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, MR 1288523
• Jessop, C. M. (2001) [1903], A treatise on the line complex, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2913-4, MR 0247995
• Klein, Felix (1870), "Zur Theorie der Liniencomplexe des ersten und zweiten Grades", Mathematische Annalen, 2 (2), Springer Berlin / Heidelberg: 198–226, doi:10.1007/BF01444020, ISSN 0025-5831, S2CID 121706710