Degenerations of cubic threefolds and matroids

Gwena, Tawanda

doi:10.1090/S0002-9939-04-07689-0

Degenerations of cubic threefolds and matroids
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by Tawanda Gwena

Proc. Amer. Math. Soc. 133 (2005), 1317-1323

DOI: https://doi.org/10.1090/S0002-9939-04-07689-0

Published electronically: November 22, 2004

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Abstract:

We present a surprising connection between cubic threefolds and the well-known regular matroid ${R}_{10}$ by making use of intermediate Jacobians of cubic threefolds realized as Prym varieties. As a corollary we obtain a new proof of the nonrationality of generic cubic threefolds.

References

V. Alexeev, Ch. Birkenhake, and K. Hulek, Degenerations of Prym varieties, J. Reine Angew. Math. 553 (2002), 73–116. MR 1944808, DOI 10.1515/crll.2002.103
Fabio Bardelli, Polarized mixed Hodge structures: on irrationality of threefolds via degeneration, Ann. Mat. Pura Appl. (4) 137 (1984), 287–369 (English, with Italian summary). MR 772264, DOI 10.1007/BF01789401
Alberto Collino, A cheap proof of the irrationality of most cubic threefolds, Boll. Un. Mat. Ital. B (5) 16 (1979), no. 2, 451–465 (English, with Italian summary). MR 546468
C. Herbert Clemens and Phillip A. Griffiths, The intermediate Jacobian of the cubic threefold, Ann. of Math. (2) 95 (1972), 281–356. MR 302652, DOI 10.2307/1970801
Vladimir Danilov and Viatcheslav Grishukhin, Maximal unimodular systems of vectors, European J. Combin. 20 (1999), no. 6, 507–526. MR 1703596, DOI 10.1006/eujc.1999.0298
Bruce Hunt, The geometry of some special arithmetic quotients, Lecture Notes in Mathematics, vol. 1637, Springer-Verlag, Berlin, 1996. MR 1438547, DOI 10.1007/BFb0094399
J. P. Murre, Algebraic equivalence modulo rational equivalence on a cubic threefold, Compositio Math. 25 (1972), 161–206. MR 352088
J. P. Murre, Algebraic equivalence modulo rational equivalence on a cubic threefold, Compositio Math. 25 (1972), 161–206. MR 352088
James G. Oxley, Matroid theory, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1992. MR 1207587
J. G. Semple and L. Roth, Introduction to Algebraic Geometry, Oxford, at the Clarendon Press, 1949. MR 0034048
Vitaly Vologodsky, The locus of indeterminacy of the Prym map, J. Reine Angew. Math. 553 (2002), 117–124. MR 1944809, DOI 10.1515/crll.2002.095