Line complex

In algebraic geometry, a line complex is a 3-fold given by the intersection of the Grassmannian G(2, 4) (embedded in projective space P⁵ by Plücker coordinates) with a hypersurface. It is called a line complex because points of G(2, 4) correspond to lines in P³, so a line complex can be thought of as a 3-dimensional family of lines in P³. The linear line complex and quadric line complex are the cases when the hypersurface has degree 1 or 2; they are both rational varieties.

References

• Griffiths, Phillip; Harris, Joseph (1994), Principles of algebraic geometry, Wiley Classics Library, New York: John Wiley & Sons, ISBN 978-0-471-05059-9 
• Jessop, C. M. (2001), A treatise on the line complex, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2913-4, http://digital.library.cornell.edu/cgi/t/text/text-idx?c=math;idno=06960001 
• Klein, Felix (1870), "Zur Theorie der Liniencomplexe des ersten und zweiten Grades", Mathematische Annalen (Springer Berlin / Heidelberg) 2 (2): 198–226, doi:10.1007/BF01444020, ISSN 0025-5831, https://zenodo.org/record/1607513