×

Cycles, derived categories, and rationality. (English) Zbl 1403.14022

Coskun, Izzet (ed.) et al., Surveys on recent developments in algebraic geometry. Bootcamp for the 2015 summer research institute on algebraic geometry, University of Utah, Salt Lake City, UT, USA, July 6–10, 2015. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-3557-8/hbk; 978-1-4704-4121-0/ebook). Proceedings of Symposia in Pure Mathematics 95, 199-266 (2017).
Summary: Our main goal is to give a sense of recent developments in the (stable) rationality problem from the point of view of unramified cohomology and 0-cycles as well as derived categories and semiorthogonal decompositions, and how these perspectives intertwine and reflect each other.
In particular, in the case of algebraic surfaces, we explain the relationship between Bloch’s conjecture. Chow-theoretic decompositions of the diagonal, categorical representability, and the existence of phantom subcategories of the derived category.
For the entire collection see [Zbl 1372.14001].

MSC:

14C25 Algebraic cycles
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
14E08 Rationality questions in algebraic geometry
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14F22 Brauer groups of schemes
14J10 Families, moduli, classification: algebraic theory
14J28 \(K3\) surfaces and Enriques surfaces
14J29 Surfaces of general type
14J30 \(3\)-folds
14J35 \(4\)-folds
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
PDFBibTeX XMLCite
Full Text: arXiv

References:

[1] Abramovich, Dan; Karu, Kalle; Matsuki, Kenji; W\l odarczyk, Jaros\l aw, Torification and factorization of birational maps, J. Amer. Math. Soc., 15, 3, 531-572 (2002) · Zbl 1032.14003 · doi:10.1090/S0894-0347-02-00396-X
[2] Addington, Nicolas, On two rationality conjectures for cubic fourfolds, Math. Res. Lett., 23, 1, 1-13 (2016) · Zbl 1375.14134 · doi:10.4310/MRL.2016.v23.n1.a1
[3] Addington, Nicolas; Thomas, Richard, Hodge theory and derived categories of cubic fourfolds, Duke Math. J., 163, 10, 1885-1927 (2014) · Zbl 1309.14014 · doi:10.1215/00127094-2738639
[4] Nicolas Addington, B. Hassett, Y. Tschinkel, and A. Varilly-Alvarado, Cubic fourfolds fibered in sextic del Pezzo surfaces, preprint arXiv:1606.05321, 2016. · Zbl 1471.14031
[5] Alekseev, V. A., On conditions for the rationality of three-folds with a pencil of del Pezzo surfaces of degree \(4\), Mat. Zametki, 41, 5, 724-730, 766 (1987) · Zbl 0623.14019
[6] Alexeev, Valery; Orlov, Dmitri, Derived categories of Burniat surfaces and exceptional collections, Math. Ann., 357, 2, 743-759 (2013) · Zbl 1282.14030 · doi:10.1007/s00208-013-0917-2
[7] Th\'eorie des topos et cohomologie \'etale des sch\'emas. Tome 1: Th\'eorie des topos, Lecture Notes in Mathematics, Vol. 269, xix+525 pp. (1972), Springer-Verlag, Berlin-New York
[8] Artin, M., Algebraization of formal moduli. II. Existence of modifications, Ann. of Math. (2), 91, 88-135 (1970) · Zbl 0177.49003 · doi:10.2307/1970602
[9] Artin, M.; Mumford, D., Some elementary examples of unirational varieties which are not rational, Proc. London Math. Soc. (3), 25, 75-95 (1972) · Zbl 0244.14017 · doi:10.1112/plms/s3-25.1.75
[10] Asher Auel, The complement of the locus of pfaffian cubic fourfolds, preprint, 2015. · Zbl 1391.11071
[11] Asher Auel and Marcello Bernardara, Semiorthogonal decompositions and birational geometry of del Pezzo surfaces over arbitrary fields, preprint arXiv:1511.07576, 2015. · Zbl 1435.14015
[12] Auel, Asher; Bernardara, Marcello; Bolognesi, Michele, Fibrations in complete intersections of quadrics, Clifford algebras, derived categories, and rationality problems, J. Math. Pures Appl. (9), 102, 1, 249-291 (2014) · Zbl 1327.14078 · doi:10.1016/j.matpur.2013.11.009
[13] Auel, Asher; Bernardara, Marcello; Bolognesi, Michele; V\'arilly-Alvarado, Anthony, Cubic fourfolds containing a plane and a quintic del Pezzo surface, Algebr. Geom., 1, 2, 181-193 (2014) · Zbl 1317.14032 · doi:10.14231/AG-2014-010
[14] Auel, Asher; B\`“ohning, Christian; Graf von Bothmer, Hans-Christian, The transcendental lattice of the sextic Fermat surface, Math. Res. Lett., 20, 6, 1017-1031 (2013) · Zbl 1331.14013 · doi:10.4310/MRL.2013.v20.n6.a2
