Barth–Nieto quintic

The Barth–Nieto quintic is the closure of the set of points (x₀:x₁:x₂:x₃:x₄:x₅) of P⁵ satisfying the equations

:$\backslash displaystyle\; x\_0+x\_1+x\_2+x\_3+x\_4+x\_5=\; 0$

:$\backslash displaystyle\; x\_0^\{-1\}+x\_1^\{-1\}+x\_2^\{-1\}+x\_3^\{-1\}+x\_4^\{-1\}+x\_5^\{-1\}\; =\; 0.$

The Barth–Nieto quintic is not rational, but has a smooth model that is a 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).{{cite book|title= Higher dimensional birational geometry (Kyoto, 1997)|contribution=The geometry of Siegel modular varieties |last1=Hulek |first1=Klaus |author1-link=Klaus Hulek|last2=Sankaran |first2=Gregory K. |series=Advanced Studies in Pure Mathematics |publisher=Math. Soc. Japan| location=Tokyo|volume=35 |year=2002 |pages=89–156|mr=1929793|doi=10.2969/aspm/03510089|isbn=978-4-931469-85-3 }}

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• {{Citation | last1=Barth | first1=Wolf |author1-link=Wolf Barth| last2=Nieto | first2=Isidro | title=Abelian surfaces of type (1,3) and quartic surfaces with 16 skew lines | mr=1257320 | year=1994 | journal=Journal of Algebraic Geometry | issn=1056-3911 | volume=3 | issue=2 | pages=173–222}}

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Category:3-folds

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