  King's conjecture states that on every smooth complete toric variety $X$
there exists a strongly exceptional collection which generates the bounded
derived category of $X$ and which consists of line bundles. We give a
counterexample to this conjecture. This example is just the Hirzebruch surface
$\mathbb{F}_2$ iteratively blown up three times, and we show by explicit
computation of cohomology vanishing that there exist no strongly exceptional
sequences of length 7.
