FanoCats

(Toric) Fano Categories Database

All varieties in this class have the same secondary fan, but as the multiplicity of the Cox degrees increases, the $\Theta$-collection expands, and with it the Ext table exhibits more complex behavior. Michalek proved in [1009.0821] that when $n>20$, such varieties do not have strongly exceptional collections of line bundles.

ID $\operatorname{rk} \mathrm{Pic}$ $\operatorname{rk} K_0$ Fan Chambers Cox degrees and $\Theta$-collection Ext table
$2$ 3 5 5: $\begin{pmatrix}1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 & 1\end{pmatrix}$
5: $\begin{pmatrix}2 & 1 & 1 & 1 & 0 \\ 1 & 1 & 0 & 1 & 0 \\ 1 & 1 & 1 & 0 & 0\end{pmatrix}$		 $\begin{pmatrix}1 & 1 & 2 & 2 & 4 \\ 0 & 1 & 1 & 1 & 3 \\ 0 & 0 & 1 & 0 & 2 \\ 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 1\end{pmatrix}$
$3$ 3 8 6: $\begin{pmatrix}1 & 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$
9: $\begin{pmatrix}3 & 2 & 1 & 2 & 2 & 1 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 0\end{pmatrix}$		 $\begin{pmatrix}1 & 2 & 3+T & 3 & 3 & 5 & 5 & 8 & 12 \\ 0 & 1 & 2 & 1 & 1 & 3 & 3 & 4 & 8 \\ 0 & 0 & 1 & 0 & 0 & 1 & 1 & 1 & 4 \\ 0 & 0 & T & 1 & 0 & 0 & 2 & 3 & 5 \\ 0 & 0 & T & 0 & 1 & 2 & 0 & 3 & 5 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 3 \\ 0 & 0 & T & 0 & 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$
$4$ 3 11 7: $\begin{pmatrix}1 & 1 & 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$
13: $\begin{pmatrix}4 & 3 & 2 & 3 & 3 & 1 & 2 & 2 & 1 & 2 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0\end{pmatrix}$		 $\begin{pmatrix}1 & 3 & 6+T & 4 & 4 & 10+5T & 9 & 9 & 16+T & 13 & 16+T & 25 & 41+T \\ 0 & 1 & 3 & 1 & 1 & 6+T & 4 & 4 & 9 & 5 & 9 & 13 & 25 \\ 0 & 0 & 1 & 0 & 0 & 3 & 1 & 1 & 4 & 1 & 4 & 5 & 13 \\ 0 & 0 & T & 1 & 0 & 5T & 3 & 0 & T & 4 & 6+T & 9 & 16+T \\ 0 & 0 & T & 0 & 1 & 5T & 0 & 3 & 6+T & 4 & T & 9 & 16+T \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 1 & 5 \\ 0 & 0 & 0 & 0 & 0 & T & 1 & 0 & 0 & 1 & 3 & 4 & 9 \\ 0 & 0 & 0 & 0 & 0 & T & 0 & 1 & 3 & 1 & 0 & 4 & 9 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 4 \\ 0 & 0 & T & 0 & 0 & 5T & 0 & 0 & T & 1 & T & 3 & 6+T \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 4 \\ 0 & 0 & 0 & 0 & 0 & T & 0 & 0 & 0 & 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$
$5$ 3 14 8: $\begin{pmatrix}1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$
