×

Equivalence of K3 surfaces from Verra threefolds. (English) Zbl 1455.14037

Summary: We study \((2,2)\) divisors in \(\mathbb{P}^2\times\mathbb{P}^2\) giving rise to pairs of nonisomorphic, derived equivalent, and \(\mathbb{L}\)-equivalent \(K3\) surfaces of degree \(2\). In particular, we confirm the existence of such fourfolds as predicted recently by A. Kuznetsov and E. Shinder [Sel. Math., New Ser. 24, No. 4, 3475–3500 (2018; Zbl 1450.11036)].

MSC:

14F08 Derived categories of sheaves, dg categories, and related constructions in algebraic geometry
14J28 \(K3\) surfaces and Enriques surfaces
14J35 \(4\)-folds
14J42 Holomorphic symplectic varieties, hyper-Kähler varieties

Citations:

Zbl 1450.11036

Software:

Macaulay2
PDFBibTeX XMLCite
Full Text: DOI arXiv Euclid

References:

[1] W. P. Barth, K. Hulek, C. A. M. Peters, and A. Van de Ven, Compact Complex Surfaces, 2nd ed., Ergeb. Math. Grenzgeb. (3) 4, Springer, Berlin, 2004. Zentralblatt MATH: 1036.14016
· Zbl 1036.14016
[2] A. Beauville, Variétés de Prym et jacobiennes intermédiaires, Ann. Sci. Éc. Norm. Supér. (4) 10 (1977), no. 3, 309-391. · Zbl 0368.14018
[3] A. Beauville, Variétés Kähleriennes dont la première classe de Chern est nulle, J. Differential Geom. 18 (1983), no. 4, 755-782. · Zbl 0537.53056
[4] A. Beauville and R. Donagi, La variété des droites d’une hypersurface cubique de dimension \(4\), C. R. Math. Acad. Sci. Paris Sér. I 301 (1985), no. 14, 703-706. Zentralblatt MATH: 0602.14041
· Zbl 0602.14041
[5] A. Bondal and D. Orlov, Reconstruction of a variety from the derived category and groups of autoequivalences, Compos. Math. 125 (2001), no. 3, 327-344. Zentralblatt MATH: 0994.18007
Digital Object Identifier: doi:10.1023/A:1002470302976
· Zbl 0994.18007 · doi:10.1023/A:1002470302976
[6] L. A. Borisov, The class of the affine line is a zero divisor in the Grothendieck ring, J. Algebraic Geom. 27 (2018), no. 2, 203-209. Zentralblatt MATH: 1415.14006
Digital Object Identifier: doi:10.1090/jag/701
· Zbl 1415.14006 · doi:10.1090/jag/701
[7] A. H. Caldararu, Derived categories of twisted sheaves on Calabi-Yau manifolds, Ph.D. dissertation, Cornell University, Ithaca, 2000.
[8] C. Camere, G. Kapustka, M. Kapustka, and G. Mongardi, Verra four-folds, twisted sheaves, and the last involution, Int. Math. Res. Not. IMRN 2019, no. 21, 6661-6710. Zentralblatt MATH: 07131137
Digital Object Identifier: doi:10.1093/imrn/rnx327
· Zbl 1453.14028 · doi:10.1093/imrn/rnx327
[9] S. Cynk and S. Rams, On a map between two \(K3\) surfaces associated to a net of quadrics, Arch. Math. (Basel) 88 (2007), no. 2, 109-122. · Zbl 1125.14021
[10] A. I. Efimov, Some remarks on L-equivalence of algebraic varieties, Selecta Math. (N.S.) 24 (2018), no. 4, 3753-3762. Zentralblatt MATH: 1397.14033
Digital Object Identifier: doi:10.1007/s00029-017-0374-y
· Zbl 1397.14033 · doi:10.1007/s00029-017-0374-y
[11] S. Galkin and E. Shinder, The Fano variety of lines and rationality problem for a cubic hypersurface, preprint, arXiv:1405.5154v2 [math.AG]. arXiv: 1405.5154v2
Zentralblatt MATH: 1408.14068
Digital Object Identifier: doi:10.1016/j.aim.2013.06.007
· Zbl 1408.14068 · doi:10.1016/j.aim.2013.06.007
[12] D. R. Grayson and M. E. Stillman, Macaulay2, a software system for research in algebraic geometry, http://www.math.uiuc.edu/Macaulay2/.
[13] B. Hassett, Some rational cubic fourfolds, J. Algebraic Geom. 8 (1999), no. 1, 103-114. Zentralblatt MATH: 0961.14029
· Zbl 0961.14029
[14] B. Hassett, Special cubic fourfolds, Compos. Math. 120 (2000), no. 1, 1-23. Zentralblatt MATH: 0956.14031
Digital Object Identifier: doi:10.1023/A:1001706324425
· Zbl 0956.14031 · doi:10.1023/A:1001706324425
