Abstract
We investigate full strongly exceptional collections on smooth, complete toric varieties. We obtain explicit results for a large family of varieties with Picard number three, containing many of the families already known. We also describe the relations between the collections and the split of the push forward of the trivial line bundle by the toric Frobenius morphism.
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References
Batyrev V.V.: On the classification of smooth projective toric varieties. Tôhoku Math. J. 43, 569–585 (1991)
Bernardi, A., Tirabassi, S.: Derived Categories of Toric Fano 3-Folds via the Frobenius Morphism. http://arxiv.org/abs/1002.1666 (preprint)
Bondal, A.I.: Derived categories of toric varieties. Oberwolfach Report 5/2006, pp. 284–286
Bondal A.I.: Representation of associative algebras and coherent sheaves. Math. USSR Izvestiya 34(1), 23–42 (1990)
Borisov L., Horja P.: On the K-theory of smooth toric DM stacks, Snowbird lectures on string geometry. Contemp. Math. 401, 21–42 (2006)
Borisov L., Hua Z.: On the conjecture of King for smooth toric Deligne-Mumford stacks. Adv. Math. 221, 277–301 (2009)
Caldararu A.: Derived categories of sheaves: a skimming, Snowbird lectures in algebraic geometry. Contemp. Math. 388, 43–75 (2005)
Costa, L., Miró-Roig, R.M.: Derived category of toric varieties with small Picard number (preprint)
Costa L., Miró-Roig R.M.: Tilting sheaves on toric varieties. Math. Z. 248(4), 849–865 (2004)
Cox, D., Little, J., Schenck, H.: Toric Varieties. http://www.cs.amherst.edu/~dac/toric.html (preprint)
Dey A., Lasoń M., Michałek M.: Derived category of toric varieties with Picard number three. Le Matematiche 64(2), 99–116 (2009)
Efimov, A.: Maximal lengths of exceptional collections of line bundles. http://arxiv.org/PS_cache/arxiv/pdf/1010/1010.3755v2.pdf (preprint)
Fulton, W.: Introduction to Toric Varieties. In: Annals of Mathematics Studies. Princeton Univeristy Press, Princeton (1993)
Gelfand S., Manin Y.: Methods of homological algebra. Springer Monographs in Mathematics, 2nd edn. Springer, Berlin (2003)
Gorodentsev A.L., Rudakov A.N.: Exceptional vector bundles on projective spaces. Duke Math. J. 54(1), 115–130 (1987)
Gronbaum B.: Convex Polytopes. Wiley, London (1967)
Hille L., Perling M.: A counterexample to King’s conjecture. Compos. Math. 142, 1507–1521 (2006)
Hille, L., Perling, M.: Exceptional Sequences of Invertible Sheaves on Rational Surfaces. http://arxiv.org/PS_cache/arxiv/pdf/0810/0810.1936v2.pdf (preprint)
Huybrechts D.: Fourier–Mukai Transforms in Algebraic Geometry. Oxford Mathematical Monographs, Oxford (2006)
Kawamata Y.: Derived categories of toric varieties. Michigan Math. J. 54(3), 517–535 (2006)
King, A.: Titling bundles on some rational surfaces. http://www.maths.bath.ac.uk/~masadk/papers/ (preprint)
Kleinschmidt P.: A classification of toric varieties with few generators. Aequatinoes Math. 35, 254–266 (1988)
Michalek M.: Family of counterexamples to King’s conjecture. Comptes Rendus Math. 349(1-2), 67–69 (2011)
Mrozek M., Batko B.: Coreduction homology algorithm. Discrete Comput. Geom. 41, 96–118 (2009)
Oda T., Park H.S.: Linear Gale transforms and Gelfand–Kapranov–Zelevinskij decompositions. Tôhoku Math. J. 43, P375–399 (1991)
Perling, M.: Examples for exceptional sequences of invertible sheaves on rational surfaces. http://arxiv.org/PS_cache/arxiv/pdf/0904/0904.0529v2.pdf (preprint)
Thomsen J.F.: Frobenius direct images of line bundles on toric varieties. J. Algebra 226, 865–874 (2000)
Acknowledgements
We would like to thank very much Rosa Maria Miró-Roig and Laura Costa for introducing us to this subject. We are also grateful for many useful and interesting talks.
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M. Lasoń is supported by the grant of Polish MNiSzW N N201 413139.
M. Michałek is supported by the grant of Polish MNiSzW N N201 413539.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Lasoń, M., Michałek, M. On the full, strongly exceptional collections on toric varieties with Picard number three. Collect. Math. 62, 275–296 (2011). https://doi.org/10.1007/s13348-011-0044-x
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DOI: https://doi.org/10.1007/s13348-011-0044-x