[15] Asher Auel, Jean-Louis Colliot-Thelene, and Raman Parimala, Universal unramified cohomology of cubic fourfolds containing a plane, Birkhauser series Progress in Mathematics vol. 320, 2017. · Zbl 1426.14005
[16] Auel, Asher; Parimala, Raman; Suresh, Venapally, Quadric surface bundles over surfaces, Doc. Math., 31-70 (2015) · Zbl 1345.14046
[17] Auslander, Maurice; Goldman, Oscar, The Brauer group of a commutative ring, Trans. Amer. Math. Soc., 97, 367-409 (1960) · Zbl 0100.26304 · doi:10.2307/1993378
[18] M. Ballard, D. Deliu, D. Favero, M.U. Isik, and L. Katzarkov, On the Derived Categories of Degree \(d\) Hypersurface Fibrations, preprint arXiv:1409.5568, 2014. · Zbl 1423.14114
[19] L. Barbieri-Viale, Cicli di codimensione 2 su varieta unirazionali complesse, K-Theory (Strasbourg, 1992), Asterisque 226 (1994), 13-41. · Zbl 0840.14033
[20] Barlow, Rebecca, Rational equivalence of zero cycles for some more surfaces with \(p_g=0\), Invent. Math., 79, 2, 303-308 (1985) · Zbl 0584.14002 · doi:10.1007/BF01388975
[21] Barth, Wolf P.; Hulek, Klaus; Peters, Chris A. M.; Van de Ven, Antonius, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics 4, xii+436 pp. (2004), Springer-Verlag, Berlin · Zbl 1036.14016 · doi:10.1007/978-3-642-57739-0
[22] Bauer, Ingrid; Catanese, Fabrizio; Pignatelli, Roberto, Surfaces of general type with geometric genus zero: a survey. Complex and differential geometry, Springer Proc. Math. 8, 1-48 (2011), Springer, Heidelberg · Zbl 1227.14040 · doi:10.1007/978-3-642-20300-8\_1
[23] Beauville, Arnaud, Vari\'et\'es de Prym et jacobiennes interm\'ediaires, Ann. Sci. \'Ecole Norm. Sup. (4), 10, 3, 309-391 (1977) · Zbl 0368.14018
[24] Arnaud Beauville, The Luroth problem, Rationality Problems in Algebraic Geometry, pp. 1-27. Springer Lecture Notes 2172 (2016). · Zbl 1374.14040
[25] Beauville, Arnaud; Colliot-Th\'el\`“ene, Jean-Louis; Sansuc, Jean-Jacques; Swinnerton-Dyer, Peter, Vari\'”et\'es stablement rationnelles non rationnelles, Ann. of Math. (2), 121, 2, 283-318 (1985) · Zbl 0589.14042 · doi:10.2307/1971174
[26] Beauville, Arnaud; Donagi, Ron, La vari\'et\'e des droites d’une hypersurface cubique de dimension \(4\), C. R. Acad. Sci. Paris S\'er. I Math., 301, 14, 703-706 (1985) · Zbl 0602.14041
[27] Be\u \i linson, A. A., Coherent sheaves on \({\bf P}^n\)and problems in linear algebra, Funktsional. Anal. i Prilozhen., 12, 3, 68-69 (1978) · Zbl 0402.14006
[28] Bernardara, Marcello, A semiorthogonal decomposition for Brauer-Severi schemes, Math. Nachr., 282, 10, 1406-1413 (2009) · Zbl 1179.14013 · doi:10.1002/mana.200610826
[29] Bernardara, Marcello; Bolognesi, Michele, Derived categories and rationality of conic bundles, Compos. Math., 149, 11, 1789-1817 (2013) · Zbl 1286.14017 · doi:10.1112/S0010437X13007392
[30] Bernardara, Marcello; Bolognesi, Michele, Categorical representability and intermediate Jacobians of Fano threefolds, EMS Ser. Congr. Rep., 1-25 (2012), Eur. Math. Soc., Z\`“urich · Zbl 1287.18010
[31] Bernardara, Marcello; Bolognesi, Michele; Faenzi, Daniele, Homological projective duality for determinantal varieties, Adv. Math., 296, 181-209 (2016) · Zbl 1362.14020 · doi:10.1016/j.aim.2016.04.003
[32] Bernardara, Marcello; Tabuada, Gon\c{c}alo, From semi-orthogonal decompositions to polarized intermediate Jacobians via Jacobians of noncommutative motives, Mosc. Math. J., 16, 2, 205-235 (2016) · Zbl 1386.14012
[33] Bittner, Franziska, The universal Euler characteristic for varieties of characteristic zero, Compos. Math., 140, 4, 1011-1032 (2004) · Zbl 1086.14016 · doi:10.1112/S0010437X03000617
[34] Bloch, Spencer, Lectures on algebraic cycles, Duke University Mathematics Series, IV, 182 pp. pp. (1980), Duke University, Mathematics Department, Durham, N.C. · Zbl 0436.14003
[35] Bloch, Spencer; Kas, A.; Lieberman, D., Zero cycles on surfaces with \(p_g=0\), Compos. Math., 33, 2, 135-145 (1976) · Zbl 0337.14006
[36] Bloch, Spencer; Ogus, Arthur, Gersten’s conjecture and the homology of schemes, Ann. Sci. \'Ecole Norm. Sup. (4), 7, 181-201 (1975) (1974) · Zbl 0307.14008
[37] Bloch, Spencer; Srinivas, V., Remarks on correspondences and algebraic cycles, Amer. J. Math., 105, 5, 1235-1253 (1983) · Zbl 0525.14003 · doi:10.2307/2374341
[38] M. Blunk, A Derived Equivalence for some Twisted Projective Homogeneous Varieties, preprint arXiv:1204.0537, 2012.
