This website, inspired by Pieter Belmans’ Fanography, is a tool for visually studying the derived categories of smooth fano toric varieties.
See here for an explainer. Please get in touch with me if you’d like to use the Macaulay2 package used for these computations.
Bug reports and contributions are welcome on GitHub.
Smooth Fano Toric Surfaces
| ID | $\operatorname{rk} \mathrm{Pic}$ | $\operatorname{rk} K_0$ | Fan | Chamers | 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: $\begin{pmatrix}0 & 1 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 1 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 & 0\end{pmatrix}$ |
$\begin{pmatrix}1 & 0 & 1 & 1 & 1 & 3 \\ 0 & 1 & 1 & 1 & 1 & 3 \\ 0 & 0 & 1 & 0 & 0 & 2 \\ 0 & 0 & 0 & 1 & 0 & 2 \\ 0 & 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$ |
Smooth Fano Toric Threefolds
| ID | $\operatorname{rk} \mathrm{Pic}$ | $\operatorname{rk} K_0$ | Fan | Chamers | Cox degrees and $\Theta$-collection | Ext table |
|---|---|---|---|---|---|---|
| $(3,0)$ | 1 | 4 |
4: $\begin{pmatrix}1 & 1 & 1 & 1\end{pmatrix}$ 4: $\begin{pmatrix}3 & 2 & 1 & 0\end{pmatrix}$ |
$\begin{pmatrix}1 & 4 & 10 & 20 \\ 0 & 1 & 4 & 10 \\ 0 & 0 & 1 & 4 \\ 0 & 0 & 0 & 1\end{pmatrix}$ | ||
| $(3,1)$ | 2 | 6 |
5: $\begin{pmatrix}0 & 0 & 0 & 1 & 1 \\ 1 & 1 & 1 & 0 & -2\end{pmatrix}$ 7: $\begin{pmatrix}1 & 1 & 1 & 1 & 0 & 0 & 0 \\ 2 & 1 & 0 & -1 & 2 & 1 & 0\end{pmatrix}$ |
$\begin{pmatrix}1 & 3 & 6 & 10 & 7 & 13 & 21 \\ 0 & 1 & 3 & 6 & 3 & 7 & 13 \\ 0 & 0 & 1 & 3 & 1 & 3 & 7 \\ T^2 & 0 & 0 & 1 & T^2 & 1 & 3 \\ 0 & 0 & 0 & 0 & 1 & 3 & 6 \\ 0 & 0 & 0 & 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$ | ||
| $(3,2)$ | 2 | 6 |
5: $\begin{pmatrix}0 & 0 & 0 & 1 & 1 \\ 1 & 1 & 1 & 0 & -1\end{pmatrix}$ 6: $\begin{pmatrix}1 & 1 & 0 & 1 & 0 & 0 \\ 2 & 1 & 2 & 0 & 1 & 0\end{pmatrix}$ |
$\begin{pmatrix}1 & 3 & 4 & 6 & 9 & 16 \\ 0 & 1 & 1 & 3 & 4 & 9 \\ 0 & 0 & 1 & 0 & 3 & 6 \\ 0 & 0 & 0 & 1 & 1 & 4 \\ 0 & 0 & 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$ | ||
| $(3,3)$ | 2 | 6 |
5: $\begin{pmatrix}1 & 1 & 1 & 0 & 0 \\ 0 & -1 & 0 & 1 & 1\end{pmatrix}$ 6: $\begin{pmatrix}2 & 2 & 1 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 & 1 & 0\end{pmatrix}$ |
$\begin{pmatrix}1 & 2 & 4 & 7 & 10 & 16 \\ 0 & 1 & 1 & 4 & 4 & 10 \\ 0 & 0 & 1 & 2 & 4 & 7 \\ 0 & 0 & 0 & 1 & 1 & 4 \\ 0 & 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$ | ||
| $(3,4)$ | 2 | 6 |
5: $\begin{pmatrix}1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1\end{pmatrix}$ 6: $\begin{pmatrix}2 & 2 & 1 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 & 1 & 0\end{pmatrix}$ |
$\begin{pmatrix}1 & 2 & 3 & 6 & 6 & 12 \\ 0 & 1 & 0 & 3 & 0 & 6 \\ 0 & 0 & 1 & 2 & 3 & 6 \\ 0 & 0 & 0 & 1 & 0 & 3 \\ 0 & 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$ | ||
| $(3,5)$ | 3 | 8 |
6: $\begin{pmatrix}1 & 1 & 0 & 0 & -1 & 0 \\ 0 & 0 & 1 & 1 & -1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1\end{pmatrix}$ 8: $\begin{pmatrix}1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0\end{pmatrix}$ |
