Smooth Fano Toric 4-folds with $\rho\leq2$
There are 10 varieties in this class:
| $(\dim,\#)$ | $\operatorname{rk} \mathrm{Pic}$ | $\operatorname{rk} K_0$ | Chambers | Cox degrees and $\Theta$-collection | Ext table |
|---|---|---|---|---|---|
| $(4,0)$ | 1 | 5 |
5: $\begin{pmatrix}1 & 1 & 1 & 1 & 1\end{pmatrix}$ 5: $\begin{pmatrix}4 & 3 & 2 & 1 & 0\end{pmatrix}$ |
$\begin{pmatrix}1 & 5 & 15 & 35 & 70 \\ 0 & 1 & 5 & 15 & 35 \\ 0 & 0 & 1 & 5 & 15 \\ 0 & 0 & 0 & 1 & 5 \\ 0 & 0 & 0 & 0 & 1\end{pmatrix}$ | |
| $(4,1)$ | 2 | 8 |
6: $\begin{pmatrix}1 & 1 & 1 & 1 & -3 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1\end{pmatrix}$ 10: $\begin{pmatrix}3 & 2 & 1 & 0 & 3 & -1 & -2 & 2 & 1 & 0 \\ 1 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 0\end{pmatrix}$ |
$\begin{pmatrix}1 & 4 & 10 & 20 & 21 & 35 & 56 & 39 & 66 & 104 \\ 0 & 1 & 4 & 10 & 10 & 20 & 35 & 21 & 39 & 66 \\ 0 & 0 & 1 & 4 & 4 & 10 & 20 & 10 & 21 & 39 \\ 0 & 0 & 0 & 1 & 1 & 4 & 10 & 4 & 10 & 21 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 4 & 10 & 20 \\ T^3 & 0 & 0 & 0 & T^3 & 1 & 4 & 1 & 4 & 10 \\ 4T^3 & T^3 & 0 & 0 & 4T^3 & 0 & 1 & T^3 & 1 & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 4 & 10 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$ | |
| $(4,2)$ | 2 | 8 |
6: $\begin{pmatrix}1 & 1 & 1 & 1 & -2 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1\end{pmatrix}$ 9: $\begin{pmatrix}3 & 2 & 1 & 0 & 3 & -1 & 2 & 1 & 0 \\ 1 & 1 & 1 & 1 & 0 & 1 & 0 & 0 & 0\end{pmatrix}$ |
$\begin{pmatrix}1 & 4 & 10 & 20 & 11 & 35 & 24 & 45 & 76 \\ 0 & 1 & 4 & 10 & 4 & 20 & 11 & 24 & 45 \\ 0 & 0 & 1 & 4 & 1 & 10 & 4 & 11 & 24 \\ 0 & 0 & 0 & 1 & 0 & 4 & 1 & 4 & 11 \\ 0 & 0 & 0 & 0 & 1 & 0 & 4 & 10 & 20 \\ T^3 & 0 & 0 & 0 & T^3 & 1 & 0 & 1 & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 4 & 10 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$ | |
| $(4,3)$ | 2 | 8 |
6: $\begin{pmatrix}1 & 1 & 1 & 1 & -1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1\end{pmatrix}$ 8: $\begin{pmatrix}3 & 2 & 1 & 3 & 0 & 2 & 1 & 0 \\ 1 & 1 & 1 & 0 & 1 & 0 & 0 & 0\end{pmatrix}$ |
$\begin{pmatrix}1 & 4 & 10 & 5 & 20 & 14 & 30 & 55 \\ 0 & 1 & 4 & 1 & 10 & 5 & 14 & 30 \\ 0 & 0 & 1 & 0 & 4 & 1 & 5 & 14 \\ 0 & 0 & 0 & 1 & 0 & 4 & 10 & 20 \\ 0 & 0 & 0 & 0 & 1 & 0 & 1 & 5 \\ 0 & 0 & 0 & 0 & 0 & 1 & 4 & 10 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$ | |
| $(4,4)$ | 2 | 8 |
6: $\begin{pmatrix}1 & 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1\end{pmatrix}$ 8: $\begin{pmatrix}3 & 2 & 3 & 1 & 2 & 0 & 1 & 0 \\ 1 & 1 & 0 & 1 & 0 & 1 & 0 & 0\end{pmatrix}$ |
$\begin{pmatrix}1 & 4 & 2 & 10 & 8 & 20 & 20 & 40 \\ 0 & 1 & 0 & 4 & 2 & 10 & 8 & 20 \\ 0 & 0 & 1 & 0 & 4 & 0 & 10 & 20 \\ 0 & 0 & 0 & 1 & 0 & 4 & 2 & 8 \\ 0 & 0 & 0 & 0 & 1 & 0 & 4 & 10 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$ | |
| $(4,5)$ | 2 | 8 |
6: $\begin{pmatrix}1 & 1 & 1 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 & 1 & 1\end{pmatrix}$ 8: $\begin{pmatrix}3 & 3 & 2 & 2 & 1 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0\end{pmatrix}$ |
