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Irrationality of generic cubic threefold via Weil’s conjectures. (English) Zbl 06975598
MSC:
14J30 \(3\)-folds
14E08 Rationality questions in algebraic geometry
14G15 Finite ground fields in algebraic geometry
14J45 Fano varieties
Software:
Macaulay2
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[1] Aubry, Y.; Haloui, S., On the number of rational points on Prym varieties over finite fields, Glasgow Math. J., 58, 55-68, (2016) · Zbl 1353.14039
[2] Bardelli, F., Polarized mixed Hodge structures: on irrationality of threefolds via degeneration, Ann. Mat. Pura Appl. (4), 137, 287-369, (1984) · Zbl 0579.14033
[3] Beauville, A., Algebraic Threefolds, 947, LES singularités du diviseur thêta de la jacobienne intermédiaire de l’hypersurface cubique dans \(\Bbb P^4\), 190-208, (1982), Springer-Verlag
[4] Beauville, A., Rationality Problems in Algebraic Geometry, 2172, The Lüroth problem, 1-27, (2016), Springer
[5] Bosch, S.; Lütkebohmert, W.; Raynaud, M., Néron Models, 21, (1990), Springer
[6] Clemens, H.; Griffiths, P., The intermediate Jacobian of the cubic threefold, Ann. of Math. (2), 95, 281-356, (1972) · Zbl 0214.48302
[7] Chai, C. L.; Oort, F., A note on the existence of absolutely simple Jacobians, J. Pure Appl. Algebra, 155, 115-120, (2001) · Zbl 1006.14006
[8] Chai, C. L.; Oort, F., Abelian varieties isogenous to a Jacobian, Ann. of Math., 176, 1, 589-635, (2012) · Zbl 1263.14032
[9] Collino, A., A cheap proof of the irrationality of most cubic threefolds, Boll. Unione Mat. Ital. B (5), 16, 451-465, (1979) · Zbl 0425.14011
[10] Debarre, O.; Laface, A.; Roulleau, X.; Bogomolov, F.; Hassett, B.; Tschinkel, Y., Geometry Over Nonclosed Fields, Lines on cubic hypersurfaces over finite fields, 19-51, (2017), Springer
[11] Deligne, P.; Mumford, D., The irreducibility of the space of curves of given genus, Publ. Math. Inst. Hautes Étud. Sci., 36, 75-109, (1969) · Zbl 0181.48803
[12] Gwena, T., Degenerations of cubic threefolds and matroids, Proc. Amer. Math. Soc., 133, 5, 1317-1323, (2005) · Zbl 1072.14033
[13] Hartshorne, R., Algebraic Geometry, 52, (1977), Springer · Zbl 0367.14001
[14] Howe, E. W.; Zhu, H. J., On the existence of absolutely simple abelian varieties of a given dimension over an arbitrary field, J. Number Theory, 92, 139-163, (2002) · Zbl 0998.11031
[15] K. Lauter, Geometric methods for improving the upper bounds on the number of rational points on algebraic curves over finite fields, J. Algebraic Geom.10(1) (2001) 19-36; Appendix by J.-P. Serre. · Zbl 0982.14015
[16] D. R. Grayson and M. E. Stillman, Macaulay2, a software system for research in algebraic geometry; Available at https://faculty.math.illinois.edu/Macaulay2/.
[17] Milne, J. S., Etale Cohomology, 33, (1980), Princeton University Press
[18] Murre, J. P., Reduction of the proof of the non-rationality of a non-singular cubic threefold to a result of Mumford, Comp. Math., 27, 1, 63-82, (1973) · Zbl 0271.14020
[19] Murre, J. P., Classification of Algebraic Varieties and Compact Manifolds, 412, Some results on cubic threefolds, 140-164, (1974), Springer
[20] Murre, J. P., Algebraic Geometry, 1124, Applications of algebraic K-theory to the theory of algebraic cycles, (1985), Springer · Zbl 0561.14002
[21] Perret, M., Number of points of Prym varieties over finite fields, Glasgow Math. J., 48, 2, 275-280, (2006) · Zbl 1124.14023
[22] J. Tate, Classes d’isogénie des variétés abéliennes sur un corps fini (d’après T. Honda), Séminaire Bourbaki, exp. 352 (1968/1969).
[23] Tsimerman, J., The existence of an abelian variety over \(\overline{\Bbb Q}\) isogenous to no Jacobian, Ann. of Math. (2), 176, 1, 637-650, (2012) · Zbl 1250.14032
[24] Tjurin, A. N., Five lectures on three-dimensional varieties, Uspehi Mat. Nauk (167), 27, 5, 3-50, (1972)
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