Homological projective duality for quadrics
Authors:
Alexander Kuznetsov and Alexander Perry
Journal:
J. Algebraic Geom. 30 (2021), 457-476
DOI:
https://doi.org/10.1090/jag/767
Published electronically:
January 15, 2021
MathSciNet review:
4283549
Full-text PDF
Abstract |
References |
Additional Information
Abstract: We show that over an algebraically closed field of characteristic not equal to 2, homological projective duality for smooth quadric hypersurfaces and for double covers of projective spaces branched over smooth quadric hypersurfaces is a combination of two operations: one interchanges a quadric hypersurface with its classical projective dual and the other interchanges a quadric hypersurface with the double cover branched along it.
References
- Francesca Carocci and Zak Turčinović, Homological projective duality for linear systems with base locus, Int. Math. Res. Not. IMRN (2018), rny222. https://academic.oup.com/imrn/article-abstract/2020/21/7829/5107178
- John Collins and Alexander Polishchuk, Gluing stability conditions, Adv. Theor. Math. Phys. 14 (2010), no. 2, 563–607. MR 2721656, DOI 10.4310/ATMP.2010.v14.n2.a6
- Anton Fonarev and Alexander Kuznetsov, Derived categories of curves as components of Fano manifolds, J. Lond. Math. Soc. (2) 97 (2018), no. 1, 24–46. MR 3764066, DOI 10.1112/jlms.12094
- Akira Ishii and Kazushi Ueda, The special McKay correspondence and exceptional collections, Tohoku Math. J. (2) 67 (2015), no. 4, 585–609. MR 3436544, DOI 10.2748/tmj/1450798075
- M. M. Kapranov, On the derived categories of coherent sheaves on some homogeneous spaces, Invent. Math. 92 (1988), no. 3, 479–508. MR 939472, DOI 10.1007/BF01393744
- Alexander Kuznetsov, Homological projective duality, Publ. Math. Inst. Hautes Études Sci. 105 (2007), 157–220. MR 2354207, DOI 10.1007/s10240-007-0006-8
- Alexander Kuznetsov, Derived categories of quadric fibrations and intersections of quadrics, Adv. Math. 218 (2008), no. 5, 1340–1369. MR 2419925, DOI 10.1016/j.aim.2008.03.007
- Alexander Kuznetsov, Semiorthogonal decompositions in algebraic geometry, Proceedings of the International Congress of Mathematicians, Vol. II (Seoul, 2014), 2014, pp. 635–660.
- Alexander Kuznetsov, Derived categories of families of sextic del Pezzo surfaces, Int. Math. Res. Not. IMRN (2019), rnz081.
- Alexander Kuznetsov and Alexander Perry, Derived categories of cyclic covers and their branch divisors, Selecta Math. (N.S.) 23 (2017), no. 1, 389–423. MR 3595897, DOI 10.1007/s00029-016-0243-0
- Alexander Kuznetsov and Alexander Perry, Derived categories of Gushel-Mukai varieties, Compos. Math. 154 (2018), no. 7, 1362–1406. MR 3826460, DOI 10.1112/s0010437x18007091
- Alexander Kuznetsov and Alexander Perry, Categorical cones and quadratic homological projective duality, https://www.ams.org/cgi-bin/mstrack/accepted_papers/jams.
- Alexander Kuznetsov and Alexander Perry, Categorical joins, arXiv:1804.00144 (2019).
