Klein cubic threefold
Encyclopedia
In algebraic geometry
, the Klein cubic threefold is the non-singular cubic threefold
in 4-dimensional projective space
given by the equation
studied by .
Its automorphism group is the group PSL2(11) of order 660 . It is unirational but not a rational variety
.
showed that it is birational to the moduli space
of (1,11)-polarized abelian surface
s.
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...
, the Klein cubic threefold is the non-singular cubic threefold
Cubic threefold
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 Jacobians to show that non-singular cubic threefolds are not rational. The space of lines on a non-singular cubic 3-fold is a...
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....
given by the equation
studied by .
Its automorphism group is the group PSL2(11) of order 660 . It is unirational but not a rational variety
Rational variety
In mathematics, a rational variety is an algebraic variety, over a given field K, which is birationally equivalent to projective space of some dimension over K...
.
showed that it is birational to the moduli space
Moduli space
In algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects...
of (1,11)-polarized abelian surface
Abelian surface
In mathematics, an abelian surface is 2-dimensional abelian variety.One dimensional complex tori are just elliptic curves and are all algebraic, but Riemann discovered that most complex tori of dimension 2 are not algebraic...
s.