Segre cubic explained

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

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 ^{3} _{5} = 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).^[1]

Notes and References

1. The Geometry of Siegel Modular Varieties . Hulek . Klaus . Sankaran . G. K. . Advanced Studies in Pure Mathematics . 35 . 2002 . 89–156.