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).

Definition

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

$\displaystyle x_{0}+x_{1}+x_{2}+x_{3}+x_{4}+x_{5}=0$

$\displaystyle x_{0}^{3}+x_{1}^{3}+x_{2}^{3}+x_{3}^{3}+x_{4}^{3}+x_{5}^{3}=0.$

Properties

The intersection of the Segre cubic 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.

The Segre cubic is rational and furthermore birationally equivalent to a compactification of the Siegel modular variety A₂(2).

References

• Hunt, Bruce (1996), The geometry of some special arithmetic quotients, Lecture Notes in Mathematics, vol. 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 1806296
• 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