A Counterexample to King's Conjecture

Details A Counterexample to King's Conjecture A Counterexample to King's Conjecture

2006-02-13

arxiv.org/abs/math/0602258v2

Compos. Math. 142, No. 6, 1507-1521 (2006)

Mathematics

Algebraic Geometry

15 pages, 4 figures, requires packages ams*, enumerate, graphicx, citation corrected

Scientific paper

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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