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Trudy Mat. Inst. Steklova, 2004, Volume 246, Pages 183–207 (Mi tm155)  

This article is cited in 40 scientific papers (total in 40 papers)

Derived Categories of Cubic and $V_{14}$

A. G. Kuznetsov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: It is shown that, after a certain natural flop, the projectivization of the exceptional rank-$2$ vector bundle on an arbitrary smooth $V_{14}$ Fano threefold turns into the projectivization of an instanton vector bundle on a smooth cubic threefold. Conversely, starting from a smooth cubic threefold with an instanton vector bundle of charge $2$ on it, we reconstruct a $V_{14}$ threefold. Based on the geometric properties of the above correspondence, we prove that the orthogonals to the exceptional pairs in the bounded derived categories of coherent sheaves on a smooth $V_{14}$ threefold and on the corresponding cubic threefold are equivalent as triangulated categories.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2004, 246, 171–194

Bibliographic databases:
UDC: 512.73
Received in February 2004

Citation: A. G. Kuznetsov, “Derived Categories of Cubic and $V_{14}$”, Algebraic geometry: Methods, relations, and applications, Collected papers. Dedicated to the memory of Andrei Nikolaevich Tyurin, corresponding member of the Russian Academy of Sciences, Trudy Mat. Inst. Steklova, 246, Nauka, MAIK Nauka/Inteperiodika, M., 2004, 183–207; Proc. Steklov Inst. Math., 246 (2004), 171–194

Citation in format AMSBIB
\by A.~G.~Kuznetsov
\paper Derived Categories of Cubic and $V_{14}$
\inbook Algebraic geometry: Methods, relations, and applications
\bookinfo Collected papers. Dedicated to the memory of Andrei Nikolaevich Tyurin, corresponding member of the Russian Academy of Sciences
\serial Trudy Mat. Inst. Steklova
\yr 2004
\vol 246
\pages 183--207
\publ Nauka, MAIK Nauka/Inteperiodika
\publaddr M.
\jour Proc. Steklov Inst. Math.
\yr 2004
\vol 246
\pages 171--194

