In algebraic geometry, the Segre cubic is a cubic threefold embedded in 4 (or sometimes 5) dimensional projective space, studied by . It is a particular Segre variety.
The Segre cubic is the set of points (x<sub>0</sub>:x<sub>1</sub>:x<sub>2</sub>:x<sub>3</sub>:x<sub>4</sub>:x<sub>5</sub>) of P<sup>5</sup> satisfying the equations
The intersection of the Segre cubic with any hyperplane x<sub>i</sub> = 0 is the Clebsch cubic surface. Its intersection with any hyperplane x<sub>i</sub> = x<sub>j</sub> is Cayley's nodal cubic surface. Its dual is the Igusa quartic 3-fold in P<sup>4</sup>. Its Hessian is the BarthâÂÂNieto quintic. A cubic hypersurface in P<sup>4</sup> 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 A<sub>2</sub>(2).