[39] Bogomolov, F. A., The Brauer group of quotient spaces of linear representations, Izv. Akad. Nauk SSSR Ser. Mat.. Math. USSR-Izv., 51 30, 3, 455-485 (1988) · Zbl 0679.14025
[40] B\`“ohning, Christian; von Bothmer, Hans-Christian Graf; Sosna, Pawel, On the derived category of the classical Godeaux surface, Adv. Math., 243, 203-231 (2013) · Zbl 1299.14015 · doi:10.1016/j.aim.2013.04.017
[41] B\`“ohning, Christian; Graf von Bothmer, Hans-Christian; Sosna, Pawel, On the Jordan-H\'”older property for geometric derived categories, Adv. Math., 256, 479-492 (2014) · Zbl 1311.14021 · doi:10.1016/j.aim.2014.02.016
[42] B\`“ohning, Christian; Graf von Bothmer, Hans-Christian; Katzarkov, Ludmil; Sosna, Pawel, Determinantal Barlow surfaces and phantom categories, J. Eur. Math. Soc. (JEMS), 17, 7, 1569-1592 (2015) · Zbl 1323.14014 · doi:10.4171/JEMS/539
[43] M. Bolognesi and F. Russo, Some loci of rational cubic fourfolds, with an appendix by G. Stagliano, preprint arXiv:1504.05863, 2016. · Zbl 1460.14089
[44] Bondal, A. I., Representations of associative algebras and coherent sheaves, Izv. Akad. Nauk SSSR Ser. Mat.. Math. USSR-Izv., 53 34, 1, 23-42 (1990) · Zbl 0692.18002
[45] Bondal, A. I.; Kapranov, M. M., Representable functors, Serre functors, and reconstructions, Izv. Akad. Nauk SSSR Ser. Mat.. Math. USSR-Izv., 53 35, 3, 519-541 (1990) · Zbl 0703.14011
[46] Bondal, A. I.; Kapranov, M. M., Framed triangulated categories, Mat. Sb.. Math. USSR-Sb., 181 70, 1, 93-107 (1991) · Zbl 0729.18008
[47] Bondal, Alexei; Orlov, Dmitri, Reconstruction of a variety from the derived category and groups of autoequivalences, Compos. Math., 125, 3, 327-344 (2001) · Zbl 0994.18007 · doi:10.1023/A:1002470302976
[48] A.I. Bondal and Dmitri Orlov, Semiorthogonal decomposition for algebraic varieties, MPIM preprint 1995-15, 1995.
[49] Bondal, Alexey I.; Larsen, Michael; Lunts, Valery A., Grothendieck ring of pretriangulated categories, Int. Math. Res. Not., 29, 1461-1495 (2004) · Zbl 1079.18008 · doi:10.1155/S1073792804140385
[50] L. Borisov, Class of the affine line is a zero divisor in the Grothendieck ring, preprint arXiv:1412.6194, 2015.
[51] Bridgeland, Tom, Equivalences of triangulated categories and Fourier-Mukai transforms, Bull. London Math. Soc., 31, 1, 25-34 (1999) · Zbl 0937.18012 · doi:10.1112/S0024609398004998
[52] Campana, F., Connexit\'e rationnelle des vari\'et\'es de Fano, Ann. Sci. \'Ecole Norm. Sup. (4), 25, 5, 539-545 (1992) · Zbl 0783.14022
[53] Kampana, F.; Peternell, T.; Pukhlikov, A. V., The generalized Tsen theorem and rationally connected Fano fibrations, Mat. Sb.. Sb. Math., 193 193, 9-10, 1443-1468 (2002) · Zbl 1080.14535 · doi:10.1070/SM2002v193n10ABEH000685
[54] L. Campedelli, Sopra alcuni piani doppi notevoli con curve di diramazione d el decimo ordine, Atti Acad. Naz. Lincei 15 (1932), 536-542. · Zbl 0004.36306
[55] G. Castelnuovo, Sulle superficie di genere zero, Memorie della Soc.It. delle Scienze (detta dei XL), ser. III, t. 10, (1896).
[56] A. Chatzistamatiou and M. Levine, Torsion orders of complete intersections, preprint arXiv:1605.01913, 2016. · Zbl 1453.14024
[57] Y. Cho and Y. Lee, Exceptional collections on Dolgachev surfaces associated with degenerations, preprint arXiv:1506.05213, 2015. · Zbl 1387.14101
[58] Clemens, C. Herbert; Griffiths, Phillip A., The intermediate Jacobian of the cubic threefold, Ann. of Math. (2), 95, 281-356 (1972) · Zbl 0214.48302 · doi:10.2307/1970801
[59] Colliot-Th\'el\`“ene, Jean-Louis, Formes quadratiques multiplicatives et vari\'”et\'es alg\'ebriques, Bull. Soc. Math. France, 106, 2, 113-151 (1978) · Zbl 0386.14012
[60] Colliot-Th\'el\`“ene, Jean-Louis, Formes quadratiques multiplicatives et vari\'”et\'es alg\'ebriques: deux compl\'ements, Bull. Soc. Math. France, 108, 2, 213-227 (1980) · Zbl 0455.14002
[61] Colliot-Th\'el\`“ene, Jean-Louis, Birational invariants, purity and the Gersten conjecture, Proc. Sympos. Pure Math. 58, 1-64 (1995), Amer. Math. Soc., Providence, RI · Zbl 0834.14009
[62] Colliot-Th\'el\`“ene, Jean-Louis, Un th\'”eor\`“eme de finitude pour le groupe de Chow des z\'”ero-cycles d’un groupe alg\'ebrique lin\'eaire sur un corps \(p\)-adique, Invent. Math., 159, 3, 589-606 (2005) · Zbl 1080.14012 · doi:10.1007/s00222-004-0393-0
[63] Colliot-Th\'el\`“ene, Jean-Louis, Descente galoisienne sur le second groupe de Chow: mise au point et applications, Doc. Math., Extra vol.: Alexander S. Merkurjev”s sixtieth birthday, 195-220 (2015) · Zbl 1352.14005
[64] Jean-Louis Colliot-Thelene, Non rationalite stable d’hypersurfaces cubiques sur des corps non algebriquement clos, preprint arXiv:1606.06982, 2016.