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    This publication is cited in the following articles:
    1. A. G. Kuznetsov, “Derived Categories of Fano Threefolds $V_{12}$”, Math. Notes, 78:4 (2005), 537–550  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. Rouquier R., “Catégories dérivées et géométrie birationnelle (d'après Bondal, Orlov, Bridgeland, Kawamata et al.) [Derived categories and birational geometry (after Bondal, Orlov, Bridgeland, Kawamata et al.)]”, Séminaire Bourbaki, Vol. 2004/2005, Exp. No. 946, Astérisque, 307, 2006, 283–307  mathscinet  zmath  isi
    3. Kuznetsov A., “Derived categories of quadric fibrations and intersections of quadrics”, Adv. Math., 218:5 (2008), 1340–1369  crossref  mathscinet  zmath  isi  elib  scopus
    4. A. G. Kuznetsov, “Derived Categories of Fano Threefolds”, Proc. Steklov Inst. Math., 264 (2009), 110–122  mathnet  crossref  mathscinet  isi  elib  elib
    5. Kuznetsov A., Markushevich D., “Symplectic structures on moduli spaces of sheaves via the Atiyah class”, J. Geom. Phys., 59:7 (2009), 843–860  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    6. Flamini F., Sernesi E., “The Curve of Lines on a Prime Fano Three-Fold of Genus 8”, Internat J Math, 21:12 (2010), 1561–1584  crossref  mathscinet  zmath  isi  scopus
    7. Kuznetsov A., Manivel L., Markushevich D., “Abel-Jacobi Maps for Hypersurfaces and Noncommutative Calabi-Yau'S”, Commun Contemp Math, 12:3 (2010), 373–416  crossref  mathscinet  zmath  isi  elib  scopus
    8. Faenzi D., Fania M.L., “Skew-symmetric matrices and Palatini scrolls”, Math Ann, 347:4 (2010), 859–883  crossref  mathscinet  zmath  isi  elib  scopus
    9. Bernardara M., Macri E., Mehrotra S., Stellari P., “A categorical invariant for cubic threefolds”, Adv Math, 229:2 (2012), 770–803  crossref  mathscinet  zmath  isi  elib  scopus
    10. Kuznetsov A., “Instanton Bundles on Fano Threefolds”, Cent. Eur. J. Math., 10:4 (2012), 1198–1231  crossref  mathscinet  zmath  isi  elib  scopus
    11. Prokhorov Yu., “Simple Finite Subgroups of the Cremona Group of Rank 3”, J. Algebr. Geom., 21:3 (2012), 563–600  crossref  mathscinet  zmath  isi  elib  scopus
    12. 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
    13. Bernardara M., Bolognesi M., “Derived Categories and Rationality of Conic Bundles”, Compos. Math., 149:11 (2013), 1789–1817  crossref  mathscinet  zmath  isi  elib  scopus
    14. Auel A., Bernardara M., Bolognesi M., “Fibrations in Complete Intersections of Quadrics, Clifford Algebras, Derived Categories, and Rationality Problems”, J. Math. Pures Appl., 102:1 (2014), 249–291  crossref  mathscinet  zmath  isi  elib  scopus
    15. Iliev A., Manivel L., “on Cubic Hypersurfaces of Dimensions 7 and 8”, Proc. London Math. Soc., 108:2 (2014), 517–540  crossref  mathscinet  zmath  isi  elib  scopus
    16. Iliev A., Manivel L., “Fano Manifolds of Calabi-Yau Hodge Type”, J. Pure Appl. Algebr., 219:6 (2015), 2225–2244  crossref  mathscinet  zmath  isi  elib  scopus
    17. Ishikawa D., “Weak Fano Bundles on the Cubic Threefold”, Manuscr. Math., 149:1-2 (2016), 171–177  crossref  mathscinet  zmath  isi  scopus
    18. Marcello Bernardara, Gonçalo Tabuada, “From semi-orthogonal decompositions to polarized intermediate Jacobians via Jacobians of noncommutative motives”, Mosc. Math. J., 16:2 (2016), 205–235  mathnet  crossref  mathscinet
    19. Bernardara M., Bolognesi M., Faenzi D., “Homological projective duality for determinantal varieties”, Adv. Math., 296 (2016), 181–209  crossref  mathscinet  zmath  isi  elib  scopus
    20. Calabrese J.R., Thomas R.P., “Derived equivalent Calabi–Yau threefolds from cubic fourfolds”, Math. Ann., 365:1-2 (2016), 155–172  crossref  mathscinet  zmath  isi  elib  scopus
    21. Kuznetsov A., “Derived Categories View on Rationality Problems”, Rationality Problems in Algebraic Geometry, Lect. Notes Math., Lecture Notes in Mathematics, 2172, eds. Pardini R., Pirola G., Springer International Publishing Ag, 2016, 67–104  crossref  mathscinet  isi  scopus
    22. Huybrechts D., “The K3 category of a cubic fourfold”, Compos. Math., 153:3 (2017), 586–620  crossref  mathscinet  zmath  isi  scopus
    23. 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
    24. Shinder E., Soldatenkov A., “On the Geometry of the Lehn-Lehn-Sorger-Van Straten Eightfold”, Kyoto J. Math., 57:4 (2017), 789–806  crossref  mathscinet  zmath  isi  scopus
    25. Laza R., Sacca G., Voisin C., “A Hyper-Kahler Compactification of the Intermediate Jacobian Fibration Associated With a Cubic 4-Fold”, Acta Math., 218:1 (2017), 55–135  crossref  mathscinet  zmath  isi  scopus
    26. Addington N., Lehn M., “On the Symplectic Eightfold Associated to a Pfaffian Cubic Fourfold”, J. Reine Angew. Math., 731 (2017), 129–137  crossref  mathscinet  zmath  isi
    27. Ouchi G., “Lagrangian Embeddings of Cubic Fourfolds Containing a Plane”, Compos. Math., 153:5 (2017), 947–972  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. 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
    30. Kuznetsov A. Perry A., “Derived Categories of Gushel-Mukai Varieties”, Compos. Math., 154:7 (2018), 1362–1406  crossref  mathscinet  isi
    31. 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  mathscinet  zmath  isi  scopus
    32. Favero D., Kelly T.L., “Fractional Calabi-Yau Categories From Landau-Ginzburg Models”, Algebraic Geom., 5:5 (2018), 596–649  crossref  mathscinet  zmath  isi  scopus
    33. Kollar J., Laza R., Sacca G., Voisin C., “Remarks on Degenerations of Hyper-Kahler Manifolds”, Ann. Inst. Fourier, 68:7 (2018), 2837–2882  crossref  isi
    34. Huybrechts D., Rennemo J.V., “Hochschild Cohomology Versus the Jacobian Ring and the Torelli Theorem For Cubic Fourfolds”, Algebraic Geom., 6:1 (2019), 76–99  crossref  mathscinet  zmath  isi  scopus
    35. Kuznetsov A., “Calabi-Yau and Fractional Calabi-Yau Categories”, J. Reine Angew. Math., 753 (2019), 239–267  crossref  mathscinet  isi
    36. Ornaghi M., Pertusi L., “Voevodsky'S Conjecture For Cubic Fourfolds and Gushel-Mukai Fourfolds Via Noncommutative K3 Surfaces”, J. Noncommutative Geom., 13:2 (2019), 499–515  crossref  mathscinet  isi
    37. Kuznetsov A., Smirnov M., “On Residual Categories For Grassmannians”, Proc. London Math. Soc., 120:5 (2020), 617–641  crossref  mathscinet  isi
    38. Przyjalkowski V., Shramov C., “Hodge Level For Weighted Complete Intersections”, Collect. Math., 71:3 (2020), 549–574  crossref  mathscinet  isi
    39. Hochenegger A., Meachan C., “Frobenius and Spherical Codomains and Neighbourhoods”, Doc. Math., 25 (2020), 483–525  mathscinet  isi
    40. Qin X., “Compactification of the Moduli Space of Minimal Instantons on the Fano 3-Fold V-5”, J. Pure Appl. Algebr., 225:3 (2021), 106526  crossref  mathscinet  isi
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