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 (''x''₀:''x''₁:''x''₂:''x''₃:''x''₄:''x''₅) of ''P''⁵ 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 ''P''⁴ 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 Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abil ...

and furthermore birationally equivalent to a compactification of the

Siegel modular variety In mathematics, a Siegel modular variety or Siegel moduli space is an algebraic variety that parametrizes certain types of abelian varieties of a fixed dimension. More precisely, Siegel modular varieties are the moduli spaces of principally pola ...

''A₂(2)''.

References

* * *{{Citation , last1=Segre , first1=Corrado , title=Sulla varietà cubica con dieci punti doppii dello spazio a quattro dimensioni. , url=https://books.google.com/books?id=_9lSAAAAYAAJ&pg=PA791 , language=Italian , jfm=19.0673.01 , year=1887 , journal=Atti della Reale Accademia delle scienze di Torino , volume=XXII , pages=791–801 3-folds