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Izv. RAN. Ser. Mat., 1994, Volume 58, Issue 3, Pages 53–87 (Mi izv788)  

This article is cited in 38 scientific papers (total in 38 papers)

Exceptional sheaves on del Pezzo surfaces

S. A. Kuleshov, D. O. Orlov

Abstract: In the present paper exceptional sheaves on del Pezzo surfaces are studied, and a description of rigid bundles on these surfaces is given. It is proved that each exceptional sheaf can be included in a complete exceptional collection. Furthermore, it is shown that all such collections can be obtained from each other by means of a sequence of standard operations called transformations.

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English version:
Russian Academy of Sciences. Izvestiya Mathematics, 1995, 44:3, 479–513

Bibliographic databases:

UDC: 512.723
MSC: 14F05, 14J26
Received: 03.06.1993

Citation: S. A. Kuleshov, D. O. Orlov, “Exceptional sheaves on del Pezzo surfaces”, Izv. RAN. Ser. Mat., 58:3 (1994), 53–87; Russian Acad. Sci. Izv. Math., 44:3 (1995), 479–513

Citation in format AMSBIB
\by S.~A.~Kuleshov, D.~O.~Orlov
\paper Exceptional sheaves on del~Pezzo surfaces
\jour Izv. RAN. Ser. Mat.
\yr 1994
\vol 58
\issue 3
\pages 53--87
\jour Russian Acad. Sci. Izv. Math.
\yr 1995
\vol 44
\issue 3
\pages 479--513

