Cubic threefold
Encyclopedia
In algebraic geometry
, a cubic threefold is a hypersurface
of degree 3 in 4-dimensional projective space
. Cubic threefolds are all unirational, but used intermediate Jacobian
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
.
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, a cubic threefold is a hypersurface
Hypersurface
In geometry, a hypersurface is a generalization of the concept of hyperplane. Suppose an enveloping manifold M has n dimensions; then any submanifold of M of n − 1 dimensions is a hypersurface...
of degree 3 in 4-dimensional projective space
Projective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
. Cubic threefolds are all unirational, but used intermediate Jacobian
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 putting a complex structure on the torus Hn/Hn for n odd...
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
Fano surface
In algebraic geometry, a Fano surface is a surface of general type whose points index the lines on a non-singular cubic threefold...
.