{tikzpicture}[scale=\tfl]
% \draw[dotted,step=1,gray] (-3,-3) grid (3,2);
\draw[line width = 1pt] (-2,-1) --
(-2,0) -- (-1,1)--(1,1)--
(2,0)--(2,-1) -- (-2,-1);
\draw[densely dashdotted, gray, line width = 1.2pt] (0,-1) -- (0,1);
\node at (2,-1) [right] {\large{$-2$}};
\node at (2,0) [right] {\large{$-2$}};
\node at (1,1) [right] {\large{$-1$}};
\node at (0,1) [above] {\large{$0$}};
\node at (-1,1) [left] {\large{$0$}};
\node at (-2,0) [left] {\large{$0$}};
\node at (-2,-1) [left] {\large{$0$}};
\node at (0,-1) [below] {\large{$0$}};
\foreach \i in {-3,...,3}{
\foreach \j in {-2,...,2}{
\draw[gray] (\i,\j) node {\tiny{$\circ$}};
}
}
\draw (0,0) node {\textbullet};
\end{tikzpicture}
\caption*{$\Phi_\infty$}
\end{subfigure}
\begin{subfigure}[b]{0.25\linewidth}
\centering
\begin{tikzpicture}[scale=\tfl]
% \draw[dotted,step=1,gray] (-3,-3) grid (3,2);
\draw[line width = 1pt] (-2,-1) --
(-2,0) -- (-1,1)--(1,1)--
(2,0)--(2,-1) -- (-2,-1); \draw[densely dashdotted, gray,line width = 1.2pt] (-2,0) -- (2,0); \draw[densely dashdotted, gray,line width = 1.2pt] (0,-1) -- (0,1) ;
\node at (-2,-1) [left] {\large{$-2$}};
\node at (-2,0) [left] {\large{$-2$}};
\node at (-1,1) [left] {\large{$-2$}};
\node at (0,1) [above] {\large{$-1$}};
\node at (1,1) [right] {\large{$-2$}};
\node at (2,0) [right] {\large{$-2$}};
\node at (2,-1) [right] {\large{$-2$}};
\node at (0,-1) [below] {\large{$0$}};
\node at (0,0) [right] {\large{$0$}};
\foreach \i in {-3,...,3}{
\foreach \j in {-2,...,2}{
\draw[gray] (\i,\j) node {\tiny{$\circ$}};
}
}
\draw (0,0) node {\textbullet};
\end{tikzpicture}
\caption*{$\deg \Phi $}
\end{subfigure}
}
\end{figure}
\begin{center}
\begin{tabularx}{\linewidth}{cc@{\hspace{5ex}}c@{\hspace{5ex}}c}
\toprule
\(y\) & \(\text{Vert}(\Delta_y)\) & \( h_y(\xi_2)\) & \( h_y(\xi_2) >\) \\
\midrule
\(0\) & \( \scalemath{0.8}{ \begin{pmatrix} -3 & -2 & 2 & 1 & -1 & 0 \\ 0 & 1 & 1 & 0 & -3 & 1 \\ 1 & 1 & -1 & -2 & 1 & 1\end{pmatrix}} \) & \(\frac{4 \, \xi_{2}^{2} + {\left(\xi_{2}^{3} + \xi_{2}^{2} - 2 \, \xi_{2} - 6\right)} e^{\xi_{2}} + 8 \, \xi_{2} + 6}{2 \, \xi_{2}^{4}}\) & \(0.6636\) \\ \midrule
\(1\) & \( \scalemath{0.8}{ \begin{pmatrix} -3 & -2 & 2 & 1 & -1 & 0 \\ 0 & 1 & 1 & 0 & -3 & 1 \\ 1 & 1 & -1 & -2 & 1 & 1\end{pmatrix}} \) & \(\frac{4 \, \xi_{2}^{2} + {\left(\xi_{2}^{3} + \xi_{2}^{2} - 2 \, \xi_{2} - 6\right)} e^{\xi_{2}} + 8 \, \xi_{2} + 6}{2 \, \xi_{2}^{4}}\) & \(0.6636\) \\ \midrule
\(\infty\) & \( \scalemath{0.8}{ \begin{pmatrix} -3 & -2 & 2 & 1 & -1 & 0 \\ 0 & 1 & 1 & 0 & -3 & 1 \\ 1 & 1 & -1 & -2 & 1 & 1\end{pmatrix}} \) & \(-\frac{{\left(4 \, \xi_{2}^{3} + {\left(\xi_{2}^{3} - 6 \, \xi_{2} - 12\right)} e^{\left(2 \, \xi_{2}\right)} + 6 \, {\left(2 \, \xi_{2}^{2} + 3 \, \xi_{2} + 2\right)} e^{\xi_{2}}\right)} e^{\left(-\xi_{2}\right)}}{3 \, \xi_{2}^{4}}\) & \(1.103\) \\ \midrule
\(\notin \{0,1,\infty\} \) & \( \scalemath{0.8}{ \begin{pmatrix} -3 & -2 & 2 & 1 & -1 & 0 \\ 0 & 1 & 1 & 0 & -3 & 1 \\ 1 & 1 & -1 & -2 & 1 & 1\end{pmatrix}} \) & \(-\frac{{\left(4 \, \xi_{2}^{3} - {\left(2 \, \xi_{2}^{3} + 3 \, \xi_{2}^{2} - 6\right)} e^{\left(2 \, \xi_{2}\right)} - 6 \, {\left(\xi_{2} + 1\right)} e^{\xi_{2}}\right)} e^{\left(-\xi_{2}\right)}}{3 \, \xi_{2}^{4}}\) & \(2.442\) \\ \midrule
\midrule
\end{tabularx}
\end{center}
%
%
%
%
%
%\begin{figure}[h]
%\centering
%
%\label{fig:data45*}
%\resizebox{!}{0.25\textheight}{
%\begin{subfigure}[b]{0.25\linewidth}
%\centering
% \begin{tikzpicture}[scale=\tfl]
% \draw[dotted,step=1,gray,] (-3,-3) grid (3,3); \draw[line width = 1pt] (3,-3) --
% (1,-2) -- (-2,1)--(-3,3)--
% (-2,3)--(3,-2) -- (3,-3); \draw[densely dashdotted, gray, line width = 1.2pt] (0,-3) -- (0,1);
% \node at (3,-3) [below] {\large{$-2$}};
% \node at (1,-2) [right] {\large{$0$}};
% \node at (-2,1) [left] {\large{$0$}};
% \node at (-3,3) [right] {\large{$0$}};
% \node at (-2,3) [right] {\large{$-2$}};
% \node at (3,-2) [right] {\large{$-2$}};
% \end{tikzpicture}
% \caption*{$\Phi_0$}
%\end{subfigure}
%\begin{subfigure}[b]{0.25\linewidth}