$\mathrm{Bl}_1\mathrm{Bl}_1\PP^2$: the double blow-up of $\PP^2$
This smooth projective variety is remarkable because it demonstrates that the walls in the secondary fan may not necessarily be complete hyperplanes.
| ID | $\operatorname{rk} \mathrm{Pic}$ | $\operatorname{rk} K_0$ | Fan | Chambers | Cox degrees and $\Theta$-collection | Ext table |
|---|---|---|---|---|---|---|
| $0$ | 3 | 5 |
5: $\begin{pmatrix}1 & -2 & 1 & 0 & 0 \\ 1 & -1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 1\end{pmatrix}$ 6: $\begin{pmatrix}1 & 0 & 0 & -1 & 1 & 0 \\ 1 & 1 & 0 & 0 & 1 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0\end{pmatrix}$ |
$\begin{pmatrix}1 & 1 & 2 & 2 & 3 & 5 \\ 0 & 1 & 1 & 2 & 2 & 4 \\ 0 & 0 & 1 & 1 & 1 & 3 \\ T & 0 & 0 & 1 & 1+T & 2 \\ 0 & 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$ |