[15] B. Hassett and K.-W. Lai, Cremona transformations and derived equivalences of K3 surfaces, Compos. Math. 154 (2018), no. 7, 1508-1533. Mathematical Reviews (MathSciNet): MR3826463
Zentralblatt MATH: 1407.14010
Digital Object Identifier: doi:10.1112/S0010437X18007145
· Zbl 1407.14010 · doi:10.1112/S0010437X18007145
[16] S. Hosono, B. H. Lian, K. Oguiso, and S.-T. Yau, Autoequivalences of derived category of a \(K3\) surface and monodromy transformations, J. Algebraic Geom. 13 (2004), no. 3, 513-545. Zentralblatt MATH: 1070.14042
Digital Object Identifier: doi:10.1090/S1056-3911-04-00364-9
· Zbl 1070.14042 · doi:10.1090/S1056-3911-04-00364-9
[17] D. Huybrechts, The K3 category of a cubic fourfold, Compos. Math. 153 (2017), no. 3, 586-620. Zentralblatt MATH: 1440.14180
Digital Object Identifier: doi:10.1112/S0010437X16008137
· Zbl 1440.14180 · doi:10.1112/S0010437X16008137
[18] A. Iliev, G. Kapustka, M. Kapustka, and K. Ranestad, Hyper-Kähler fourfolds and Kummer surfaces, Proc. Lond. Math. Soc. (3) 115 (2017), no. 6, 1276-1316. Zentralblatt MATH: 1408.14033
Digital Object Identifier: doi:10.1112/plms.12063
· Zbl 1408.14033 · doi:10.1112/plms.12063
[19] A. Ito, M. Miura, S. Okawa, and K. Ueda, The class of the affine line is a zero divisor in the Grothendieck ring: Via \(G_2\)-Grassmannians, J. Algebraic Geom. 28 (2019), no. 2, 245-250. Zentralblatt MATH: 1420.14019
Digital Object Identifier: doi:10.1090/jag/731
· Zbl 1420.14019 · doi:10.1090/jag/731
[20] A. Ito, M. Miura, S. Okawa, and K. Ueda, Derived equivalence and Grothendieck ring of varieties: The case of K3 surfaces of degree 12 and abelian varieties, Selecta Math. (N.S.) 26 (2020), no. 3, art. ID 38. Zentralblatt MATH: 07213680
Digital Object Identifier: doi:10.1007/s00029-020-00561-x
· Zbl 1467.14051 · doi:10.1007/s00029-020-00561-x
[21] G. Kapustka and A. Verra, On the Morin problem, in preparation.
[22] M. Kapustka and M. Rampazzo, Torelli problem for Calabi-Yau threefolds with GLSM description, Commun. Number Theory Phys. 13 (2019), no. 4, 725-761. Zentralblatt MATH: 07142778
Digital Object Identifier: doi:10.4310/CNTP.2019.v13.n4.a2
· Zbl 1451.14024 · doi:10.4310/CNTP.2019.v13.n4.a2
[23] A. Kuznetsov, Hyperplane sections and derived categories, Izv. Ross. Akad. Nauk Ser. Mat. 70 (2006), no. 3, 23-128. Zentralblatt MATH: 1133.14016
Digital Object Identifier: doi:10.1070/IM2006v070n03ABEH002318
· Zbl 1133.14016 · doi:10.1070/IM2006v070n03ABEH002318
[24] A. Kuznetsov, Derived categories of quadric fibrations and intersections of quadrics, Adv. Math. 218 (2008), no. 5, 1340-1369. Zentralblatt MATH: 1168.14012
Digital Object Identifier: doi:10.1016/j.aim.2008.03.007
· Zbl 1168.14012 · doi:10.1016/j.aim.2008.03.007
[25] A. Kuznetsov, “Derived categories of cubic fourfolds” in Cohomological and Geometric Approaches to Rationality Problems, Progr. Math. 282, Birkhäuser Boston, Boston, 2010, 219-243. Zentralblatt MATH: 1202.14012
· Zbl 1202.14012
[26] A. Kuznetsov and E. Shinder, Grothendieck ring of varieties, D- and L-equivalence, and families of quadrics, Selecta Math. (N.S.) 24 (2018), no. 4, 3475-3500. Zentralblatt MATH: 06941785
Digital Object Identifier: doi:10.1007/s00029-017-0344-4
· Zbl 1450.11036 · doi:10.1007/s00029-017-0344-4
[27] M. Larsen and V. A. Lunts, Motivic measures and stable birational geometry, Mosc. Math. J. 3 (2003), no. 1, 85-95. Zentralblatt MATH: 1056.14015
Digital Object Identifier: doi:10.17323/1609-4514-2003-3-1-85-95
· Zbl 1056.14015 · doi:10.17323/1609-4514-2003-3-1-85-95
[28] Y. Laszlo, Théorème de Torelli générique pour les intersections complètes de trois quadriques de dimension paire, Invent. Math. 98 (1989), no. 2, 247-264. · Zbl 0661.14006
