In
algebraic geometry, a cubic threefold is a
hypersurface
In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidea ...
of degree 3 in 4-dimensional
projective space. Cubic threefolds are all
unirational In mathematics, a rational variety is an algebraic variety, over a given field ''K'', which is birationally equivalent to a projective space of some dimension over ''K''. This means that its function field is isomorphic to
:K(U_1, \dots , U_d),
t ...
, but used
intermediate Jacobian
In mathematics, the intermediate Jacobian of a compact Kähler manifold or Hodge structure is a complex torus that is a common generalization of the Jacobian variety of a curve and the Picard variety and the Albanese variety. It is obtained by ...
s to show that non-singular cubic threefolds are not rational. The space of lines on a non-singular cubic 3-fold is a
Fano surface
In algebraic geometry, a Fano surface is a surface of general type (in particular, not a Fano variety) whose points index the lines on a non-singular cubic threefold. They were first studied by .
Hodge diamond:
Fano surfaces are perhaps the s ...
.
Examples
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Koras–Russell cubic threefold In algebraic geometry, the Koras–Russell cubic threefolds are smooth affine complex threefolds diffeomorphic to \mathbf^3studied by . They have a hyperbolic action of a one-dimensional torus \mathbf^*with a unique fixed point, such that the ...
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Klein cubic threefold In algebraic geometry, the Klein cubic threefold is the non-singular cubic threefold in 4-dimensional projective space given by the equation
:V^2W+W^2X+X^2Y+Y^2Z+Z^2V =0 \,
studied by .
Its automorphism group is the group PSL2(11) of order 660 . ...
*
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 satisfyin ...
References
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*{{Citation , last1=Murre , first1=J. P. , author-link1=Jaap Murre , title=Algebraic equivalence modulo rational equivalence on a cubic threefold , url=http://www.numdam.org/item?id=CM_1972__25_2_161_0 , mr=0352088 , year=1972 , journal=Compositio Mathematica , issn=0010-437X , volume=25 , pages=161–206
Algebraic varieties
3-folds