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.

Geometric invariant theory (GIT) gives a moduli space of smooth cubic threefolds, with one point for each isomorphism class of smooth cubic threefolds. (Allcock 2003) compactified this moduli space by means of the GIT quotient of all polystable cubic forms in 5 variables. In particular, this compactification is isomorphic to the K-moduli space of cubic threefolds, as proven by (Liu Xu).

Examples

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

References

• Allcock, Daniel (2003), "The moduli space of cubic threefolds", J. Algebraic Geom. 12: 201–223, https://doi.org/10.1090/S1056-3911-02-00313-2 
• 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 
• Liu, Yuchen; Xu, Chenyang (2019), "K-stability of cubic threefolds", Duke Math. J. 168 (11): 2029–2073, https://doi.org/10.1215/00127094-2019-0006 
• 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