Hostname: page-component-7c8c6479df-p566r Total loading time: 0 Render date: 2024-03-28T11:34:07.130Z Has data issue: false hasContentIssue false

Exceptional sequences of invertible sheaves on rational surfaces

Published online by Cambridge University Press:  18 March 2011

Lutz Hille
Affiliation:
Mathematisches Institut, Fachbereich Mathematik und Informatik der Universität Münster, Einsteinstraße 62, 48149 Münster, Germany (email: lhill_01@uni-muenster.de)
Markus Perling
Affiliation:
Fakultät für Mathematik, Ruhr-Universität Bochum, Universitätsstraße 150, 44780 Bochum, Germany (email: Markus.Perling@rub.de)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this article we consider exceptional sequences of invertible sheaves on smooth complete rational surfaces. We show that to every such sequence one can associate a smooth complete toric surface in a canonical way. We use this structural result to prove various theorems on exceptional and strongly exceptional sequences of invertible sheaves on rational surfaces. We construct full strongly exceptional sequences for a large class of rational surfaces. For the case of toric surfaces we give a complete classification of full strongly exceptional sequences of invertible sheaves.

Type
Research Article
Copyright
Copyright © Foundation Compositio Mathematica 2011

References

[AH99]Altmann, K. and Hille, L., Strong exceptional sequences provided by quivers, Algebr. Represent. Theory 2 (1999), 117.Google Scholar
[Asp08]Aspinwall, P. S, D-Branes on toric Calabi–Yau varieties (2008), arXiv:math/0806.2612.Google Scholar
[Bae88]Baer, D., Tilting sheaves in representation theory of algebras, Manuscripta Math. 60 (1988), 323347.Google Scholar
[Be{ĭ}78]Beĭlinson, A. A., Coherent sheaves on ℙn and problems of linear algebra, Funct. Anal. Appl. 12 (1978), 214216.Google Scholar
[BP06]Bergman, A. and Proudfoot, N., Moduli spaces for D-branes at the tip of a cone, JHEP, 0603:73 (2006).Google Scholar
[BP08]Bergman, A. and Proudfoot, N., Moduli spaces for Bondal quivers, Pacific J. Math. 237 (2008), 201221.Google Scholar
[Bon90]Bondal, A. I., Representation of associative algebras and coherent sheaves, Math. USSR Izv. 34 (1990), 2342.Google Scholar
[BP94]Bondal, A. I. and Polishchuk, A. E., Homological properties of associative algebras: the method of helices, Russ. Acad. Sci. Izv. Math. 42 (1994), 219260.Google Scholar
[BH09]Borisov, L. and Hua, Z., On the conjecture of King for smooth toric Deligne–Mumford stacks, Adv. Math. 221 (2009), 277301.Google Scholar
[Bri05]Bridgeland, T., t-structures on some local Calabi–Yau varieties, J. Algebra 289 (2005), 453483.Google Scholar
[Bri06]Bridgeland, T., Derived categories of coherent sheaves, in Proceedings of the international congress of mathematicians, Madrid, Spain, 2006, Madrid, 22–30 August 2006, vol. II (European Mathematical Society, Zürich, 2006), 563582.Google Scholar
[BS10]Bridgeland, T. and Stern, D., Helices on del Pezzo surfaces and tilting Calabi–Yau algebras, Adv. Math. 224 (2010), 16721716.Google Scholar
[Bro06]Broomhead, N., Cohomology of line bundles on a toric variety and constructible sheaves on its polytope (2006), arXiv:math/0611469.Google Scholar
[CM04]Costa, L. and Miró-Roig, R. M., Tilting sheaves on toric varieties, Math. Z. 248 (2004), 849865.Google Scholar
[CM05]Costa, L. and Miró-Roig, R. M., Derived categories of projective bundles, Proc. Amer. Math. Soc. 133 (2005), 25332537.Google Scholar
[CS08]Craw, A. and Smith, G. G., Projective toric varieties as fine moduli spaces of quiver representations, Amer. J. Math. 130 (2008), 15091534.Google Scholar
[Dem80]Demazure, M., Surfaces de Del Pezzo. I. II. III. IV. V, in Semin. sur les singularites des surfaces, Cent. Math. Ec. Polytech., Palaiseau 1976–77, Lecture Notes in Mathematics, vol. 777 (Springer, Berlin, 1980), 2169.Google Scholar
[Dou01]Douglas, M. R., D-branes, categories and N=1 supersymmetry, J. Math. Phys. 42 (2001), 28182843.Google Scholar
[DL85]Drezet, J. M. and Le Potier, J., Fibrés stables et fibrés exceptionnels sur ℙ2, Ann. Sci. École. Norm. Sup. (4) 18 (1985), 193244.Google Scholar
[Ful93]Fulton, W., Introduction to toric varieties (Princeton University Press, Princeton, NJ, 1993).Google Scholar
[GKZ94]Gelfand, I. M., Kapranov, M. M. and Zelevinsky, A. V., Discriminants, resultants and multidimensional determinants, Mathematics: Theory & Applications (Birkhäuser, Boston, MA, 1994).Google Scholar
[Gor89]Gorodentsev, A. L., Exceptional bundles on surfaces with a moving anticanonical class, Math. USSR Izv. 33 (1989), 6783.Google Scholar
