title: Barth–Nieto quintic
text: # 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
  
  The Barth–Nieto quintic is the closure of the set of points $(x_0:x_1:x_2:x_3:x_4:x_5)$ of $P^5$ satisfying the equations
  
      $\displaystyle x_{0}+x_{1}+x_{2}+x_{3}+x_{4}+x_{5}=0$
      $\displaystyle x_{0}^{-1}+x_{1}^{-1}+x_{2}^{-1}+x_{3}^{-1}+x_{4}^{-1}+x_{5}^{-1}=0.$
  
  ## 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 $A_{1,3}(2).$
  
  