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

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \displaystyle x_0+x_1+x_2+x_3+x_4+x_5= 0

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \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.