$\begin{pmatrix}1 & 2 & 2 & 4 & 5 & 8 & 8 & 13 \\ 0 & 1 & 0 & 2 & 2 & 5 & 3 & 8 \\ 0 & 0 & 1 & 2 & 2 & 3 & 5 & 8 \\ 0 & 0 & 0 & 1 & 1 & 2 & 2 & 5 \\ 0 & 0 & 0 & 0 & 1 & 2 & 2 & 4 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$ | ||
| $(3,6)$ | 3 | 8 |
6: $\begin{pmatrix}1 & 1 & 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 1 & -1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1\end{pmatrix}$ 8: $\begin{pmatrix}1 & 0 & 1 & 1 & 0 & 0 & 1 & 0 \\ 1 & 1 & 0 & 1 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 & 1 & 0 & 0 & 0\end{pmatrix}$ |
$\begin{pmatrix}1 & 2 & 3 & 4 & 5 & 7 & 9 & 14 \\ 0 & 1 & 1 & 1 & 3 & 4 & 4 & 9 \\ 0 & 0 & 1 & 1 & 2 & 2 & 4 & 7 \\ 0 & 0 & 0 & 1 & 0 & 2 & 3 & 5 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & 4 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 3 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$ | ||
| $(3,7)$ | 3 | 8 |
6: $\begin{pmatrix}1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1\end{pmatrix}$ 8: $\begin{pmatrix}1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0\end{pmatrix}$ |
$\begin{pmatrix}1 & 2 & 2 & 4 & 2 & 4 & 4 & 8 \\ 0 & 1 & 0 & 2 & 0 & 2 & 0 & 4 \\ 0 & 0 & 1 & 2 & 0 & 0 & 2 & 4 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 & 1 & 2 & 2 & 4 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$ | ||
| $(3,8)$ | 3 | 8 |
6: $\begin{pmatrix}1 & -1 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1\end{pmatrix}$ 8: $\begin{pmatrix}1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0\end{pmatrix}$ |
$\begin{pmatrix}1 & 2 & 2 & 4 & 3 & 6 & 5 & 10 \\ 0 & 1 & 0 & 2 & 0 & 3 & 0 & 5 \\ 0 & 0 & 1 & 2 & 1 & 2 & 3 & 6 \\ 0 & 0 & 0 & 1 & 0 & 1 & 0 & 3 \\ 0 & 0 & 0 & 0 & 1 & 2 & 2 & 4 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$ | ||
| $(3,9)$ | 3 | 8 |
6: $\begin{pmatrix}1 & 1 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 & 0 & -1\end{pmatrix}$ 8: $\begin{pmatrix}1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0\end{pmatrix}$ |
$\begin{pmatrix}1 & 2 & 2 & 4 & 4 & 7 & 7 & 12 \\ 0 & 1 & 0 & 2 & 1 & 4 & 2 & 7 \\ 0 & 0 & 1 & 2 & 1 & 2 & 4 & 7 \\ 0 & 0 & 0 & 1 & 0 & 1 & 1 & 4 \\ 0 & 0 & 0 & 0 & 1 & 2 & 2 & 4 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$ | ||
| $(3,10)$ | 3 | 8 |
6: $\begin{pmatrix}1 & 1 & -2 & 1 & 0 & 0 \\ 0 & 0 & 1 & -1 & 1 & 0 \\ 0 & 0 & 0 & 1 & -1 & 1\end{pmatrix}$ 9: $\begin{pmatrix}2 & 1 & 1 & 0 & 0 & 2 & -1 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0\end{pmatrix}$ |
$\begin{pmatrix}1 & 1 & 3 & 3 & 6 & 4 & 6 & 9 & 16 \\ 0 & 1 & 1 & 3 & 3 & 3 & 6 & 7 & 13 \\ 0 & 0 & 1 & 1 & 3 & 1 & 3 & 4 & 9 \\ 0 & 0 & 0 & 1 & 1 & 1 & 3 & 3 & 7 \\ 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 4 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 3 & 6 \\ T^2 & 0 & 0 & 0 & 0 & T^2 & 1 & 1 & 3 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$ | ||
