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Izv. RAN. Ser. Mat., 2006, Volume 70, Issue 3, Pages 23–128 (Mi izv556)  

This article is cited in 41 scientific papers (total in 41 papers)

Hyperplane sections and derived categories

A. G. Kuznetsov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: We give a generalization of the theorem of Bondal and Orlov about the derived categories of coherent sheaves on intersections of quadrics, revealing the relation of this theorem to projective duality. As an application, we describe the derived categories of coherent sheaves on Fano 3-folds of index 1 and degrees 12, 16 and 18.

DOI: https://doi.org/10.4213/im556

Full text: PDF file (1400 kB)
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English version:
Izvestiya: Mathematics, 2006, 70:3, 447–547

Bibliographic databases:

Document Type: Article
UDC: 512.73
MSC: 18E30, 14A22
Received: 26.04.2005
Revised: 12.09.2005

Citation: A. G. Kuznetsov, “Hyperplane sections and derived categories”, Izv. RAN. Ser. Mat., 70:3 (2006), 23–128; Izv. Math., 70:3 (2006), 447–547

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Samokhin A., “Some remarks on the derived categories of coherent sheaves on homogeneous spaces”, J. Lond. Math. Soc. (2), 76:1 (2007), 122–134  crossref  mathscinet  zmath  isi  scopus
    2. Kuznetsov A., “Derived categories of quadric fibrations and intersections of quadrics”, Adv. Math., 218:5 (2008), 1340–1369  crossref  mathscinet  zmath  isi  elib  scopus
    3. Sawon J., “Twisted Fourier-Mukai transforms for holomorphic symplectic four-folds”, Adv. Math., 218:3 (2008), 828–864  crossref  mathscinet  zmath  isi  elib  scopus
    4. Kuznetsov A., “Lefschetz decompositions and categorical resolutions of singularities”, Selecta Math. (N.S.), 13:4 (2008), 661–696  crossref  mathscinet  zmath  isi  elib  scopus
    5. A. G. Kuznetsov, “Derived Categories of Fano Threefolds”, Proc. Steklov Inst. Math., 264 (2009), 110–122  mathnet  crossref  mathscinet  isi  elib  elib
    6. Macri E., Stellari P., “Infinitesimal Derived Torelli Theorem for K3 Surfaces”, with an Appendix by Sukhendu Mehrotra, Internat. Math. Res. Notices, 2009, no. 17, 3190–3220  crossref  mathscinet  zmath  isi  elib  scopus
    7. Baranovsky V., Pecharich J., “On equivalences of derived and singular categories”, Cent. Eur. J. Math., 8:1 (2010), 1–14  crossref  mathscinet  zmath  adsnasa  isi  scopus
    8. Han F., “Geometry of the genus 9 Fano 4-folds”, Ann. Inst. Fourier (Grenoble), 60:4 (2010), 1401–1434  crossref  mathscinet  zmath  isi  scopus
    9. Polishchuk A., “$K$-theoretic exceptional collections at roots of unity”, J. K-Theory, 7:1 (2011), 169–201  crossref  mathscinet  zmath  isi  elib  scopus
    10. Huybrechts D., Macrì E., Stellari P., “Formal deformations and their categorical general fibre”, Comment. Math. Helv., 86:1 (2011), 41–71  crossref  mathscinet  zmath  isi  elib  scopus
    11. Kuznetsov A., “Base change for semiorthogonal decompositions”, Compos. Math., 147:3 (2011), 852–876  crossref  mathscinet  zmath  isi  elib  scopus
    12. MichałKapustka, Kristian Ranestad, “Vector bundles on Fano varieties of genus ten”, Math. Ann, 2012  crossref  mathscinet  isi  scopus
    13. Bernardara M. Bolognesi M., “Categorical Representability and Intermediate Jacobians of Fano Threefolds”, Derived Categories in Algebraic Geometry - Tokyo 2011, EMS Ser. Congr. Rep., ed. Kawamata Y., Eur. Math. Soc., 2012, 1–25  mathscinet  zmath  isi
    14. Asher Auel, Marcello Bernardara, Michele Bolognesi, “Fibrations in complete intersections of quadrics, Clifford algebras, derived categories, and rationality problems”, Journal de Mathématiques Pures et Appliquées, 2013  crossref  mathscinet  scopus
    15. Galkin S. Shinder E., “Exceptional Collections of Line Bundles on the Beauville Surface”, Adv. Math., 244 (2013), 1033–1050  crossref  mathscinet  zmath  isi  elib  scopus
    16. Krug A., “Extension Groups of Tautological Sheaves on Hilbert Schemes”, J. Algebr. Geom., 23:3 (2014), 571–598  crossref  mathscinet  zmath  isi  scopus
    17. Bayer A., Macri E., “Projectivity and Birational Geometry of Bridgeland Moduli Spaces”, J. Am. Math. Soc., 27:3 (2014), 707–752  crossref  mathscinet  zmath  isi  scopus
