Hille‒Perling’s Counterexample to King’s Conjecture
We begin with Hirzebruch surface of type 2 and consider 3 consecutive blow-ups. Hille and Perling showed in [arxiv:math/0602258] that this toric variety does not have a strong exceptional collection of line bundles.
| ID | $\operatorname{rk} \mathrm{Pic}$ | $\operatorname{rk} K_0$ | Fan | Chamers | Cox degrees and $\Theta$-collection | Ext table |
|---|---|---|---|---|---|---|
| $0$ | 2 | 4 |
4: $\begin{pmatrix}1 & -2 & 1 & 0 \\ 0 & 1 & 0 & 1\end{pmatrix}$ 5: $\begin{pmatrix}1 & 0 & -1 & 1 & 0 \\ 1 & 1 & 1 & 0 & 0\end{pmatrix}$ |
$\begin{pmatrix}1 & 2 & 3 & 4 & 6 \\ 0 & 1 & 2 & 2 & 4 \\ T & 0 & 1 & 1+T & 2 \\ 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 1\end{pmatrix}$ | ||
| $1$ | 3 | 5 |
5: $\begin{pmatrix}1 & -2 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 \\ -1 & 1 & 0 & 0 & 1\end{pmatrix}$ 6: $\begin{pmatrix}1 & 0 & 0 & -1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 0\end{pmatrix}$ |
$\begin{pmatrix}1 & 1 & 2 & 2 & 3 & 5 \\ 0 & 1 & 1 & 2 & 2 & 4 \\ 0 & 0 & 1 & 1 & 1 & 3 \\ T & 0 & 0 & 1 & 1+T & 2 \\ 0 & 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$ | ||
| $2$ | 4 | 6 |
6: $\begin{pmatrix}1 & -2 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 0 \\ -1 & 1 & 0 & 0 & 1 & 0 \\ -2 & 1 & 0 & 0 & 0 & 1\end{pmatrix}$ 11: $\begin{pmatrix}1 & 1 & 0 & 0 & 0 & 0 & -1 & -1 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 \\ 0 & -1 & 1 & 0 & 0 & -1 & 1 & 0 & 0 & -1 & 0\end{pmatrix}$ |
$\begin{pmatrix}1 & 1 & 1 & 1 & 2 & 2 & 2 & 2 & 3 & 3 & 5 \\ 0 & 1 & 1+T & 1 & 1 & 2 & 2+T & 2 & 2 & 3 & 4 \\ 0 & 0 & 1 & 1 & 1 & 1 & 2 & 2 & 2 & 2 & 4 \\ T & 0 & 0 & 1 & 1+T & 1 & 1 & 2 & 1 & 2 & 3 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 3 \\ T & 0 & T & 0 & 0 & 1 & 1+T & 1 & 1+T & 1 & 2 \\ T & T & 0 & 0 & 0 & 0 & 1 & 1 & 1+T & 1+T & 2 \\ 2T & T & T & 0 & T & 0 & 0 & 1 & T & 1+T & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 2 \\ T & 0 & T & 0 & T & 0 & T & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$ | ||
| $3$ | 5 | 7 |
7: $\begin{pmatrix}1 & -2 & 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 0 & 0 \\ -1 & 1 & 0 & 0 & 1 & 0 & 0 \\ -2 & 1 & 0 & 0 & 0 & 1 & 0 \\ -3 & 1 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$ 21: $\begin{pmatrix}1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & 1 & -1 & 1 & -1 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & -1 & -1 & 1 & 1 & 0 & 0 & 0 & 0 & -1 & -1 & 1 & 1 & 0 & 0 & 0 & 0 & -1 & -1 & 0 \\ 0 & -1 & -1 & -2 & 1 & 0 & 0 & 0 & -1 & -1 & -1 & -2 & 1 & 0 & 0 & 0 & -1 & -1 & -1 & -2 & 0\end{pmatrix}$ |
Tip: click on an ID, then use j and l to go back and forth and compare the data.