[65] Jean-Louis Colliot-Thelene, \( \CH_0\)-trivialite universelle d’hypersurfaces cubiques presque diagonales, arXiv:1607.05673, to appear in Algebraic Geometry. · Zbl 1401.14074
[66] Colliot-Th\'el\`“ene, Jean-Louis; Coray, Daniel, L”\'equivalence rationnelle sur les points ferm\'es des surfaces rationnelles fibr\'ees en coniques, Compositio Math., 39, 3, 301-332 (1979) · Zbl 0386.14003
[67] Colliot-Th\'el\`“ene, Jean-Louis; Kahn, Bruno, Cycles de codimension 2 et \(H^3\) non ramifi\'”e pour les vari\'et\'es sur les corps finis, J. K-Theory, 11, 1, 1-53 (2013) · Zbl 1263.19004 · doi:10.1017/is012009001jkt194
[68] Colliot-Th\'el\`“ene, Jean-Louis; Ojanguren, Manuel, Vari\'”et\'es unirationnelles non rationnelles: au-del\`“a de l”exemple d’Artin et Mumford, Invent. Math., 97, 1, 141-158 (1989) · Zbl 0686.14050 · doi:10.1007/BF01850658
[69] Colliot-Th\'el\`“ene, Jean-Louis; Pirutka, Alena, Hypersurfaces quartiques de dimension 3: non-rationalit\'”e stable, Ann. Sci. \'Ec. Norm. Sup\'er. (4), 49, 2, 371-397 (2016) · Zbl 1371.14028
[70] Colliot-Th\'el\`“ene, Jean-Louis; Raskind, Wayne, \({\mathcal{K}}_2\)-cohomology and the second Chow group, Math. Ann., 270, 2, 165-199 (1985) · Zbl 0536.14004 · doi:10.1007/BF01456181
[71] Colliot-Th\'el\`“ene, Jean-Louis; Sansuc, Jean-Jacques, \(La R\)-\'”equivalence sur les tores, Ann. Sci. \'Ecole Norm. Sup. (4), 10, 2, 175-229 (1977) · Zbl 0356.14007
[72] Colliot-Th\'el\`“ene, Jean-Louis; Sansuc, Jean-Jacques, Cohomologie des groupes de type multiplicatif sur les sch\'”emas r\'eguliers, C. R. Acad. Sci. Paris S\'er. A-B, 287, 6, A449-A452 (1978) · Zbl 0399.14011
[73] Colliot-Th\'el\`“ene, Jean-Louis; Sansuc, Jean-Jacques, Fibr\'”es quadratiques et composantes connexes r\'eelles, Math. Ann., 244, 2, 105-134 (1979) · Zbl 0418.14016 · doi:10.1007/BF01420486
[74] Colliot-Th\'el\`“ene, Jean-Louis; Voisin, Claire, Cohomologie non ramifi\'”ee et conjecture de Hodge enti\`“ere, Duke Math. J., 161, 5, 735-801 (2012) · Zbl 1244.14010 · doi:10.1215/00127094-1548389
[75] Koll\'ar, J\'anos; Smith, Karen E.; Corti, Alessio, Rational and nearly rational varieties, Cambridge Studies in Advanced Mathematics 92, vi+235 pp. (2004), Cambridge University Press, Cambridge · Zbl 1060.14073 · doi:10.1017/CBO9780511734991
[76] Coughlan, Stephen, Enumerating exceptional collections of line bundles on some surfaces of general type, Doc. Math., 20, 1255-1291 (2015) · Zbl 1349.14057
[77] de Fernex, Tommaso; Fusi, Davide, Rationality in families of threefolds, Rend. Circ. Mat. Palermo (2), 62, 1, 127-135 (2013) · Zbl 1264.14019 · doi:10.1007/s12215-013-0110-1
[78] Deligne, Pierre; Illusie, Luc, Rel\`“evements modulo \(p^2\) et d\'”ecomposition du complexe de de Rham, Invent. Math., 89, 2, 247-270 (1987) · Zbl 0632.14017 · doi:10.1007/BF01389078
[79] Dolgachev, Igor, Algebraic surfaces with \(q=p_g=0\), C.I.M.E. Summer Sch. 76, 97-215 (2010), Springer, Heidelberg · doi:10.1007/978-3-642-11087-0\_3
[80] Enriques, Federigo, Sulle irrazionalit\`“a da cui pu\`o farsi dipendere la risoluzione d”un’ equazione algebrica \(f(xyz)=0\) con funzioni razionali di due parametri, Math. Ann., 49, 1, 1-23 (1897) · JFM 28.0559.02 · doi:10.1007/BF01445357
[81] Enriques, Federigo, Memorie scelte di geometria. Vol. I: 1893-1898, Pubblicate a cura dell’Accademia Nazionale dei Lincei, xxii+541 pp. (1 plate) pp. (1956), Nicola Zanichelli Editore, Bologna · Zbl 0073.15602
[82] Esnault, H\'el\`“ene, Varieties over a finite field with trivial Chow group of 0-cycles have a rational point, Invent. Math., 151, 1, 187-191 (2003) · Zbl 1092.14010 · doi:10.1007/s00222-002-0261-8
[83] Fano, Gino, Sulle forme cubiche dello spazio a cinque dimensioni contenenti rigate razionali del \(4^\circ\) ordine, Comment. Math. Helv., 15, 71-80 (1943) · Zbl 0027.13102 · doi:10.1007/BF02565634
[84] Fulton, William, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] 2, xi+470 pp. (1984), Springer-Verlag, Berlin · Zbl 0885.14002 · doi:10.1007/978-3-662-02421-8
