Hirzebruch Surfaces
| ID | $\operatorname{rk} \mathrm{Pic}$ | $\operatorname{rk} K_0$ | Fan | Chamers | Cox degrees and $\Theta$-collection | Ext table |
|---|---|---|---|---|---|---|
| $0$ | 2 | 4 |
4: $\begin{pmatrix}1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1\end{pmatrix}$ 4: $\begin{pmatrix}1 & 0 & 1 & 0 \\ 1 & 1 & 0 & 0\end{pmatrix}$ |
$\begin{pmatrix}1 & 2 & 2 & 4 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 1\end{pmatrix}$ | ||
| $1$ | 2 | 4 |
4: $\begin{pmatrix}1 & -1 & 1 & 0 \\ 0 & 1 & 0 & 1\end{pmatrix}$ 4: $\begin{pmatrix}1 & 0 & 1 & 0 \\ 1 & 1 & 0 & 0\end{pmatrix}$ |
$\begin{pmatrix}1 & 2 & 3 & 5 \\ 0 & 1 & 1 & 3 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 1\end{pmatrix}$ | ||
| $2$ | 2 | 4 |
4: $\begin{pmatrix}1 & -2 & 1 & 0 \\ 0 & 1 & 0 & 1\end{pmatrix}$ 5: $\begin{pmatrix}1 & 0 & -1 & 1 & 0 \\ 1 & 1 & 1 & 0 & 0\end{pmatrix}$ |
$\begin{pmatrix}1 & 2 & 3 & 4 & 6 \\ 0 & 1 & 2 & 2 & 4 \\ T & 0 & 1 & 1+T & 2 \\ 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 1\end{pmatrix}$ | ||
| $3$ | 2 | 4 |
4: $\begin{pmatrix}1 & -3 & 1 & 0 \\ 0 & 1 & 0 & 1\end{pmatrix}$ 6: $\begin{pmatrix}1 & 0 & -1 & -2 & 1 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0\end{pmatrix}$ |
$\begin{pmatrix}1 & 2 & 3 & 4 & 5 & 7 \\ 0 & 1 & 2 & 3 & 3 & 5 \\ T & 0 & 1 & 2 & 2+T & 3 \\ 2T & T & 0 & 1 & 1+2T & 2+T \\ 0 & 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$ | ||
| $4$ | 2 | 4 |
4: $\begin{pmatrix}1 & -4 & 1 & 0 \\ 0 & 1 & 0 & 1\end{pmatrix}$ 7: $\begin{pmatrix}1 & 0 & -1 & -2 & -3 & 1 & 0 \\ 1 & 1 & 1 & 1 & 1 & 0 & 0\end{pmatrix}$ |
$\begin{pmatrix}1 & 2 & 3 & 4 & 5 & 6 & 8 \\ 0 & 1 & 2 & 3 & 4 & 4 & 6 \\ T & 0 & 1 & 2 & 3 & 3+T & 4 \\ 2T & T & 0 & 1 & 2 & 2+2T & 3+T \\ T & 2T & T & 0 & 1 & 1+T & 2+2T \\ 0 & 0 & 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$ | ||
| $5$ | 2 | 4 |
4: $\begin{pmatrix}1 & -5 & 1 & 0 \\ 0 & 1 & 0 & 1\end{pmatrix}$ 8: $\begin{pmatrix}1 & 0 & -1 & -2 & -3 & -4 & 1 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0\end{pmatrix}$ |
$\begin{pmatrix}1 & 2 & 3 & 4 & 5 & 6 & 7 & 9 \\ 0 & 1 & 2 & 3 & 4 & 5 & 5 & 7 \\ T & 0 & 1 & 2 & 3 & 4 & 4+T & 5 \\ 2T & T & 0 & 1 & 2 & 3 & 3+2T & 4+T \\ T & 2T & T & 0 & 1 & 2 & 2+T & 3+2T \\ 0 & T & 2T & T & 0 & 1 & 1 & 2+T \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$ |
Tip: click on an ID, then use j and l to go back and forth and compare the data.