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 Mat. Zametki, 2005, Volume 78, Issue 4, Pages 579–594 (Mi mz2615)

Derived Categories of Fano Threefolds $V_{12}$

A. G. Kuznetsov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: In the present paper, we give a description of the derived category of coherent sheaves on a Fano threefold of index 1 and degree 12 (the variety $V_{12}$). It can easily be shown that if $X$ is a $V_{12}$ variety, then its derived category contains an exceptional pair of vector bundles $(\mathscr U,\mathscr O_X)$, where $\mathscr O_X$ is the trivial bundle, and $\mathscr U$ is the Mukai bundle of rank 5 (which induces the embedding $X\to\operatorname{Gr}(5,10)$). The orthogonal subcategory $\mathscr A_X= ^\perp<\mathscr U,\mathscr O>\subset\mathscr D^b(X)$ can be treated as the nontrivial part of the derived category of $X$. The main result of the present paper is the construction of the category equivalence $\mathscr A_X\cong\mathscr D^b(C^\vee)$, where $C^\vee$ is the curve of genus 7 which can be canonically associated to $X$ according to the results due to Iliev and Markushevich. In the construction of the equivalence, we make use of the geometric results due to Iliev and Markushevich, as well as the Bondal and Orlov results about derived categories. As an application, we prove that the Fano surface of $X$ (which is the surface parametrizing conics on $X$) is isomorphicto $S^2C^\vee$, the symmetric square of the corresponding curve of genus 7.

DOI: https://doi.org/10.4213/mzm2615

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English version:
Mathematical Notes, 2005, 78:4, 537–550

Bibliographic databases:

UDC: 514.762

Citation: A. G. Kuznetsov, “Derived Categories of Fano Threefolds $V_{12}$”, Mat. Zametki, 78:4 (2005), 579–594; Math. Notes, 78:4 (2005), 537–550

Citation in format AMSBIB
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\by A.~G.~Kuznetsov
\paper Derived Categories of Fano Threefolds $V_{12}$
\jour Mat. Zametki
\yr 2005
\vol 78
\issue 4
\pages 579--594
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\crossref{https://doi.org/10.4213/mzm2615}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2226730}
\zmath{https://zbmath.org/?q=an:1111.14038}
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\transl
\jour Math. Notes
\yr 2005
\vol 78
\issue 4
\pages 537--550
\crossref{https://doi.org/10.1007/s11006-005-0152-6}
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• http://mi.mathnet.ru/eng/mz2615
• https://doi.org/10.4213/mzm2615
• http://mi.mathnet.ru/eng/mz/v78/i4/p579

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. G. Kuznetsov, “Hyperplane sections and derived categories”, Izv. Math., 70:3 (2006), 447–547
2. Kuznetsov, A, “Derived categories of quadric fibrations and intersections of quadrics”, Advances in Mathematics, 218:5 (2008), 1340
3. Brambilla M.Ch., Faenzi D., “Moduli Spaces of Rank-2 ACM Bundles on Prime Fano Threefolds”, Michigan Math J, 60:1 (2011), 113–148
4. 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
5. 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
6. Brambilla M.Ch., Faenzi D., “Vector Bundles on Fano Threefolds of Genus 7 and Brill-Noether Loci”, Int. J. Math., 25:3 (2014), 1450023
7. 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
8. 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
9. 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
10. A. G. Kuznetsov, “On linear sections of the spinor tenfold. I”, Izv. Math., 82:4 (2018), 694–751
11. Laterveer R., “Zero-Cycles on Self-Products of Surfaces: Some New Examples Verifying Voisin'S Conjecture”, Rend. Circ. Mat. Palermo, 68:2 (2019), 419–431
12. Hosono Sh., Takagi H., “Derived Categories of Artin-Mumford Double Solids”, Kyoto J. Math., 60:1 (2020), 107–177
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