| $(3,11)$ | 3 | 8 |
6: $\begin{pmatrix}-1 & 1 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 & -1 & 1 \\ 0 & 0 & 1 & 1 & 1 & -1\end{pmatrix}$ 8: $\begin{pmatrix}1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 \\ 2 & 1 & 2 & 1 & 0 & 1 & 0 & 0\end{pmatrix}$ |
$\begin{pmatrix}1 & 1 & 2 & 3 & 3 & 6 & 6 & 12 \\ 0 & 1 & 1 & 1 & 3 & 4 & 3 & 9 \\ 0 & 0 & 1 & 0 & 0 & 3 & 0 & 6 \\ 0 & 0 & 0 & 1 & 1 & 2 & 3 & 6 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & 4 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 3 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$ | ||
| $(3,12)$ | 4 | 10 |
7: $\begin{pmatrix}1 & -1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & -1 & 1 \\ 0 & 0 & 1 & 1 & 0 & -1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 1 & -1\end{pmatrix}$ 10: $\begin{pmatrix}1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0\end{pmatrix}$ |
$\begin{pmatrix}1 & 2 & 1 & 2 & 2 & 4 & 3 & 5 & 6 & 10 \\ 0 & 1 & 0 & 1 & 0 & 2 & 1 & 3 & 2 & 6 \\ 0 & 0 & 1 & 2 & 1 & 2 & 2 & 3 & 5 & 8 \\ 0 & 0 & 0 & 1 & 0 & 1 & 1 & 2 & 2 & 5 \\ 0 & 0 & 0 & 0 & 1 & 2 & 0 & 0 & 3 & 5 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 & 2 & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$ | ||
| $(3,13)$ | 4 | 10 |
7: $\begin{pmatrix}0 & 0 & 0 & 0 & 1 & -1 & 1 \\ 0 & 0 & 1 & 1 & 0 & 0 & -1 \\ 1 & -1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 & 1 & -1\end{pmatrix}$ 10: $\begin{pmatrix}1 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 0 \\ 1 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0\end{pmatrix}$ |
$\begin{pmatrix}1 & 2 & 2 & 3 & 3 & 3 & 5 & 5 & 8 & 12 \\ 0 & 1 & 1 & 2 & 1 & 1 & 3 & 3 & 4 & 8 \\ 0 & 0 & 1 & 2 & 1 & 1 & 2 & 2 & 4 & 7 \\ 0 & 0 & 0 & 1 & 0 & 0 & 1 & 1 & 1 & 4 \\ 0 & 0 & 0 & 0 & 1 & 0 & 2 & 0 & 3 & 5 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 2 & 3 & 5 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 3 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$ | ||
| $(3,14)$ | 4 | 10 |
7: $\begin{pmatrix}1 & -1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & -1 & 1 \\ 0 & 0 & 1 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 1 & -1\end{pmatrix}$ 10: $\begin{pmatrix}1 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\end{pmatrix}$ |
$\begin{pmatrix}1 & 1 & 2 & 2 & 2 & 4 & 2 & 4 & 4 & 8 \\ 0 & 1 & 0 & 2 & 1 & 2 & 1 & 2 & 3 & 6 \\ 0 & 0 & 1 & 1 & 0 & 2 & 0 & 2 & 0 & 4 \\ 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 3 \\ 0 & 0 & 0 & 0 & 1 & 2 & 0 & 0 & 2 & 4 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 & 2 & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$ | ||
| $(3,15)$ | 4 | 10 |
7: $\begin{pmatrix}1 & -1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & -1 & 1 \\ 0 & 0 & 1 & 1 & -1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 1 & -1\end{pmatrix}$ 10: $\begin{pmatrix}1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0\end{pmatrix}$ |
