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A Counterexample to King's Conjecture

Lutz Hille·Markus Perling
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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 $\mathbb{F}_2$ iteratively blown up three times, and we show by explicit computation of cohomology vanishing that there exist no strongly exceptional sequences of length 7.

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Lutz Hille, Markus Perling. 2006-02-16. A Counterexample to King's Conjecture. https://arxiv.org/abs/math/0602258

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