In algebraic geometry, the Klein cubic threefold is the non-singular cubic threefold in 4-dimensional projective space given by the equation

${\displaystyle V^{2}W+W^{2}X+X^{2}Y+Y^{2}Z+Z^{2}V=0\,}$

studied by Klein (1879). Its automorphism group is the group PSL₂(11) of order 660 (Adler 1978). It is unirational but not a rational variety. Gross & Popescu (2001) showed that it is birational to the moduli space of (1,11)-polarized abelian surfaces.

References

• Adler, Allan (1978), "On the automorphism group of a certain cubic threefold", American Journal of Mathematics, 100 (6): 1275–1280, doi:10.2307/2373973, ISSN 0002-9327, JSTOR 2373973, MR 0522700
• Gross, Mark; Popescu, Sorin (2001), "The moduli space of (1,11)-polarized abelian surfaces is unirational", Compositio Mathematica, 126 (1): 1–23, doi:10.1023/A:1017518027822, ISSN 0010-437X, MR 1827859
• Klein, Felix (1879), "Ueber die Transformation elfter Ordnung der elliptischen Functionen", Mathematische Annalen, 15 (3): 533–555, doi:10.1007/BF02086276, ISSN 0025-5831