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1 results

  • A counterexample to King's conjecture

  • Lutz Hille, Markus Perling
  • Journal:
    Compositio Mathematica / Volume 142 / Issue 6 / November 2006
    Published online by Cambridge University Press:
    24 November 2006, pp. 1507-1521
    Print publication:
    November 2006
    • Article
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  • 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 which consist of line bundles.