$\begin{pmatrix}1 & 2 & 5 & 9 & 15 & 25 & 35 & 55 \\ 0 & 1 & 1 & 5 & 5 & 15 & 15 & 35 \\ 0 & 0 & 1 & 2 & 5 & 9 & 15 & 25 \\ 0 & 0 & 0 & 1 & 1 & 5 & 5 & 15 \\ 0 & 0 & 0 & 0 & 1 & 2 & 5 & 9 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 5 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$ | |
| $(4,6)$ | 2 | 9 |
6: $\begin{pmatrix}1 & 1 & 1 & 0 & 0 & 0 \\ -2 & 0 & 0 & 1 & 1 & 1\end{pmatrix}$ 11: $\begin{pmatrix}2 & 2 & 2 & 1 & 2 & 1 & 1 & 0 & 1 & 0 & 0 \\ 2 & 1 & 0 & 2 & -1 & 1 & 0 & 2 & -1 & 1 & 0\end{pmatrix}$ |
$\begin{pmatrix}1 & 3 & 6 & 8 & 10 & 16 & 27 & 30 & 41 & 50 & 76 \\ 0 & 1 & 3 & 3 & 6 & 8 & 16 & 16 & 27 & 30 & 50 \\ 0 & 0 & 1 & 1 & 3 & 3 & 8 & 8 & 16 & 16 & 30 \\ 0 & 0 & 0 & 1 & 0 & 3 & 6 & 8 & 10 & 16 & 27 \\ T^2 & 0 & 0 & 2T^2 & 1 & 1 & 3 & 3+3T^2 & 8 & 8 & 16 \\ 0 & 0 & 0 & 0 & 0 & 1 & 3 & 3 & 6 & 8 & 16 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 3 & 3 & 8 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 3 & 6 \\ 0 & 0 & 0 & T^2 & 0 & 0 & 0 & 2T^2 & 1 & 1 & 3 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$ | |
| $(4,7)$ | 2 | 9 |
6: $\begin{pmatrix}1 & 1 & 1 & 0 & 0 & 0 \\ -1 & 0 & 0 & 1 & 1 & 1\end{pmatrix}$ 9: $\begin{pmatrix}2 & 2 & 1 & 2 & 1 & 0 & 1 & 0 & 0 \\ 2 & 1 & 2 & 0 & 1 & 2 & 0 & 1 & 0\end{pmatrix}$ |
$\begin{pmatrix}1 & 3 & 5 & 6 & 12 & 15 & 22 & 31 & 53 \\ 0 & 1 & 1 & 3 & 5 & 5 & 12 & 15 & 31 \\ 0 & 0 & 1 & 0 & 3 & 5 & 6 & 12 & 22 \\ 0 & 0 & 0 & 1 & 1 & 1 & 5 & 5 & 15 \\ 0 & 0 & 0 & 0 & 1 & 1 & 3 & 5 & 12 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 3 & 6 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 5 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$ | |
| $(4,8)$ | 2 | 9 |
6: $\begin{pmatrix}1 & 1 & 1 & 0 & 0 & 0 \\ -1 & -1 & 0 & 1 & 1 & 1\end{pmatrix}$ 10: $\begin{pmatrix}2 & 1 & 2 & 1 & 2 & 0 & 1 & 0 & 1 & 0 \\ 1 & 2 & 0 & 1 & -1 & 2 & 0 & 1 & -1 & 0\end{pmatrix}$ |
$\begin{pmatrix}1 & 2 & 3 & 7 & 6 & 11 & 15 & 25 & 26 & 45 \\ 0 & 1 & 0 & 3 & 0 & 7 & 6 & 15 & 10 & 26 \\ 0 & 0 & 1 & 2 & 3 & 3 & 7 & 11 & 15 & 25 \\ 0 & 0 & 0 & 1 & 0 & 2 & 3 & 7 & 6 & 15 \\ 0 & T^2 & 0 & 0 & 1 & T^2 & 2 & 3 & 7 & 11 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 3 & 0 & 6 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 & 3 & 7 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 3 \\ 0 & T^2 & 0 & 0 & 0 & T^2 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$ | |
| $(4,9)$ | 2 | 9 |
6: $\begin{pmatrix}1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 1\end{pmatrix}$ 9: $\begin{pmatrix}2 & 1 & 2 & 0 & 2 & 1 & 0 & 1 & 0 \\ 2 & 2 & 1 & 2 & 0 & 1 & 1 & 0 & 0\end{pmatrix}$ |
$\begin{pmatrix}1 & 3 & 3 & 6 & 6 & 9 & 18 & 18 & 36 \\ 0 & 1 & 0 & 3 & 0 & 3 & 9 & 6 & 18 \\ 0 & 0 & 1 & 0 & 3 & 3 & 6 & 9 & 18 \\ 0 & 0 & 0 & 1 & 0 & 0 & 3 & 0 & 6 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 3 & 6 \\ 0 & 0 & 0 & 0 & 0 & 1 & 3 & 3 & 9 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 3 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$ |