In
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, a line complex is a 3-fold given by the intersection of the Grassmannian ''G''(2, 4) (embedded in projective space ''P''
5 by
Plücker coordinates
In geometry, Plücker coordinates, introduced by Julius Plücker in the 19th century, are a way to assign six homogeneous coordinates to each line in projective 3-space, P3. Because they satisfy a quadratic constraint, they establish a one-to- ...
) with a hypersurface. It is called a line complex because points of ''G''(2, 4) correspond to lines in ''P''
3, so a line complex can be thought of as a 3-dimensional family of lines in ''P''
3. The linear line complex and quadric line complex are the cases when the hypersurface has degree 1 or 2; they are both
rational varieties 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 ...
.
References
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Algebraic varieties
3-folds
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