Looking for reference or proof to some facts stated on Anand Pillay's book

In my current work I am using facts 2.1.11 and 2.1.12 from Anand Pillay's book Geometric Stability Theory.

The facts are stated as follows:

Fact 2.1.11. Let $(S,\mbox{cl})$ be a locally projective, locally finite, infinite homogeneous geometry. Then $(S,\mbox{cl})$ is isomorphic to some affine or projective geometry over a finite field.

and

Fact 2.1.12. If $(S,\mbox{cl})$ is a projective geometry of dimension at least 4, such that all $2$-dimensional closed sets have at least three elements, then $(S,\mbox{cl})$ is a projective geometry over some division ring.

The author states this fact without reference, and I was not able to find one myself. Any directions would be very welcomed.

• 2.1.12 is a classical result in projective geometry, any good book should have it. e.g. see math.stackexchange.com/questions/549099/… for a list of books. – Dima Pasechnik May 19 '15 at 11:48
• 2.1.11 is also a result that is very old, at least in geometric terms (I don't know what an "infinite homogeneous" means though). Does it mean that the rank is infinite? – Dima Pasechnik May 19 '15 at 11:54
• These are two separate conditions. It simply means it is infinite as a set, and that for any two points outside a closed set there is a $\mbox{cl}$-automorphism which fixes the set pointwise while mapping one of the points to the other – Shai Deshe May 19 '15 at 11:58
• you need infiniteness to avoid a sporadic example related to the Mathieu group $M_{22}$, I suppose. Well, I don't know how to deal with the case of 2-dimensional closed sets (a.k.a. lines) being of size 2, and locally being an infinite projective plane. – Dima Pasechnik May 19 '15 at 12:10
• if your 2-dimensional closed sets have at least 3 elements then one doesn't need any group actions and infiniteness, it's just pure synthetic geometry to show that you sill get an affine or projective geometry. – Dima Pasechnik May 19 '15 at 13:50

1 Answer

In the notes on chapters at the end of Geometric Stability Theory, I give references. For Fact 1.11 it is Doyen and Hubaut, Finite regular locally projective geometries, Math. Zeitschrift, 1971. Zilber's book, Uncountably categorical theories, Translations of Math. monographs, vol 117, AMS, 1993, also mentions this on p. 27.

• Dear Professor Pillay, Welcome to MathOverFlow! – Mostafa Mirabi May 24 '15 at 11:04