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
[edit]The Barth–Nieto quintic is the closure of the set of points (x0:x1:x2:x3:x4:x5) of P5 satisfying the equations
Properties
[edit]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]
References
[edit]- ^ Hulek, Klaus; Sankaran, Gregory K. (2002). "The geometry of Siegel modular varieties". Higher dimensional birational geometry (Kyoto, 1997). Advanced Studies in Pure Mathematics. Vol. 35. Tokyo: Math. Soc. Japan. pp. 89–156. doi:10.2969/aspm/03510089. MR 1929793.
- 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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