Cubic threefold

In algebraic geometry, a cubic threefold is a hypersurface of degree 3 in 4-dimensional projective space. Cubic threefolds are all unirational, but (Clemens Griffiths) used intermediate Jacobians to show that non-singular cubic threefolds are not rational. The space of lines on a non-singular cubic 3-fold is a Fano surface.

Examples

• Koras–Russell cubic threefold
• Klein cubic threefold
• Segre cubic

References

• Bombieri, Enrico; Swinnerton-Dyer, H. P. F. (1967), "On the local zeta function of a cubic threefold", Ann. Scuola Norm. Sup. Pisa (3) 21: 1–29, http://www.numdam.org/item?id=ASNSP_1967_3_21_1_1_0 
• Clemens, C. Herbert; Griffiths, Phillip A. (1972), "The intermediate Jacobian of the cubic threefold", Annals of Mathematics, Second Series 95 (2): 281–356, doi:10.2307/1970801, ISSN 0003-486X 
• Murre, J. P. (1972), "Algebraic equivalence modulo rational equivalence on a cubic threefold", Compositio Mathematica 25: 161–206, ISSN 0010-437X, http://www.numdam.org/item?id=CM_1972__25_2_161_0