Segre cubic

In algebraic geometry, the Segre cubic is a cubic threefold embedded in 4 (or sometimes 5) dimensional projective space, studied by Corrado Segre (1887).

The Segre cubic is the set of points (x₀:x₁:x₂:x₃:x₄:x₅) of P⁵ satisfying the equations

\displaystyle x_0+x_1+x_2+x_3+x_4+x_5= 0

\displaystyle x_0^3+x_1^3+x_2^3+x_3^3+x_4^3+x_5^3 = 0.

Its intersection with any hyperplane x_i = 0 is the Clebsch cubic surface. Its intersection with any hyperplane x_i = x_j is Cayley's nodal cubic surface. Its dual is the Igusa quartic 3-fold in P⁴. Its Hessian is the Barth–Nieto quintic. A cubic hypersurface in P⁴ has at most 10 nodes, and up to isomorphism the Segre cubic is the unique one with 10 nodes. Its nodes are the points conjugate to (1:1:1:−1:−1:−1) under permutations of coordinates.

References

Hunt, Bruce (1996), The geometry of some special arithmetic quotients, Lecture Notes in Mathematics 1637, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0094399, ISBN 978-3-540-61795-2, MR 1438547
Hunt, Bruce (2000), "Nice modular varieties", Experimental Mathematics 9 (4): 613–622, doi:10.1080/10586458.2000.10504664, ISSN 1058-6458, MR R1806296
Segre, Corrado (1887), "Sulla varietà cubica con dieci punti doppii dello spazio a quattro dimensioni.", Atti della Reale Accademia delle scienze di Torino (in Italian) XXII: 791–801, JFM 19.0673.01