[85] Fulton, William, Rational equivalence on singular varieties, Inst. Hautes \'Etudes Sci. Publ. Math., 45, 147-167 (1975) · Zbl 0332.14002
[86] Galkin, Sergey; Shinder, Evgeny, Exceptional collections of line bundles on the Beauville surface, Adv. Math., 244, 1033-1050 (2013) · Zbl 1408.14068 · doi:10.1016/j.aim.2013.06.007
[87] Sergey Galkin and Evgeny Shinder, The Fano variety of lines and rationality problem for a cubic hypersurface, preprint arXiv:1405.5154, 2014. · Zbl 1408.14068
[88] Galkin, Sergey; Katzarkov, Ludmil; Mellit, Anton; Shinder, Evgeny, Derived categories of Keum’s fake projective planes, Adv. Math., 278, 238-253 (2015) · Zbl 1327.14081 · doi:10.1016/j.aim.2015.03.001
[89] L. Godeaux, Les involutions cycliques appartenant a une surface algebrique, Actual. Sci. Ind. 270, Hermann, Paris, 1935. · Zbl 0013.41303
[90] Gorodentsev, A. L., Exceptional bundles on surfaces with a moving anticanonical class, Izv. Akad. Nauk SSSR Ser. Mat.. Math. USSR-Izv., 52 33, 1, 67-83 (1989) · Zbl 0736.14003
[91] Gorodentsev, A. L.; Rudakov, A. N., Exceptional vector bundles on projective spaces, Duke Math. J., 54, 1, 115-130 (1987) · Zbl 0646.14014 · doi:10.1215/S0012-7094-87-05409-3
[92] Gorchinskiy, Sergey; Orlov, Dmitri, Geometric phantom categories, Publ. Math. Inst. Hautes \'Etudes Sci., 117, 329-349 (2013) · Zbl 1285.14018 · doi:10.1007/s10240-013-0050-5
[93] Graber, Tom; Harris, Joe; Starr, Jason, Families of rationally connected varieties, J. Amer. Math. Soc., 16, 1, 57-67 (2003) · Zbl 1092.14063 · doi:10.1090/S0894-0347-02-00402-2
[94] A. Grothendieck, Le groupe de Brauer. I, II, III, Dix Exposes sur la Cohomologie des Schemas (J. Giraud et al), North-Holland, 1968. · Zbl 0192.57801
[95] Hassett, Brendan, Special cubic fourfolds, Compositio Math., 120, 1, 1-23 (2000) · Zbl 0956.14031 · doi:10.1023/A:1001706324425
[96] Hassett, Brendan, Some rational cubic fourfolds, J. Algebraic Geom., 8, 1, 103-114 (1999) · Zbl 0961.14029
[97] Brendan Hassett, Cubic fourfolds, K3 surfaces, and rationality questions, lecture notes for a 2015 CIME-CIRM summer school available on http://www.math.brown.edu/bhassett/papers.html. · Zbl 1454.14111
[98] Hassett, Brendan; Kresch, Andrew; Tschinkel, Yuri, Stable rationality and conic bundles, Math. Ann., 365, 3-4, 1201-1217 (2016) · Zbl 1353.14019 · doi:10.1007/s00208-015-1292-y
[99] Brendan Hassett and K.-W. Lai, Cremona transformations and derived equivalences of K3 surfaces, preprint arXiv:1612.07751, 2017. · Zbl 1407.14010
[100] Hassett, Brendan; Tschinkel, Yuri, Rational curves on holomorphic symplectic fourfolds, Geom. Funct. Anal., 11, 6, 1201-1228 (2001) · Zbl 1081.14515 · doi:10.1007/s00039-001-8229-1
[101] Brendan Hassett and Yuri Tschinkel, On stable rationality of Fano threefolds and del Pezzo fibrations, arXiv:1601.07074, to appear in Journal fur die reine und angewandte Mathematik. · Zbl 1503.14014
[102] Hille, Lutz; Perling, Markus, Exceptional sequences of invertible sheaves on rational surfaces, Compos. Math., 147, 4, 1230-1280 (2011) · Zbl 1237.14043 · doi:10.1112/S0010437X10005208
[103] S. Hosono and H. Takagi, Derived categories of Artin-Mumford double solids, preprint ArXiv 1506.02744 (2015). · Zbl 1475.14033
[104] Huybrechts, Daniel, Fourier-Mukai transforms in algebraic geometry, Oxford Mathematical Monographs, viii+307 pp. (2006), The Clarendon Press, Oxford University Press, Oxford · Zbl 1095.14002 · doi:10.1093/acprof:oso/9780199296866.001.0001
[105] Ingalls, Colin; Kuznetsov, Alexander, On nodal Enriques surfaces and quartic double solids, Math. Ann., 361, 1-2, 107-133 (2015) · Zbl 1408.14069 · doi:10.1007/s00208-014-1066-y
[106] Iskovskih, V. A.; Manin, Ju. I., Three-dimensional quartics and counterexamples to the L\`“uroth problem, Mat. Sb. (N.S.), 86(128), 140-166 (1971) · Zbl 0222.14009
[107] B. Kahn, Torsion order of smooth projective surfaces, with an appendix by J.-L. Colliot-Thelene, preprint arXiv:1605.01762, 2016.
