Smooth Fano Toric Surfaces (aka del Pezzo Surfaces)
There are 5 varieties in this class:
| $(\dim,\#)$ | $\operatorname{rk} \mathrm{Pic}$ | $\operatorname{rk} K_0$ | Chambers | Cox degrees and $\Theta$-collection | Ext table |
|---|---|---|---|---|---|
| $(2,0)$ | 1 | 3 |
3: $\begin{pmatrix}1 & 1 & 1\end{pmatrix}$ 3: $\begin{pmatrix}2 & 1 & 0\end{pmatrix}$ |
$\begin{pmatrix}1 & 3 & 6 \\ 0 & 1 & 3 \\ 0 & 0 & 1\end{pmatrix}$ | |
| $(2,1)$ | 2 | 4 |
4: $\begin{pmatrix}1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1\end{pmatrix}$ 4: $\begin{pmatrix}1 & 0 & 1 & 0 \\ 1 & 1 & 0 & 0\end{pmatrix}$ |
$\begin{pmatrix}1 & 2 & 2 & 4 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 1\end{pmatrix}$ | |
| $(2,2)$ | 2 | 4 |
4: $\begin{pmatrix}1 & -1 & 1 & 0 \\ 0 & 1 & 0 & 1\end{pmatrix}$ 4: $\begin{pmatrix}1 & 0 & 1 & 0 \\ 1 & 1 & 0 & 0\end{pmatrix}$ |
$\begin{pmatrix}1 & 2 & 3 & 5 \\ 0 & 1 & 1 & 3 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 1\end{pmatrix}$ | |
| $(2,3)$ | 3 | 5 |
5: $\begin{pmatrix}1 & -1 & 1 & 0 & 0 \\ 0 & 1 & -1 & 1 & 0 \\ 0 & 0 & 1 & -1 & 1\end{pmatrix}$ 5: $\begin{pmatrix}1 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0\end{pmatrix}$ |
$\begin{pmatrix}1 & 1 & 2 & 2 & 4 \\ 0 & 1 & 1 & 1 & 3 \\ 0 & 0 & 1 & 0 & 2 \\ 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 1\end{pmatrix}$ | |
| $(2,4)$ | 4 | 6 |
6: $\begin{pmatrix}1 & -1 & 1 & 0 & 0 & 0 \\ -1 & 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & -1 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 1 & -1\end{pmatrix}$ 6: $\begin{pmatrix}0 & 1 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 1 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 & 0\end{pmatrix}$ |
$\begin{pmatrix}1 & 0 & 1 & 1 & 1 & 3 \\ 0 & 1 & 1 & 1 & 1 & 3 \\ 0 & 0 & 1 & 0 & 0 & 2 \\ 0 & 0 & 0 & 1 & 0 & 2 \\ 0 & 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$ |