A Counterexample to King's Conjecture
Abstract
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 F2 iteratively blown up three times, and we show by explicit computation of cohomology vanishing that there exist no strongly exceptional sequences of length 7.
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