[108] Kapranov, M. M., Derived category of coherent bundles on a quadric, Funktsional. Anal. i Prilozhen., 20, 2, 67 pp. (1986) · Zbl 0607.18004
[109] Karpenko, Nikita A.; Merkurjev, Alexander S., On standard norm varieties, Ann. Sci. \'Ec. Norm. Sup\'er. (4), 46, 1, 175-214 (2013) (2013) · Zbl 1275.14006
[110] K. Kawatani and S. Okawa Nonexistence of semiorthogonal decompositions and sections of the canonical bundle, preprint ArXiv:1508.00682
[111] Keller, Bernhard, On differential graded categories. International Congress of Mathematicians. Vol. II, 151-190 (2006), Eur. Math. Soc., Z\`“urich · Zbl 1140.18008
[112] J. Keum, A vanishing theorem on fake projective planes with enough automorphisms, preprint: arXiv:1407.7632v3. · Zbl 1407.14035
[113] Koll\'ar, J\'anos, Nonrational hypersurfaces, J. Amer. Math. Soc., 8, 1, 241-249 (1995) · Zbl 0839.14031 · doi:10.2307/2152888
[114] Koll\'ar, J\'anos; Miyaoka, Yoichi; Mori, Shigefumi, Rational connectedness and boundedness of Fano manifolds, J. Differential Geom., 36, 3, 765-779 (1992) · Zbl 0759.14032
[115] Kulikov, Vik. S., A remark on the nonrationality of a generic cubic fourfold, Mat. Zametki. Math. Notes, 83 83, 1-2, 57-64 (2008) · Zbl 1147.14018 · doi:10.1134/S0001434608010070
[116] Kuznetsov, Alexander, Derived categories of quadric fibrations and intersections of quadrics, Adv. Math., 218, 5, 1340-1369 (2008) · Zbl 1168.14012 · doi:10.1016/j.aim.2008.03.007
[117] A. Kuznetsov, Exceptional Collection of Vector Bundles on V22 Fano Threefolds, Vestn. Mosk. Univ., Ser. 1: Mat., Mekh., No. 3, 41-44 (1996), english translation in Moscow Univ. Math. Bull. 51 (3), 35-37 (1996). · Zbl 0913.14010
[118] Kuznetsov, Alexander, Derived categories of the Fano threefolds \(V_{12} \), Mat. Zametki. Math. Notes, 78 78, 3-4, 537-550 (2005) · Zbl 1111.14038 · doi:10.1007/s11006-005-0152-6
[119] Kuznetsov, Alexander, Derived category of a cubic threefold and the variety \(V_{14} \), Tr. Mat. Inst. Steklova. Proc. Steklov Inst. Math., 246, 3(246), 171-194 (2004) · Zbl 1107.14028
[120] Kuznetsov, Alexander, Hyperplane sections and derived categories, Izv. Ross. Akad. Nauk Ser. Mat.. Izv. Math., 70 70, 3, 447-547 (2006) · Zbl 1133.14016 · doi:10.1070/IM2006v070n03ABEH002318
[121] Kuznetsov, Alexander, Homological projective duality, Publ. Math. Inst. Hautes \'Etudes Sci., 105, 157-220 (2007) · Zbl 1131.14017 · doi:10.1007/s10240-007-0006-8
[122] Kuznetsov, Alexander, Derived categories of Fano threefolds, Tr. Mat. Inst. Steklova. Proc. Steklov Inst. Math., 264 264, 1, 110-122 (2009) · Zbl 1312.14055 · doi:10.1134/S0081543809010143
[123] Kuznetsov, Alexander, Derived categories of cubic fourfolds, Progr. Math. 282, 219-243 (2010), Birkh\`“auser Boston, Inc., Boston, MA · Zbl 1202.14012 · doi:10.1007/978-0-8176-4934-0\_9
[124] Alexander Kuznetsov, A simple counterexample to the Jordan-Holder property for derived categories preprint math.AG/1304.0903 (2013).
[125] Alexander Kuznetsov, Semiorthogonal decompositions in algebraic geometry, in Proceedings of the International Congress of Mathematicians (Seoul, 2014), vol. II (2014), pp. 635-660. arXiv:1404.3143. · Zbl 1373.18009
[126] Alexander Kuznetsov, Derived categories view on rationality problems, Lecture notes for the CIME-CIRM summer school (Levico Terme, 2015). arXiv:1509.09115.
[127] Kuznetsov, Alexander; Perry, Alexander, Derived categories of cyclic covers and their branch divisors, Selecta Math. (N.S.), 23, 1, 389-423 (2017) · Zbl 1365.14021 · doi:10.1007/s00029-016-0243-0
[128] Kuznetsov, Alexander; Polishchuk, Alexander, Exceptional collections on isotropic Grassmannians, J. Eur. Math. Soc. (JEMS), 18, 3, 507-574 (2016) · Zbl 1338.14021 · doi:10.4171/JEMS/596
[129] Alexander Kuznetsov and E. Shinder, Grothendieck ring of varieties, D- and L-equivalence, and families of quadrics, preprint arXiv:1612.07193, 2016. · Zbl 1450.11036
[130] K.-W. Lai, New cubic fourfolds with odd degree unirational parametrizations, preprint arXiv:1606.03853, 2016.
