In algebraic geometry, the Barth–Nieto quintic is a quintic 3-fold in 4 (or sometimes 5) dimensional projective space studied by 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
\displaystylex0+x1+x2+x3+x4+x5=0
\displaystyle
| -1 | |
| x | |
| 0 |
| -1 | |
| +x | |
| 1 |
| -1 | |
| +x | |
| 2 |
| -1 | |
| +x | |
| 3 |
| -1 | |
| +x | |
| 4 |
| -1 | |
| +x | |
| 5 |
=0.
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 A1,3(2).[1]