In
algebraic geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
, the Barth–Nieto quintic is a
quintic 3-fold in 4 (or sometimes 5) dimensional
projective space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
studied by that is the
Hessian of the
Segre cubic In algebraic geometry, the Segre cubic is a cubic threefold embedded in 4 (or sometimes 5) dimensional projective space, studied by .
Definition
The Segre cubic is the set of points (''x''0:''x''1:''x''2:''x''3:''x''4:''x''5) of ''P''5 satisfying ...
.
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
:
:
Properties
The Barth–Nieto quintic is not
rational
Rationality is the quality of being guided by or based on reason. In this regard, a person acts rationally if they have a good reason for what they do, or a belief is rational if it is based on strong evidence. This quality can apply to an ...
, but has a smooth model that is a modular
Calabi–Yau manifold
In algebraic and differential geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has certain properties, such as Ricci flatness, yielding applications in theoretical physics. P ...
with
Kodaira dimension
In algebraic geometry, the Kodaira dimension measures the size of the canonical model of a projective variety .
Soviet mathematician Igor Shafarevich in a seminar introduced an important numerical invariant of surfaces with the notation . ...
zero. Furthermore, it is
birationally equivalent
In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational fu ...
to a compactification of the
Siegel modular variety
In mathematics, a Siegel modular variety or Siegel moduli space is an algebraic variety that parametrizes certain types of abelian varieties of a fixed dimension. More precisely, Siegel modular varieties are the moduli spaces of principally pola ...
''A
1,3(2)''.
References
*
3-folds
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