[HHV06]Hanany, A., Herzog, C. P. and Vegh, B., Brane tilings and exceptional collections, JHEP, 0607:1 (2006).Google Scholar
[Hap88]Happel, D., Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Note Series, vol. 119 (Cambridge University Press, Cambridge, 1988).Google Scholar
[Har77]Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics, vol. 52 (Springer, Berlin, 1977).Google Scholar
[HK06]Herzog, C. P. and Karp, R. L., Exceptional collections and D-branes probing toric singularities, JHEP, 0602:61 (2006).Google Scholar
[Hil04]Hille, L., Exceptional sequences of line bundles on toric varieties, in: Mathematisches Institut Universität Göttingen, Seminars WS03-04, ed. Y. Tschinkel (2004), 175–190.Google Scholar
[HP06]Hille, L. and Perling, M., A counterexample to King’s conjecture, Compositio Math. 142 (2006), 15071521.Google Scholar
[Hv07]Hille, L. and van den Bergh, M., Fourier–Mukai transforms, in Handbook of tilting theory, London Mathematical Society Lecture Note Series, vol. 332 (Cambridge University Press, Cambridge, 2007), 147177.Google Scholar
[Huy06]Huybrechts, D., Fourier–Mukai transforms in algebraic geometry, Oxford Mathematical Monographs (Oxford University Press, Oxford, 2006).Google Scholar
[Kap86]Kapranov, M. M., Derived category of coherent bundles on a quadric, Funct. Anal. Appl. 20 (1986), 141142.Google Scholar
[Kap88]Kapranov, M. M., On the derived categories of coherent sheaves on some homogeneous spaces, Invent. Math. 92 (1988), 479508.Google Scholar
[KN98]Karpov, B. V. and Nogin, D. Yu., Three-block exceptional sets on del Pezzo surfaces, Izv. Math. 62 (1998), 429463.Google Scholar
[Kaw06]Kawamata, Y., Derived categories of toric varieties, Michigan Math. J. 54 (2006), 517535.Google Scholar
[Kaw09]Kawamata, Y., Derived categories and birational geometry, in Algebraic geometry, Seattle 2005, Proceedings of the 2005 Summer Research Institute, Seattle, WA, 25 July–12 August 2005, Proceedings of Symposia in Pure Mathematics, vol. 80 (American Mathematical Society, Providence, RI, 2009), 655665.Google Scholar
[Kin97]King, A., Tilting bundles on some rational surfaces, Unpublished manuscript, 1997.Google Scholar
[Kon95]Kontsevich, M., Homological algebra of mirror symmetry, in Proceedings of the International Congress of Mathematicians (Zürich 1994), Vols. 1 and 2 (Birkhäuser, Basel, 1995), 120139.Google Scholar
[Kul97]Kuleshov, S. A., Exceptional and rigid sheaves on surfaces with anticanonical class without base components, J. Math. Sci. (N.Y.) 86 (1997), 29513003.Google Scholar
[KO95]Kuleshov, S. A. and Orlov, D. O., Exceptional sheaves on del Pezzo surfaces, Russ. Acad. Sci. Izv. Math. 44 (1995), 337375.Google Scholar
[Kuz08]Kuznetsov, A., Exceptional collections for Grassmannians of isotropic lines, Proc. Lond. Math. Soc. (3) 97 (2008), 155182.Google Scholar
[Man86]Manin, Y. I., Cubic forms, in Algebra, geometry, arithmetic, North-Holland Mathematical Library, vol. 4, second edition (North-Holland, Amsterdam, 1986).Google Scholar
[MO78]Miyake, K. and Oda, T., Lectures on torus embeddings and applications (Springer, Berlin, 1978).Google Scholar
[Nog94]Nogin, D. Y., Helices on some Fano threefolds: constructivity of semiorthogonal bases of K 0, Ann. Sci. École Norm. Sup. (4) 27 (1994), 129172.Google Scholar
[Oda88]Oda, T., Convex bodies and algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 15 (Springer, Berlin, 1988).Google Scholar
[OP91]Oda, T. and Park, H. S., Linear Gale transforms and Gelfand–Kapranov–Zelevinskij decomposition, Tohoku Math. J. (2) 43 (1991), 375399.Google Scholar
[Orl93]Orlov, D. O., Projective bundles, monoidal transformations, and derived categories of coherent sheaves, Russ. Acad. Sci. Izv. Math. 41 (1993), 133141.Google Scholar
[Per07]Perling, M., Divisorial cohomology vanishing on toric varieties, submitted (2007), arXiv:0711.4836.Google Scholar
[Rud90]Rudakov, A. N., Helices and vector bundles: seminaire Rudakov, London Mathematical Society Lecture Note Series, vol. 148 (Cambridge University Press, Cambridge, 1990).Google Scholar
[Sam05]Samokhin, A., On the derived category of coherent sheaves on a 5-dimensional Fano variety, C. R. Math. Acad. Sci. Paris 340 (2005), 889893.Google Scholar
[Sam07]Samokhin, A., Some remarks on the derived categories of coherent sheaves on homogeneous space, J. Lond. Math. Soc. (2) 76 (2007), 122134.Google Scholar
[van04a]van den Bergh, M., Non-commutative crepant resolutions, in The Legacy of Niels Henrik Abel: The Abel Bicentennial, Oslo, 2002, Oslo, 3–8 June 2002 (Springer, Berlin, 2004).Google Scholar
[van04b]van den Bergh, M., Three-dimensional flops and noncommutative rings, Duke Math. J. 122 (2004), 423455.Google Scholar