Supriya Ghosh (Editor)

Segre cubic

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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

x 0 + x 1 + x 2 + x 3 + x 4 + x 5 = 0 x 0 3 + x 1 3 + x 2 3 + x 3 3 + x 4 3 + x 5 3 = 0.

Its intersection with any hyperplane xi = 0 is the Clebsch cubic surface. Its intersection with any hyperplane xi = xj 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

Segre cubic Wikipedia
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