$\mathrm{Bl}_2\PP^n$: the blow-up of $\PP^n$ at 2 points.
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 | Chamers | 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: $\begin{pmatrix}6 & 5 & 4 & 5 & 5 & 3 & 4 & 4 & 3 & 2 & 4 & 3 & 2 & 1 & 3 & 2 & 1 & 2 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0\end{pmatrix}$ |
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| $7$ | 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: $\begin{pmatrix}7 & 6 & 5 & 6 & 6 & 4 & 5 & 5 & 4 & 3 & 5 & 4 & 3 & 2 & 4 & 3 & 2 & 1 & 3 & 2 & 1 & 2 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0\end{pmatrix}$ |
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| $8$ | 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: $\begin{pmatrix}8 & 7 & 6 & 7 & 7 & 5 & 6 & 6 & 5 & 4 & 6 & 5 & 4 & 3 & 5 & 4 & 3 & 2 & 4 & 3 & 2 & 1 & 3 & 2 & 1 & 2 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0\end{pmatrix}$ |
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| $9$ | 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: $\begin{pmatrix}9 & 8 & 7 & 8 & 8 & 6 & 7 & 7 & 6 & 5 & 7 & 6 & 5 & 4 & 6 & 5 & 4 & 3 & 5 & 4 & 3 & 2 & 4 & 3 & 2 & 1 & 3 & 2 & 1 & 2 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0\end{pmatrix}$ |
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| $10$ | 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: $\begin{pmatrix}10 & 9 & 8 & 9 & 9 & 7 & 8 & 8 & 7 & 6 & 8 & 7 & 6 & 5 & 7 & 6 & 5 & 4 & 6 & 5 & 4 & 3 & 5 & 4 & 3 & 2 & 4 & 3 & 2 & 1 & 3 & 2 & 1 & 2 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0\end{pmatrix}$ |
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| $11$ | 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.