In algebraic geometry, a line complex is a set of lines that can be specified by a list of homogeneous polynomial equations. That is, a projective variety of lines. A linear line complex is defined by a list of degree-1 polynomials. A quadratic line complex is defined by a list of degree-2 polynomials. Similarly for cubic, quartic, quintic, sextic, etc. They were first studied by Julius Plücker in Neue Geometrie des Raumes (1868). Other important figures include Felix Klein, Sophus Lie, Arthur Cayley, William Hamilton, and Alfred Clebsch.
Setup By the standard trick in projective geometry, a line in 3-dimensional space is lifted to a plane through the origin in 4-dimensional space. In other words, the space of lines in R 3 {\displaystyle \mathbb {R} ^{3}} is lifted to the space of planes through the origin in R 4 {\displaystyle \mathbb {R} ^{4}} , which is the Grassmannian G ( 2 , 4 ) {\displaystyle G(2,4)} . It is then embedded to the projective space P ( ∧ 2 R 4 ) {\displaystyle \mathbb {P} (\wedge ^{2}\mathbb {R} ^{4})} via exterior product. Note that during the projective embedding, we get lines that does not exist R 3 {\displaystyle \mathbb {R} ^{3}} : the lines at infinity.
P ( ∧ 2 R 4 ) {\displaystyle \mathbb {P} (\wedge ^{2}\mathbb {R} ^{4})} is the projectivized space of bivectors in R 4 {\displaystyle \mathbb {R} ^{4}} , where ∧ {\displaystyle \wedge } is the exterior product. The space has homogeneous coordinates (Plücker coordinates) [ p 12 , p 13 , p 14 , p 23 , p 24 , p 34 ] {\displaystyle [p_{12},p_{13},p_{14},p_{23},p_{24},p_{34}]} . By convention, if p i j {\displaystyle p_{ij}} where i > j {\displaystyle i>j} is written, then p i j := − p j i {\displaystyle p_{ij}:=-p_{ji}} .
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