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 Template:Harvs that is the Hessian of the Segre cubic.

Definition

The Barth–Nieto quintic is the closure of the set of points (x₀:x₁:x₂:x₃:x₄:x₅) of P⁵ satisfying the equations

$x_0+x_1+x_2+x_3+x_4+x_5=0$

$x_0^{-1}+x_1^{-1}+x_2^{-1}+x_3^{-1}+x_4^{-1}+x_5^{-1}=0.$

Properties

The Barth–Nieto quintic is not rational, but has a smooth model that is a modular Calabi–Yau manifold with Kodaira dimension zero. Furthermore, it is birationally equivalent to a compactification of the Siegel modular variety A_1,3(2).

References

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