# 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