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    This publication is cited in the following articles:
    1. Abe T., “Semistable Sheaves With Symmetric C1 on Del Pezzo Surfaces of Degree 5 and 6”, Eur. J. Math.  crossref  isi
    2. B. V. Karpov, “On the algebraic geometry of $S$-duality”, Math. Notes, 61:2 (1997), 133–145  mathnet  crossref  crossref  mathscinet  zmath  isi
    3. B. V. Karpov, D. Yu. Nogin, “Three-block exceptional collections over Del Pezzo surfaces”, Izv. Math., 62:3 (1998), 429–463  mathnet  crossref  crossref  mathscinet  zmath  isi
    4. B. V. Karpov, “On the structure of rigid semistable sheaves on algebraic surfaces”, Math. Notes, 64:5 (1998), 600–606  mathnet  crossref  crossref  mathscinet  zmath  isi
    5. B. V. Karpov, “$S$-duality testing and exceptional bundles”, Izv. Math., 63:1 (1999), 103–117  mathnet  crossref  crossref  mathscinet  zmath  isi
    6. D. Kussin, H. Meltzer, “The braid group action for exceptional curves”, Arch. Math. (Basel), 79:5 (2002), 335–344  crossref  mathscinet  zmath  isi
    7. C. P. Herzog, J. Walcher, “Dibaryons from exceptional collections”, J. High Energy Phys., 2003, no. 9, 060, 37 pp.  crossref  mathscinet  isi
    8. A. L. Gorodentsev, S. A. Kuleshov, “Helix theory”, Mosc. Math. J., 4:2 (2004), 377–440  mathnet  crossref  mathscinet  zmath
    9. P. S. Aspinwall, I. V. Melnikov, “D-branes on vanishing del Pezzo surfaces”, J. High Energy Phys., 2004, no. 12, 042, 30 pp.  crossref  mathscinet  isi
    10. C. P. Herzog, “Seiberg duality is an exceptional mutation”, J. High Energy Phys., 2004, no. 8, 064, 31 pp.  crossref  mathscinet  isi
    11. C. P. Herzog, “Exceptional collections and del Pezzo gauge theories”, J. High Energy Phys., 2004, no. 4, 069, 20 pp.  crossref  mathscinet  isi
    12. H. Meltzer, “Exceptional vector bundles, tilting sheaves and tilting complexes for weighted projective lines”, Mem. Am. Math. Soc., 171, no. 808, 2004, viii+139 pp.  mathscinet  isi
    13. D. Auroux, L. Katzarkov, D. Orlov, “Mirror symmetry for del Pezzo surfaces: vanishing cycles and coherent sheaves”, Invent. Math., 166:3 (2006), 537–582  crossref  mathscinet  zmath  adsnasa  isi  elib
    14. A. Elagin, “Exceptional Sets on Del Pezzo Surfaces with One Log-Terminal Singularity”, Math. Notes, 82:1 (2007), 33–46  mathnet  crossref  crossref  mathscinet  isi  elib
    15. E. Macrì, “Stability conditions on curves”, Math. Res. Lett., 14:4 (2007), 657–672  crossref  mathscinet  zmath  isi
    16. E. Segal, “The $A_\infty$ deformation theory of a point and the derived categories of local Calabi-Yaus”, J. Algebra, 320:8 (2008), 3232–3268  crossref  mathscinet  zmath  isi
    17. D. Kussin, Noncommutative curves of genus zero: related to finite dimensional algebras, Mem. Amer. Math. Soc., 201, no. 942, 2009, x+128 pp.  mathscinet
    18. T. Bridgeland, D. Stern, “Helices on del Pezzo surfaces and tilting Calabi-Yau algebras”, Adv. Math., 224:4 (2010), 1672–1716  crossref  mathscinet  zmath  isi
    19. Y. Indexed Kawamata, “Derived categories and minimal models”, Sugaku Expositions. Sugaku Expositions, 23:2 (2010), 235–259  mathscinet
    20. A. Polishchuk, “K-theoretic exceptional collections at roots of unity”, J. K-Theory, 7:1 (2011), 169–201  crossref  mathscinet  zmath  isi  elib
    21. L. Hille, M. Perling, “Exceptional sequences of invertible sheaves on rational surfaces”, Compos. Math., 147:4 (2011), 1230–1280  crossref  mathscinet  zmath  isi  elib
    22. U. V. Dubey, V. M. Mallick, “Reconstruction of a superscheme from its derived category”, J. Ramanujan Math. Soc., 27:4 (2012), 411–424  mathscinet  zmath  isi
    23. M. Ballard, D. Favero, “Hochschild dimensions of tilting objects”, Int. Math. Res. Not. IMRN, 2012, no. 11, 2607–2645  mathscinet  zmath
    24. S. A. Kuleshov, “Isometries of semi-orthogonal forms on a $\mathbb Z$-module of rank 3”, Izv. Math., 77:1 (2013), 44–86  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    25. V. Alexeev, D. Orlov, “Derived categories of Burniat surfaces and exceptional collections”, Math. Ann., 357:2 (2013), 743–759  crossref  mathscinet  zmath  isi  elib
    26. U. V. Dubey, V. M. Mallick, “Reconstruction of a superscheme from its derived category”, J. Ramanujan Math. Soc., 28:2 (2013), 179–193  mathscinet  zmath  isi  elib
    27. P. Hacking, “Exceptional bundles associated to degenerations of surfaces”, Duke Math. J., 162:6 (2013), 1171–1202  crossref  mathscinet  zmath  isi  elib
    28. Ch. Böhning, Y.-Ch. Graf von Bothmer, P. Sosna, “On the Jordan-Hölder property for geometric derived categories”, Adv. Math., 256 (2014), 479–492  crossref  mathscinet  zmath  isi  elib
    29. Coughlan S., “Enumerating Exceptional Collections of Line Bundles on Some Surfaces of General Type”, Doc. Math., 20 (2015), 1255–1291  isi
    30. Okawa Sh. Uehara H., “Exceptional Sheaves on the Hirzebruch Surface _{2}”, Int. Math. Res. Notices, 2015, no. 23, 12781–12803  crossref  mathscinet  zmath  isi  scopus
    31. Alexey Elagin, Valery Lunts, “On full exceptional collections of line bundles on del Pezzo surfaces”, Mosc. Math. J., 16:4 (2016), 691–709  mathnet  crossref  mathscinet
    32. Uehara H., “Autoequivalences of derived categories of elliptic surfaces with non-zero Kodaira dimension”, Algebraic Geom., 3:5 (2016), 543–577  crossref  mathscinet  isi
    33. Shiraishi Yu. Takahashi A. Wada K., “On Weyl groups and Artin groups associated to orbifold projective lines”, J. Algebra, 453 (2016), 249–290  crossref  mathscinet  zmath  isi  elib  scopus
    34. Chunyi Li, “Deformations of the Hilbert scheme of points on a del Pezzo surface”, Mosc. Math. J., 17:2 (2017), 291–321  mathnet  crossref  mathscinet
    35. Belmans P. Presotto D., “Construction of Non-Commutative Surfaces With Exceptional Collections of Length 4”, J. Lond. Math. Soc.-Second Ser., 98:1 (2018), 85–103  crossref  isi
    36. Auel A. Bernardara M., “Semiorthogonal Decompositions and Birational Geometry of Del Pezzo Surfaces Over Arbitrary Fields”, Proc. London Math. Soc., 117:1 (2018), 1–64  crossref  isi
    37. Uehara H., “A Trichotomy For the Autoequivalence Groups on Smooth Projective Surfaces”, Trans. Am. Math. Soc., 371:5 (2019), 3529–3547  crossref  mathscinet  zmath  isi  scopus
    38. Przyjalkowski V. Shramov C., “Hodge Level For Weighted Complete Intersections”, Collect. Math., 71:3 (2020), 549–574  crossref  isi
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