    18. Abuaf R., “Wonderful Resolutions and Categorical Crepant Resolutions of Singularities”, J. Reine Angew. Math., 708 (2015), 115–141  crossref  mathscinet  zmath  isi  elib  scopus
    19. Meachan C., “Derived Autoequivalences of Generalised Kummer Varieties”, Math. Res. Lett., 22:4 (2015), 1193–1221  crossref  mathscinet  zmath  isi  elib  scopus
    20. Kuznetsov A., “Küchle fivefolds of type c5”, Math. Z., 284:3-4 (2016), 1245–1278  crossref  mathscinet  zmath  isi  scopus
    21. Lombardi L., Tirabassi S., “Deformations of minimal cohomology classes on abelian varieties”, Commun. Contemp. Math., 18:4 (2016), 1550066  crossref  mathscinet  zmath  isi  elib  scopus
    22. Bernardara M. Bolognesi M. Faenzi D., “Homological projective duality for determinantal varieties”, Adv. Math., 296 (2016), 181–209  crossref  mathscinet  zmath  isi  elib  scopus
    23. Kuznetsov A. Polishchuk A., “Exceptional collections on isotropic Grassmannians”, J. Eur. Math. Soc., 18:3 (2016), 507–574  crossref  mathscinet  zmath  isi  elib  scopus
    24. Kuznetsov A., “Derived Categories View on Rationality Problems”, Rationality Problems in Algebraic Geometry, Lect. Notes Math., Lecture Notes in Mathematics, 2172, ed. Pardini R. Pirola G., Springer International Publishing Ag, 2016, 67–104  crossref  mathscinet  isi  scopus
    25. Kuznetsov A. Perry A., “Derived categories of cyclic covers and their branch divisors”, Sel. Math.-New Ser., 23:1 (2017), 389–423  crossref  mathscinet  zmath  isi  scopus
    26. Hirano Yu., “Derived Knorrer Periodicity and Orlov'S Theorem For Gauged Landau-Ginzburg Models”, Compos. Math., 153:5 (2017), 973–1007  crossref  mathscinet  zmath  isi  scopus
    27. Vial Ch., “Exceptional Collections, and the Neron-Severi Lattice For Surfaces”, Adv. Math., 305 (2017), 895–934  crossref  mathscinet  zmath  isi  scopus
    28. Auel A. Bernardara M., “Cycles, Derived Categories, and Rationality”, Surveys on Recent Developments in Algebraic Geometry, Proceedings of Symposia in Pure Mathematics, 95, ed. Coskun I. DeFernex T. Gibney A., Amer Mathematical Soc, 2017, 199–266  crossref  mathscinet  zmath  isi  scopus
    29. Fonarev A. Kuznetsov A., “Derived Categories of Curves as Components of Fano Manifolds”, J. Lond. Math. Soc.-Second Ser., 97:1 (2018), 24–46  crossref  mathscinet  zmath  isi  scopus
    30. Bondal A., Kapranov M., Schechtman V., “Perverse Schobers and Birational Geometry”, Sel. Math.-New Ser., 24:1, SI (2018), 85–143  crossref  mathscinet  zmath  isi  scopus
    31. Kuznetsov A.G., Prokhorov Yu.G., Shramov C.A., “Hilbert Schemes of Lines and Conics and Automorphism Groups of Fano Threefolds”, Jap. J. Math., 13:1 (2018), 109–185  crossref  mathscinet  zmath  isi  scopus
    32. Tabuada G., “A Note on the Schur-Finiteness of Linear Sections”, Math. Res. Lett., 25:1 (2018), 237–253  crossref  mathscinet  zmath  isi
    33. Hassett B., Lai K.-W., “Cremona Transformations and Derived Equivalences of K3 Surfaces”, Compos. Math., 154:7 (2018), 1508–1533  crossref  mathscinet  isi
    34. Kuznetsov A. Perry A., “Derived Categories of Gushel-Mukai Varieties”, Compos. Math., 154:7 (2018), 1362–1406  crossref  mathscinet  isi
    35. Ottem J.Ch., Rennemo J.V., “A Counterexample to the Birational Torelli Problem For Calabi-Yau Threefolds”, J. Lond. Math. Soc.-Second Ser., 97:3 (2018), 427–440  crossref  zmath  isi  scopus
    36. Krug A. Ploog D. Sosna P., “Derived Categories of Resolutions of Cyclic Quotient Singularities”, Q. J. Math., 69:2 (2018), 509–548  crossref  mathscinet  isi  scopus
    37. Kuznetsov A., “Derived Equivalence of Ito-Miura-Okawa-Ueda Calabi-Yau 3-Folds”, J. Math. Soc. Jpn., 70:3 (2018), 1007–1013  crossref  mathscinet  isi  scopus
    38. 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  zmath  isi  scopus
    39. A. G. Kuznetsov, “On linear sections of the spinor tenfold. I”, Izv. Math., 82:4 (2018), 694–751  mathnet  crossref  crossref  adsnasa  isi  elib
    40. Moschetti R., “The Derived Category of a Non Generic Cubic Fourfold Containing a Plane”, Math. Res. Lett., 25:5 (2018), 1525–1545  isi
    41. Manivel L., “Double Spinor Calabi-Yau Varieties”, Epijournal Geom. Algebr., 3 (2019), 2  isi
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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