[131] Laza, Radu, The moduli space of cubic fourfolds via the period map, Ann. of Math. (2), 172, 1, 673-711 (2010) · Zbl 1201.14026 · doi:10.4007/annals.2010.172.673
[132] Looijenga, Eduard, The period map for cubic fourfolds, Invent. Math., 177, 1, 213-233 (2009) · Zbl 1177.32010 · doi:10.1007/s00222-009-0178-6
[133] Lee, Kyoung-Seog, Derived categories of surfaces isogenous to a higher product, J. Algebra, 441, 180-195 (2015) · Zbl 1327.14084 · doi:10.1016/j.jalgebra.2015.06.022
[134] Lunts, Valery A.; Orlov, Dmitri, Uniqueness of enhancement for triangulated categories, J. Amer. Math. Soc., 23, 3, 853-908 (2010) · Zbl 1197.14014 · doi:10.1090/S0894-0347-10-00664-8
[135] Marcolli, Matilde; Tabuada, Gon\c{c}alo, From exceptional collections to motivic decompositions via noncommutative motives, J. Reine Angew. Math., 701, 153-167 (2015) · Zbl 1349.14021 · doi:10.1515/crelle-2013-0027
[136] Markushevich, D.; Tikhomirov, A. S., The Abel-Jacobi map of a moduli component of vector bundles on the cubic threefold, J. Algebraic Geom., 10, 1, 37-62 (2001) · Zbl 0987.14028
[137] Matsusaka, Teruhisa, On a characterization of a Jacobian variety, Memo. Coll. Sci. Univ. Kyoto. Ser. A. Math., 32, 1-19 (1959) · Zbl 0094.34103
[138] Merkurjev, Alexander, Unramified elements in cycle modules, J. Lond. Math. Soc. (2), 78, 1, 51-64 (2008) · Zbl 1155.14017 · doi:10.1112/jlms/jdn011
[139] Morel, Fabien, Milnor’s conjecture on quadratic forms and mod 2 motivic complexes, Rend. Sem. Mat. Univ. Padova, 114, 63-101 (2006) (2005) · Zbl 1165.14309
[140] Mumford, David, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, No. 5, viii+242 pp. (1970), Published for the Tata Institute of Fundamental Research, Bombay; Oxford University Press, London · Zbl 0223.14022
[141] Mumford, David, Rational equivalence of \(0\)-cycles on surfaces, J. Math. Kyoto Univ., 9, 195-204 (1968) · Zbl 0184.46603
[142] Murre, J. P., On the Hodge conjecture for unirational fourfolds, Nederl. Akad. Wetensch. Proc. Ser. A {\bf 80}=Indag. Math., 39, 3, 230-232 (1977) · Zbl 0352.14006
[143] H. Nuer, Unirationality of moduli spaces of special cubic fourfolds and K3 surfaces, preprint arXiv:1503.05256, 2015. · Zbl 1371.14052
[144] Okawa, Shinnosuke, Semi-orthogonal decomposability of the derived category of a curve, Adv. Math., 228, 5, 2869-2873 (2011) · Zbl 1230.14020 · doi:10.1016/j.aim.2011.06.043
[145] Dmitri Orlov, An Exceptional Collection of Vector Bundles on the variety V5, Vestn. Mosk. Univ., Ser. 1: Mat., Mekh., No. 5, 69-71 (1991), english translation in Moscow Univ. Math. Bull. 46 (5), 48-50 (1991). · Zbl 0784.14010
[146] Orlov, Dmitri, Projective bundles, monoidal transformations, and derived categories of coherent sheaves, Izv. Ross. Akad. Nauk Ser. Mat.. Russian Acad. Sci. Izv. Math., 56 41, 1, 133-141 (1993) · Zbl 0798.14007 · doi:10.1070/IM1993v041n01ABEH002182
[147] Orlov, Dmitri, Derived categories of coherent sheaves on abelian varieties and equivalences between them, Izv. Ross. Akad. Nauk Ser. Mat.. Izv. Math., 66 66, 3, 569-594 (2002) · Zbl 1031.18007 · doi:10.1070/IM2002v066n03ABEH000389
[148] Orlov, Dmitri, Derived categories of coherent sheaves and triangulated categories of singularities, Progr. Math. 270, 503-531 (2009), Birkh\`“auser Boston, Inc., Boston, MA · Zbl 1200.18007 · doi:10.1007/978-0-8176-4747-6\_16
[149] M. Perling, Combinatorial aspects of exceptional sequences on (rational) surfaces, preprint ArXiv 1311.7349. · Zbl 1427.14041
[150] Peyre, Emmanuel, Unramified cohomology and rationality problems, Math. Ann., 296, 2, 247-268 (1993) · Zbl 0790.12001 · doi:10.1007/BF01445105
[151] A. Pirutka, Invariants birationnels dans la suite spectrale de Bloch-Ogus, J. K-theory 10 (2012), 565-582. · Zbl 1275.14007
[152] T. Reye, Die Geometrie der Lage, Vortrage, Baumgartner, Leipzig, 1882. · JFM 24.0582.04
[153] Rost, Markus, Chow groups with coefficients, Doc. Math., 1, No. 16, 319-393 (1996) · Zbl 0864.14002
[154] Helices and vector bundles. Seminaire Rudakov, London Mathematical Society Lecture Note Series, 148. A.N. Rudakov, ed. Cambridge University Press, Cambridge, 1990.
