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 Wolf Barth and Isidro Nieto (1994) 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

${\displaystyle \displaystyle x_{0}+x_{1}+x_{2}+x_{3}+x_{4}+x_{5}=0}$

${\displaystyle \displaystyle x_{0}^{-1}+x_{1}^{-1}+x_{2}^{-1}+x_{3}^{-1}+x_{4}^{-1}+x_{5}^{-1}=0.}$

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).^[1]