- Giorgio Ottaviani, Spinor bundles on quadrics, Trans. Amer. Math. Soc. 307 (1988), no. 1, 301–316. MR 936818, DOI 10.1090/S0002-9947-1988-0936818-5
- Alexander Perry, Noncommutative homological projective duality, Adv. Math. 350 (2019), 877–972. MR 3948688, DOI 10.1016/j.aim.2019.04.052
- Jørgen Vold Rennemo, The homological projective dual of $\textrm {Sym}^2\Bbb P(V)$, Compos. Math. 156 (2020), no. 3, 476–525. MR 4053459, DOI 10.1112/s0010437x19007772
- Jørgen Vold Rennemo and Ed Segal, Hori-mological projective duality, Duke Math. J. 168 (2019), no. 11, 2127–2205. MR 3992034, DOI 10.1215/00127094-2019-0014
- Richard P. Thomas, Notes on homological projective duality, Algebraic geometry: Salt Lake City 2015, Proc. Sympos. Pure Math., vol. 97, Amer. Math. Soc., Providence, RI, 2018, pp. 585–609. MR 3821163
References
- Francesca Carocci and Zak Turčinović, Homological projective duality for linear systems with base locus, Int. Math. Res. Not. IMRN (2018), rny222. https://academic.oup.com/imrn/article-abstract/2020/21/7829/5107178
- John Collins and Alexander Polishchuk, Gluing stability conditions, Adv. Theor. Math. Phys. 14 (2010), no. 2, 563–607. MR 2721656
- Anton Fonarev and Alexander Kuznetsov, Derived categories of curves as components of Fano manifolds, J. Lond. Math. Soc. (2) 97 (2018), no. 1, 24–46. MR 3764066, DOI 10.1112/jlms.12094
- Akira Ishii and Kazushi Ueda, The special McKay correspondence and exceptional collections, Tohoku Math. J. (2) 67 (2015), no. 4, 585–609. MR 3436544, DOI 10.2748/tmj/1450798075
- M. M. Kapranov, On the derived categories of coherent sheaves on some homogeneous spaces, Invent. Math. 92 (1988), no. 3, 479–508. MR 939472, DOI 10.1007/BF01393744
- Alexander Kuznetsov, Homological projective duality, Publ. Math. Inst. Hautes Études Sci. 105 (2007), 157–220. MR 2354207, DOI 10.1007/s10240-007-0006-8
- Alexander Kuznetsov, Derived categories of quadric fibrations and intersections of quadrics, Adv. Math. 218 (2008), no. 5, 1340–1369. MR 2419925, DOI 10.1016/j.aim.2008.03.007
- Alexander Kuznetsov, Semiorthogonal decompositions in algebraic geometry, Proceedings of the International Congress of Mathematicians, Vol. II (Seoul, 2014), 2014, pp. 635–660.
- Alexander Kuznetsov, Derived categories of families of sextic del Pezzo surfaces, Int. Math. Res. Not. IMRN (2019), rnz081.
- Alexander Kuznetsov and Alexander Perry, Derived categories of cyclic covers and their branch divisors, Selecta Math. (N.S.) 23 (2017), no. 1, 389–423. MR 3595897, DOI 10.1007/s00029-016-0243-0
- Alexander Kuznetsov and Alexander Perry, Derived categories of Gushel-Mukai varieties, Compos. Math. 154 (2018), no. 7, 1362–1406. MR 3826460, DOI 10.1112/s0010437x18007091
- Alexander Kuznetsov and Alexander Perry, Categorical cones and quadratic homological projective duality, https://www.ams.org/cgi-bin/mstrack/accepted_papers/jams.
- Alexander Kuznetsov and Alexander Perry, Categorical joins, arXiv:1804.00144 (2019).
- Giorgio Ottaviani, Spinor bundles on quadrics, Trans. Amer. Math. Soc. 307 (1988), no. 1, 301–316. MR 936818, DOI 10.2307/2000764
- Alexander Perry, Noncommutative homological projective duality, Adv. Math. 350 (2019), 877–972. MR 3948688, DOI 10.1016/j.aim.2019.04.052
- Jørgen Vold Rennemo, The homological projective dual of $\mathrm {Sym}^2\mathbb {P}(V)$, Compos. Math. 156 (2020), no. 3, 476–525. MR 4053459, DOI 10.1112/s0010437x19007772
- Jørgen Vold Rennemo and Ed Segal, Hori-mological projective duality, Duke Math. J. 168 (2019), no. 11, 2127–2205. MR 3992034, DOI 10.1215/00127094-2019-0014
- Richard P. Thomas, Notes on homological projective duality, Algebraic geometry: Salt Lake City 2015, Proc. Sympos. Pure Math., vol. 97, Amer. Math. Soc., Providence, RI, 2018, pp. 585–609. MR 3821163
Additional Information
Alexander Kuznetsov
Affiliation:
Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkin str., Moscow 119991 Russia; and National Research University Higher School of Economics, Moscow, Russia
MR Author ID:
359553
Email:
akuznet@mi-ras.ru
Alexander Perry
Affiliation:
Department of Mathematics, Columbia University, New York, New York 10027
MR Author ID:
907593
Email:
aperry@math.columbia.edu
Received by editor(s):
March 4, 2019
Published electronically:
January 15, 2021
Additional Notes:
The first author was partially supported by the HSE University Basic Research Program, Russian Academic Excellence Project “5-100”. The second author was partially supported by NSF postdoctoral fellowship DMS-1606460, NSF grant DMS-1902060, and the Institute for Advanced Study.
Article copyright:
© Copyright 2021
University Press, Inc.
