Barth–Nieto quintic
# 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 $(x_0:x_1:x_2:x_3:x_4:x_5)$ of $P^5$ satisfying the equations

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

## 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 $A_{1,3}(2).$

