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Can someone point me to a modern treatment of the Veblen-Young characterization of projective spaces of dimension greater than $2$ as $P(V)$ for some vector space $V$?

[Added: see here for a statement of the Veblen-Young Theorem. The article includes a reference to the two-volume work of Veblen and Young in which the result was first proved. --PLC]

A precise reference to the original work where this appeared would also be very useful!

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  • $\begingroup$ @Mariano: well, I guess I'm not your guy for this, but...I really don't understand the question at all. I would have thought that every projective space is the projectivization of a vector space by definition. Are you talking about some kind of abstract projective geometry, or...what? $\endgroup$
    – Pete L. Clark
    Commented Jun 3, 2011 at 6:19
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    $\begingroup$ @Pete: based on the comment "dimension greater than $2$" I am guessing that yes, this refers to an abstract notion of projective space: en.wikipedia.org/wiki/… . $\endgroup$
    – Qiaochu Yuan
    Commented Jun 3, 2011 at 13:26
  • $\begingroup$ @Qiaochu: thanks. I took the liberty of editing that link into the question. I hope Mariano will not mind. $\endgroup$
    – Pete L. Clark
    Commented Jun 3, 2011 at 14:59
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    $\begingroup$ Isn't this in Artin's "Geometric Algebra"? (for some value of "modern") $\endgroup$
    – Felipe Voloch
    Commented Jun 3, 2011 at 16:07

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The 2-volume book by Veblen and Young is an obvious reference for the original proof. A modern reformulation of their result is the work of Tits classifying buildings of rank at least 3 in terms of algebraic groups over fields.

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  • $\begingroup$ Ah. I thought the result might have been published before the book in 'paper' form. Thanks! $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Jun 3, 2011 at 13:13
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    $\begingroup$ In fact, I think that's a good way to view this work of Tits: as a massive generalization of this theorem to other types of incidence geometries. $\endgroup$
    – Todd Trimble
    Commented Jun 3, 2011 at 20:54
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Here are some modern treatments:

  1. Projective Geometries: From Foundations to Applications, Albrecht Beutelspacher (1998) - see chapter 3 for the Veblen-Young characterisation.

  2. Points and Lines, Ernest E. Shult (2011) - covers the characterisation results of other types of incidence geometries as well.

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