FanoCats

(Toric) Fano Categories Database

Smooth Fano Toric Surfaces (aka del Pezzo Surfaces)

ID $\operatorname{rk} \mathrm{Pic}$ $\operatorname{rk} K_0$ Fan 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: $\begin{pmatrix}0 & 1 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 1 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 & 0\end{pmatrix}$		 $\begin{pmatrix}1 & 0 & 1 & 1 & 1 & 3 \\ 0 & 1 & 1 & 1 & 1 & 3 \\ 0 & 0 & 1 & 0 & 0 & 2 \\ 0 & 0 & 0 & 1 & 0 & 2 \\ 0 & 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$