Combinatorial aspects of exceptional sequences on (rational) surfaces



We investigate combinatorial aspects of exceptional sequences in the derived category of coherent sheaves on certain smooth and complete algebraic surfaces. We show that to any such sequence there is canonically associated a complete toric surface whose torus fixpoints are either smooth or cyclic T-singularities (in the sense of Wahl) of type \(\frac{1}{r^2}(1, kr - 1)\). We also show that any exceptional sequence can be transformed by mutation into an exceptional sequence which consists only of objects of rank one.

Mathematics Subject Classification

Primary 14F05 14J26 14M25 Secondary 14J29 32S25 


  1. 1.
    Aigner, M.: Markov’s Theorem and 100 Years of the Uniqueness Conjecture. Springer, Berlin (2013)CrossRefMATHGoogle Scholar
  2. 2.
    Böhning, C., Graf von Bothmer, H.-C., Katzarkov, L., Sosna, P.: Determinantal Barlow surfaces and phantom categories. J. Eur. Math. Soc. 17(7), 1569–1592 (2012)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bondal, A.I.: Representation of associative algebras and coherent sheaves. Math. USSR Izv. 34(1), 23–42 (1990)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Cox, D.A., Little, J.B., Schenck, H.K.: Toric Varieties, Volume 124 of Graduate Studies in Mathematics. American Mathematical Society, Providence (2011)Google Scholar
  5. 5.
    Drezet, J .M., Le Potier, J.: Fibrés stables et fibrés exceptionnels sur \(\mathbb{P}_2\). Ann. scient. Éc. Norm. Sup. 4e série 18, 193–244 (1985)MATHGoogle Scholar
  6. 6.
    Hacking, P.: Exceptional bundles associated to degenerations of surfaces. Duke Math. J. 162(6), 1171–1202 (2013)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Hacking, P., Prokhorov, Y.: Smoothable del Pezzo surfaces with quotient singularities. Compos. Math. 146(1), 169–192 (2010)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Hille, L., Perling, M.: Exceptional sequences of invertible sheaves on rational surfaces. Compos. Math 147(4), 1230–1280 (2011)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Kasprzyk, A., Nill, B., Prince, T.: Minimality and mutation-equivalence of polygons, preprint (2015). arxiv:1501.05335v1
  10. 10.
    Kollár, J., Shepherd-Barron, N.: Threefolds and deformations of surface singularities. Invent. Math. 91(2), 299–338 (1988)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Nogin, D.Y.: Spirals of period four and equations of Markov type. Math. USSR Izv. 37(1), 209–226 (1991)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Orlov, D.O.: Projective bundles, monoidal transformations, and derived categories of coherent sheaves. Russ. Acad. Sci. Ivz. Math. 41(1), 133–141 (1993)MathSciNetGoogle Scholar
  13. 13.
    Rudakov, A.N.: Markov numbers and exceptional bundles on \(\mathbb{P}^2\). Math. USSR Izv. 32(1), 99–112 (1989)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Rudakov, A.N.: Exceptional vector bundles on a quadric. Math. USSR Izv. 33(1), 115–138 (1990)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Rudakov, A.N.: Helices and Vector Bundles: Seminaire Rudakov. Cambridge University Press, Cambridge (1990)CrossRefMATHGoogle Scholar
  16. 16.
    Thomason, R.W.: The classification of triangulated subcategories. Compos. Math. 105(1), 1–27 (1997)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    van den Bergh, M.: Non-commutative crepant resolutions (with some corrections) (2002/2009). arXiv:math/0211064v2
  18. 18.
    Vial, C.: Exceptional collections and the Néron-Severi lattice for surfaces. Adv. Math. 305, 895–934 (2017)Google Scholar
  19. 19.
    Wahl, J.: Smoothings of normal surface singularities. Topology 20(3), 219–246 (1981)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität BielefeldBielefeldGermany

Personalised recommendations