Barth–Nieto quintic

In algebraic geometry, the Barth–Nieto quintic is a quintic 3-fold in 4 (or sometimes 5) dimensional projective space studied by Barth & Nieto (1994) that is the Hessian of the Segre cubic. The Barth–Nieto quintic is the closure of 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_{0}^{-1}+x_{1}^{-1}+x_{2}^{-1}+x_{3}^{-1}+x_{4}^{-1}+x_{5}^{-1}=0.

The Barth–Nieto quintic is not rational, but has a smooth model that is a Calabi–Yau manifold.

References

Barth, W.; Nieto, I. (1994), "Abelian surfaces of type (1,3) and quartic surfaces with 16 skew lines", Journal of Algebraic Geometry, 3 (2): 173–222, ISSN 1056-3911, MR 1257320

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