Talk:Collineation

Only basic

 * Section title added. —Nils von Barth (nbarth) (talk) 00:48, 2 November 2009 (UTC)

This is only the basic statement of the FToPG. Lacking : a proof, references, implications.

different statement
I know the article is incomplete but I was taught at uni that the fundamental theorem says something different. Suppose you have two projective spaces and a collineation from one to the other. Then their dimension is equal, and not only are the two fields isomorphic, there is an semilinear nonsingular map from the first space to the second inducing the collineation. This nonsingular map is unique to multiplication with a scalar from the second field. Basically, when the geometric dimension is n, a set of n+2 points no three of which collinear, can be mapped onto another by a collineation in the second space , in just so many ways as the fields have automorphisms.

Do you agree?


 * Well, that sounds like an extension. Do we have a good discussion of collineation yet? By the way, please sign with ~ for a time-stamped signature. Charles Matthews 12:35, 4 February 2006 (UTC)

Apparently not. I can do that within a week. It is good that some people check what I do to change some basic stuff as they are my first pages. Just a question, am I violating copyright when taking definitions of stuff like polar space or generalized quadrangle, from my syllabus at Ghent university? A counterargument might be that while rare, definitions appear elsewhere, but details like degenerations or finite/infinite might make it recognisable. Evilbu 15:14, 4 February 2006 (UTC)

Although I have seen this statement of the Fundamental Theorem in some texts, I do not think that it is a good statement. The group theoretic description of the group's structure is certainly fundamental from a group theorist's pov. However, a geometer is more interested in the action of the group on the points of the projective geometry, so you often also see something in the statement to the effect that the group contains a subgroup (PGL) which acts regularly on the frames of the geometry. I am concerned about this issue because I was writing up a page (oval (projective plane)) and had a statement involving the FTPG. I had wanted to make a link to this page, but the statement here was not sufficient to cover my meaning. I would like to expand the statement so that I could make this link, but thought that I should ask for other opinions first. Wcherowi (talk) 20:47, 16 August 2011 (UTC)

Projectivity vs. collineation
I’m not totally clear on the distinction between projectivity and collineation; PlanetMath notes that in some definitions there is a difference, but in others there is not. An example would clarify matters, if anyone has one forthcoming.
 * —Nils von Barth (nbarth) (talk) 01:55, 2 November 2009 (UTC)