[155] Saltman, David J., Noether’s problem over an algebraically closed field, Invent. Math., 77, 1, 71-84 (1984) · Zbl 0546.14014 · doi:10.1007/BF01389135
[156] Segre, Beniamino, Sur un probl\`“eme de M. Zariski, Colloques Internationaux du Centre National de la Recherche Scientifique, no. 24, 135-138 (1950), Centre National de la Recherche Scientifique, Paris · Zbl 0040.08201
[157] Segre, Beniamino, Sull’esistenza, sia nel campo razionale che nel campo reale, di involuzioni piane non birazionali, Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. (8), 10, 94-97 (1951) · Zbl 0043.14901
[158] Shen, Mingmin; Vial, Charles, The Fourier transform for certain hyperk\`“ahler fourfolds, Mem. Amer. Math. Soc., 240, 1139, vii+163 pp. (2016) · Zbl 1386.14025 · doi:10.1090/memo/1139
[159] Shepherd-Barron, N. I., Stably rational irrational varieties, 693-700 (2004), Univ. Torino, Turin · Zbl 1067.14051
[160] V.V. Shokurov, Prym varieties: theory and applications, Math. USSR-Izv. 23 (1984), 83-147. · Zbl 0572.14025
[161] P. Sosna, Some remarks on phantom categories and motives, preprint arXiv:1511.07711. · Zbl 1281.14001
[162] Emmy Noether in Bryn Mawr, Proceedings of a Symposium in honor of her 100th birthday held at Bryn Mawr College, Bryn Mawr, Pa., March 17-19, 1982, viii+182 pp. (1983), Springer-Verlag, New York-Berlin · Zbl 0511.00003
[163] Tabuada, Gon\c{c}alo, Noncommutative motives, University Lecture Series 63, x+114 pp. (2015), American Mathematical Society, Providence, RI · Zbl 1333.14002 · doi:10.1090/ulect/063
[164] S. Tanimoto and A. Varilly-Alvarado, Kodaira dimension of moduli of special cubic fourfolds, preprint arXiv:1509.01562, 2015. · Zbl 1439.14118
[165] To\`“en, Bertrand, The homotopy theory of \(dg\)-categories and derived Morita theory, Invent. Math., 167, 3, 615-667 (2007) · Zbl 1118.18010 · doi:10.1007/s00222-006-0025-y
[166] Totaro, Burt, Hypersurfaces that are not stably rational, J. Amer. Math. Soc., 29, 3, 883-891 (2016) · Zbl 1376.14017 · doi:10.1090/jams/840
[167] Totaro, Burt, The integral cohomology of the Hilbert scheme of two points, Forum Math. Sigma, 4, e8, 20 pp. (2016) · Zbl 1375.14018 · doi:10.1017/fms.2016.5
[168] Vial, Charles, Exceptional collections, and the N\'eron-Severi lattice for surfaces, Adv. Math., 305, 895-934 (2017) · Zbl 1387.14061 · doi:10.1016/j.aim.2016.10.012
[169] Voevodsky, V., A nilpotence theorem for cycles algebraically equivalent to zero, Internat. Math. Res. Notices, 4, 187-198 (1995) · Zbl 0861.14006 · doi:10.1155/S1073792895000158
[170] Claire Voisin, Remarks on zero-cycles of self-products of varieties, in Moduli of vector bundles (Proceedings of the Taniguchi congress on vector bundles), Maruyama Ed., Decker, 1994, pp. 265-285. · Zbl 0912.14003
[171] Voisin, Claire, Abel-Jacobi map, integral Hodge classes and decomposition of the diagonal, J. Algebraic Geom., 22, 1, 141-174 (2013) · Zbl 1259.14006 · doi:10.1090/S1056-3911-2012-00597-9
[172] Voisin, Claire, Bloch’s conjecture for Catanese and Barlow surfaces, J. Differential Geom., 97, 1, 149-175 (2014) · Zbl 1386.14145
[173] Voisin, Claire, Degree 4 unramified cohomology with finite coefficients and torsion codimension 3 cycles, EMS Ser. Congr. Rep., 347-368 (2012), Eur. Math. Soc., Z\`“urich · Zbl 1317.14044 · doi:10.4171/119-1/20
[174] Voisin, Claire, Th\'eor\`“eme de Torelli pour les cubiques de \({\bf P}^5\), Invent. Math., 86, 3, 577-601 (1986) · Zbl 0622.14009 · doi:10.1007/BF01389270
[175] Voisin, Claire, Hodge theory and complex algebraic geometry. I, Cambridge Studies in Advanced Mathematics 76, x+322 pp. (2007), Cambridge University Press, Cambridge · Zbl 1129.14019
[176] Voisin, Claire, Some aspects of the Hodge conjecture, Jpn. J. Math., 2, 2, 261-296 (2007) · Zbl 1159.14005 · doi:10.1007/s11537-007-0639-x
[177] Voisin, Claire, Unirational threefolds with no universal codimension \(2\) cycle, Invent. Math., 201, 1, 207-237 (2015) · Zbl 1327.14223 · doi:10.1007/s00222-014-0551-y
[178] Claire Voisin, On the universal \(\CH_0\) group of cubic hypersurfaces, arXiv:1407.7261, to appear in Journal of the European Mathematical Society. · Zbl 1366.14009
[179] Voisin, Claire, Chow rings, decomposition of the diagonal, and the topology of families, Annals of Mathematics Studies 187, viii+163 pp. (2014), Princeton University Press, Princeton, NJ · Zbl 1288.14001 · doi:10.1515/9781400850532
[180] Claire Voisin, Stable birational invariants and the Luroth problem, in Advances in geometry and mathematical physics (Lectures given at the Geometry and Topology conference, Harvard University, 2014), H.-D. Cau and S.-T. Yau, eds., Surveys in Differential Geometry 21, International Press, 2016, pp. 313-332. · Zbl 1451.14132
[181] Zariski, Oscar, On Castelnuovo’s criterion of rationality \(p_a=P_2=0\) of an algebraic surface, Illinois J. Math., 2, 303-315 (1958) · Zbl 0085.36203
[182] Zucker, Steven, The Hodge conjecture for cubic fourfolds, Compositio Math., 34, 2, 199-209 (1977) · Zbl 0347.14005
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.