[29] C. Lehn, M. Lehn, C. Sorger, and D. van Straten, Twisted cubics on cubic fourfolds, J. Reine Angew. Math. 731 (2017), 87-128. Zentralblatt MATH: 1376.53096
· Zbl 1376.53096
[30] E. Looijenga, The period map for cubic fourfolds, Invent. Math. 177 (2009), no. 1, 213-233. Zentralblatt MATH: 1177.32010
Digital Object Identifier: doi:10.1007/s00222-009-0178-6
· Zbl 1177.32010 · doi:10.1007/s00222-009-0178-6
[31] E. Macrì and P. Stellari, Fano varieties of cubic fourfolds containing a plane, Math. Ann. 354 (2012), no. 3, 1147-1176. Zentralblatt MATH: 1266.18016
Digital Object Identifier: doi:10.1007/s00208-011-0776-7
· Zbl 1266.18016 · doi:10.1007/s00208-011-0776-7
[32] C. Madonna and V. V. Nikulin, On the classical correspondence between \(K3\) surfaces, Proc. Steklov Inst. Math. 2003, no. 2(241), 120-153. Zentralblatt MATH: 1076.14046
· Zbl 1076.14046
[33] C. Madonna and V. V. Nikulin, “On a classical correspondence between \(K3\) surfaces, II” in Strings and Geometry, Clay Math. Proc. 3, Amer. Math. Soc., Providence, 2004, 285-300. Zentralblatt MATH: 1156.14318
· Zbl 1156.14318
[34] L. Manivel, Double spinor Calabi-Yau varieties, Épijournal Geom. Algebrique 3 (2019), art. ID 2. Zentralblatt MATH: 1443.14042
· Zbl 1443.14042
[35] N. Martin, The class of the affine line is a zero divisor in the Grothendieck ring: An improvement, C. R. Math. Acad. Sci. Paris 354 (2016), no. 9, 936-939. Zentralblatt MATH: 1378.14009
Digital Object Identifier: doi:10.1016/j.crma.2016.05.016
· Zbl 1378.14009 · doi:10.1016/j.crma.2016.05.016
[36] V. V. Nikulin, Integer symmetric bilinear forms and some of their geometric applications (in Russian), Izv. Math. 14 (1979), no. 1, 103-167; English translation in Math USSR-Izv. 14 (1979), no. 1, 103-167. Zentralblatt MATH: 0427.10014
Digital Object Identifier: doi:10.1070/IM1980v014n01ABEH001060
· Zbl 0427.10014 · doi:10.1070/IM1980v014n01ABEH001060
[37] V. V. Nikulin, Quotient-groups of groups of automorphisms of hyperbolic forms of subgroups generated by \(2\)-reflections (in Russian), Dokl. Math. 248 (1979), no. 6, 1307-1309; English translation in Soviet Math. Dokl. 20 (1979), no. 5, 1156-1158. Zentralblatt MATH: 0445.10020
· Zbl 0445.10020
[38] K. G. O’Grady, Dual double EPW-sextics and their periods, Pure Appl. Math. Q. 4 (2008), no. 2, 427-468. Zentralblatt MATH: 1152.14010
Digital Object Identifier: doi:10.4310/PAMQ.2008.v4.n2.a6
· Zbl 1152.14010 · doi:10.4310/PAMQ.2008.v4.n2.a6
[39] D. O. Orlov, Equivalences of derived categories and \(K3\) surfaces, J. Math. Sci. (N.Y.) 84 (1997), no. 5, 1361-1381. Zentralblatt MATH: 0938.14019
Digital Object Identifier: doi:10.1007/BF02399195
· Zbl 0938.14019 · doi:10.1007/BF02399195
[40] J. C. Ottem and J. V. Rennemo, A counterexample to the birational Torelli problem for Calabi-Yau threefolds, J. Lond. Math. Soc. (2) 97 (2018), no. 3, 427-440. Zentralblatt MATH: 1393.14006
Digital Object Identifier: doi:10.1112/jlms.12111
· Zbl 1393.14006 · doi:10.1112/jlms.12111
[41] B. van Geemen, Some remarks on Brauer groups of \(K3\) surfaces, Adv. Math. 197 (2005), no. 1, 222-247. Zentralblatt MATH: 1082.14040
Digital Object Identifier: doi:10.1016/j.aim.2004.10.004
· Zbl 1082.14040 · doi:10.1016/j.aim.2004.10.004
[42] A. Verra, “The Prym map has degree two on plane sextics” in The Fano Conference, Univ. Torino, Turin, 2004, 735-759. Zentralblatt MATH: 1169.14310
· Zbl 1169.14310
[43] C. Voisin, Théorème de Torelli pour les cubiques de \(\mathbf{P}^5 \), Invent. Math. 86 (1986), no. 3, 577-601. · Zbl 0622.14009
[44] C. Voisin, Hodge Theory and Complex Algebraic Geometry, I, Cambridge Stud. Adv. Math. 76, Cambridge Univ. Press, Cambridge, 2007. · Zbl 1129.14019
[45] K. · Zbl 1118.14013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.