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A counterexample to King's conjecture

Published online by Cambridge University Press:  24 November 2006

Lutz Hille  and
Markus Perling
Lutz Hille
Affiliation:
Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germanyhille@math.uni-bielefeld.de
Markus Perling
Affiliation:
Institut Fourier — UMR5582, 100 rue des Maths, BP 74, 38402 St. Martin d'Heres, Franceperling@mozart.ujf-grenoble.fr
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Abstract

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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.

Keywords

toric varietiesderived categoriesexceptional sequences14M2518E30 (primary)14J81 (secondary)
Type
Research Article
Information
Compositio Mathematica , Volume 142 , Issue 6 , November 2006 , pp. 1507 - 1521
Copyright
Foundation Compositio Mathematica 2006