$\begin{pmatrix}1 & 2 & 1 & 2 & 3 & 5 & 2 & 4 & 6 & 10 \\ 0 & 1 & 0 & 1 & 1 & 3 & 0 & 2 & 2 & 6 \\ 0 & 0 & 1 & 2 & 1 & 2 & 1 & 2 & 4 & 7 \\ 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 1 & 4 \\ 0 & 0 & 0 & 0 & 1 & 2 & 0 & 0 & 2 & 4 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 & 3 & 5 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 3 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$ | ||
| $(3,16)$ | 5 | 12 |
8: $\begin{pmatrix}1 & -1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & -1 & 1 & 0 & 0 \\ 0 & 0 & 1 & -1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 1 & -1 & 1 & 0 & 0 & 0 & 0\end{pmatrix}$ 12: $\begin{pmatrix}0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \\ 1 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0\end{pmatrix}$ |
$\begin{pmatrix}1 & 0 & 2 & 0 & 1 & 2 & 1 & 2 & 1 & 2 & 3 & 6 \\ 0 & 1 & 0 & 2 & 1 & 2 & 1 & 2 & 1 & 2 & 3 & 6 \\ 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 3 \\ 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 3 \\ 0 & 0 & 0 & 0 & 1 & 2 & 0 & 0 & 0 & 0 & 2 & 4 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 & 0 & 0 & 2 & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 & 2 & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$ | ||
| $(3,17)$ | 5 | 12 |
8: $\begin{pmatrix}1 & -1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & -1 & 1 & 0 & 0 \\ 0 & 0 & 1 & -1 & 1 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 1 & -1 & 1 & 0 & 0 & 0 & 0\end{pmatrix}$ 12: $\begin{pmatrix}0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \\ 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0\end{pmatrix}$ |
$\begin{pmatrix}1 & 2 & 0 & 0 & 1 & 2 & 1 & 2 & 1 & 2 & 4 & 7 \\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 1 & 4 \\ 0 & 0 & 1 & 2 & 1 & 2 & 2 & 3 & 1 & 2 & 5 & 8 \\ 0 & 0 & 0 & 1 & 0 & 1 & 1 & 2 & 0 & 1 & 2 & 5 \\ 0 & 0 & 0 & 0 & 1 & 2 & 0 & 0 & 0 & 0 & 3 & 5 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 & 0 & 0 & 2 & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 & 3 & 5 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 3 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$ |
Smooth Fano Toric 4,5,6-folds
The are respectively 124, 866, and 7622 smooth fano toric varieties in dimensions 4, 5, and 6, thus they are further broken up by Picard rank.
Note: this data is computed, but only available online in dim $\leq4$ or Picard rank $\leq3$, so gray links don’t work yet.
| $\dim$ | $\operatorname{rk} \mathrm{Pic}$ | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| $2$ (5) | ≤2 (1+2) 3 (1) 4 (1) | |||||||||||
| $3$ (18) | ≤2 (1+4) 3 (7) 4 (4) 5 (2) | |||||||||||
| $4$ (124) | ≤2 (1+9) 3 (28) 4 (47) 5 (27) 6 (10) 7 (1) 8 (1) | |||||||||||
| $5$ (866) | ≤2 (1+15) 3 (91) 4 (268) 5 (312) 6 (137) 7 (35) 8 (5) 9 (2) | |||||||||||
| $6$ (7622) | ≤2 (1+26) 3 (257) 4 (1318) 5 (2807) 6 (2204) 7 (771) 8 (186) 9 (39) 10 (11) 11 (1) 12 (1) | |||||||||||