17: $\begin{pmatrix}5 & 4 & 3 & 4 & 4 & 2 & 3 & 3 & 2 & 1 & 3 & 2 & 1 & 2 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0\end{pmatrix}$		 $\begin{pmatrix}1 & 4 & 10+T & 5 & 5 & 20+6T & 14 & 14 & 30+T & 35+21T & 19 & 30+T & 55+6T & 44 & 55+6T & 85+T & 146+6T \\ 0 & 1 & 4 & 1 & 1 & 10+T & 5 & 5 & 14 & 20+6T & 6 & 14 & 30+T & 19 & 30+T & 44 & 85+T \\ 0 & 0 & 1 & 0 & 0 & 4 & 1 & 1 & 5 & 10+T & 1 & 5 & 14 & 6 & 14 & 19 & 44 \\ 0 & 0 & T & 1 & 0 & 6T & 4 & 0 & T & 21T & 5 & 10+T & 6T & 14 & 20+6T & 30+T & 55+6T \\ 0 & 0 & T & 0 & 1 & 6T & 0 & 4 & 10+T & 21T & 5 & T & 20+6T & 14 & 6T & 30+T & 55+6T \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 4 & 0 & 1 & 5 & 1 & 5 & 6 & 19 \\ 0 & 0 & 0 & 0 & 0 & T & 1 & 0 & 0 & 6T & 1 & 4 & T & 5 & 10+T & 14 & 30+T \\ 0 & 0 & 0 & 0 & 0 & T & 0 & 1 & 4 & 6T & 1 & 0 & 10+T & 5 & T & 14 & 30+T \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & T & 0 & 0 & 4 & 1 & 0 & 5 & 14 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 1 & 6 \\ 0 & 0 & T & 0 & 0 & 6T & 0 & 0 & T & 21T & 1 & T & 6T & 4 & 6T & 10+T & 20+6T \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & T & 0 & 1 & 0 & 1 & 4 & 5 & 14 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 5 \\ 0 & 0 & 0 & 0 & 0 & T & 0 & 0 & 0 & 6T & 0 & 0 & T & 1 & T & 4 & 10+T \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 5 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & T & 0 & 0 & 0 & 0 & 0 & 1 & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$
$6$ 3 17 9: $\begin{pmatrix}1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$
21: $\begin{pmatrix}6 & 5 & 4 & 5 & 5 & 3 & 4 & 4 & 3 & 2 & 4 & 3 & 2 & 1 & 3 & 2 & 1 & 2 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0\end{pmatrix}$
$7$ 3 20 10: $\begin{pmatrix}1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$
25: $\begin{pmatrix}7 & 6 & 5 & 6 & 6 & 4 & 5 & 5 & 4 & 3 & 5 & 4 & 3 & 2 & 4 & 3 & 2 & 1 & 3 & 2 & 1 & 2 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0\end{pmatrix}$
$8$ 3 23 11: $\begin{pmatrix}1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$
29: $\begin{pmatrix}8 & 7 & 6 & 7 & 7 & 5 & 6 & 6 & 5 & 4 & 6 & 5 & 4 & 3 & 5 & 4 & 3 & 2 & 4 & 3 & 2 & 1 & 3 & 2 & 1 & 2 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0\end{pmatrix}$
$9$ 3 26 12: $\begin{pmatrix}1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$
33: $\begin{pmatrix}9 & 8 & 7 & 8 & 8 & 6 & 7 & 7 & 6 & 5 & 7 & 6 & 5 & 4 & 6 & 5 & 4 & 3 & 5 & 4 & 3 & 2 & 4 & 3 & 2 & 1 & 3 & 2 & 1 & 2 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0\end{pmatrix}$
$10$ 3 29 13: $\begin{pmatrix}1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$
37: $\begin{pmatrix}10 & 9 & 8 & 9 & 9 & 7 & 8 & 8 & 7 & 6 & 8 & 7 & 6 & 5 & 7 & 6 & 5 & 4 & 6 & 5 & 4 & 3 & 5 & 4 & 3 & 2 & 4 & 3 & 2 & 1 & 3 & 2 & 1 & 2 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0\end{pmatrix}$
$11$ 3 32 14: $\begin{pmatrix}1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$
41: $\begin{pmatrix}11 & 10 & 9 & 10 & 10 & 8 & 9 & 9 & 8 & 7 & 9 & 8 & 7 & 6 & 8 & 7 & 6 & 5 & 7 & 6 & 5 & 4 & 6 & 5 & 4 & 3 & 5 & 4 & 3 & 2 & 4 & 3 & 2 & 1 & 3 & 2 & 1 & 2 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0\end{pmatrix}$

Tip: use j and l to go back and forth and compare the data.