Meeting time and place:

We meet on Tuesdays,14-16 Uhr (c.t.), in WSC-S-U-3.01


This semester we will hold a seminar on notions of rationality of algebraic varieties, and related topics, such as aspects of the integral Hodge conjecture, unramified cohomology, . Here is an preliminary outline:


Preliminary Schedule

Lecture 1. 14.4 Giuseppe Ancona: Artin-Mumford paper [AM]

Lecture 2. 21.4 Gabriele Guzman: two papers of Murre [M1], [M2]

Lecture 3. 28.4 Toan Nguyen: paper of Hasset [H]

Lecture 4. 5.5 Adeel Khan: papers of Atiyah-Hirzebruch [AH] and Totaro [T1], Kollár's lemma [K1]

Lecture 5. 12.5 Girja Tripathi: Soule-Voisin [SV]

Lecture 6. 19.5 Jin Cao.: the paper of Bloch-Srinivas [BS]

Lecture 7. 26.5 Lorenzo Mantovani: Colliot-Thélène,-Voisin [CT/V]

Lecture 8. 2.6 Aurélien Rodriguez: paper of Voisin [V3]

Lecture 9. 9.6/16.6 Federico Binda: Paper of Voisin [V1]

Lecture 10. 23.6 Marc Levine: Paper of Colliot-Thélène,/Pirutka [CT/P]

Lecture 11. 30.6 Andre Chatzistamatiou: Papers of Totaro [T2] and Kollár [K2]

Lecture 12. 7.7 Toan Nguyen: Papers of Beauville [B1], [B2]

Lecture 13. 14.7  N.N.: Papers of Colliot-Thélène, Swinnerton-Dyer [CT/SW]


[AM] Artin, M.; Mumford, D., Some elementary examples of unirational varieties which are not rational. Proc. London Math. Soc. (3) 25 (1972), 75–95.

[M1] Murre, J. P., Algebraic equivalence modulo rational equivalence on a cubic threefold. Compositio Math. 25 (1972), 161–206.

[M2] Murre, J. P., Reduction of the proof of the non-rationality of a non-singular cubic threefold to a result of Mumford. Compositio Math. 27 (1973), 63–82.

[CG] Clemens, C. Herbert; Griffiths, Phillip A., The intermediate Jacobian of the cubic threefold. Ann. of Math. (2) 95 (1972), 281–356.

[C] Clemens, Herbert, The quartic double solid revisited. Complex geometry and Lie theory (Sundance, UT, 1989), 89–101, Proc. Sympos. Pure Math., 53, Amer. Math. Soc., Providence, RI, 1991.

[H] Hassett, Brendan, Some rational cubic fourfolds. J. Algebraic Geom. 8 (1999), no. 1, 103–114.

[CT/V] Colliot-Thélène, Jean-Louis; Voisin, Claire, Cohomologie non ramifiée et conjecture de Hodge entière. Duke Math. J. 161 (2012), no. 5, 735–801.

[BS] Bloch, S.; Srinivas, V., Remarks on correspondences and algebraic cycles. Amer. J. Math. 105 (1983), no. 5, 1235–1253.

[V3] Voisin, Claire, Abel-Jacobi map, integral Hodge classes and decomposition of the diagonal. J. Algebraic Geom. 22 (2013), no. 1, 141–174.

[V2] Voisin, Claire, On integral Hodge classes on uniruled or Calabi-Yau threefolds. Moduli spaces and arithmetic geometry, 43–73, Adv. Stud. Pure Math., 45, Math. Soc. Japan, Tokyo, 2006.

[SV] Soulé, C.; Voisin, C., Torsion cohomology classes and algebraic cycles on complex projective manifolds. Adv. Math. 198 (2005), no. 1, 107–127.

[AH] M.F. Atiyah, F. Hirzebruch, Analytic cycles on complex manifolds, Topology 1 (1962) 25–45

[T1] Totaro, Burt, Torsion algebraic cycles and complex cobordism. J. Amer. Math. Soc. 10 (1997), no. 2, 467–493.

[T2] Burt Totaro, Hypersurfaces that are not stably rational arXiv:1502.04040 [math.AG]

[CT/P] Jean-Louis Colliot-Thélène, Alena Pirutka, Hypersurfaces quartiques de dimension 3 : non rationalité stable arXiv:1402.4153 [math.AG]

[V1] Claire Voisin, Unirational threefolds with no universal codimension 2 cycle arXiv:1312.2122 [math.AG]

[B2] Arnaud Beauville, A very general sextic double solid is not stably rational arXiv:1411.7484 [math.AG]

[B1] Arnaud Beauville, A very general quartic double fourfold or fivefold is not stably rational arXiv:1411.3122 [math.AG]

[CT/SW] Colliot-Thélène, J.-L.; Swinnerton-Dyer, Peter, Zero-cycles and rational points on some surfaces over a global function field. Acta Arith. 155 (2012), no. 1, 63–70.

[CT1] Colliot-Thélène, Jean-Louis, Variétés presque rationnelles, leurs points rationnels et leurs dégénérescences. Arithmetic geometry, 1–44, Lecture Notes in Math., 2009, Springer, Berlin, 2011.

[K2] Kollár, János, Nonrational hypersurfaces. J. Amer. Math. Soc. 8 (1995), no. 1, 241–249.

[K1] Kollár, János, Lemma p. 134, Classification of irregular varieties (Trento, 1990), 100–105, Lecture Notes in Math., 1515, Springer, Berlin, 1992.