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]

References

• Barth, Wolf; Nieto, Isidro (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

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