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Once I found by accident an article by MacPherson: "Classical projective geometry and modular varieties", in "Algebraic analysis, geometry, and number theory" (Baltimore, MD, 1988), whose introduction thrilled me and I'm still curious about how that developed and if now a general theory of configurations as continuation of classical geometry exists. Do you know something about it?

copy from the article: "Classical projective geometry was a beautiful field in mathematics. It died, in our opinion, not because it ran out of theorems to prove, but because it lacked organizing principles by which to select theorems that were important. Also, it was isolated from the rest of mathematics. Much of what we do may be regarded as direct continuation of nineteenth century synthetic geometry. In fact, we hope the new motivation of studying C-complexes will provide projective geometry with one organizational principle, and with one relation tying it to "mainstream" mathematics. We note … representable matroids, arrangements of hyperplanes, and motivic cohomology. A large part of this paper's exposition is motivated by this dream of continuing classical projective geometry."

Edit: Mnev's theorem, that every scheme over Z "is" a moduli space for point-configurations in the plane, makes me ask about applications and if versions for other number rings exist?

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We model theorists have been studying such things for the past couple of decades, so I know something about this.

Suppose that (S, cl) is a matroid -- i.e. a set S endowed with a closure operator satisfying a couple of natural axioms; canonical examples are where S is a vector space and cl(X) = Span(X), or when S a "projective space" over some field (i.e. the set of all 1-dimensional subspaces of a vector space, and cl is the closure operator induced by linear span). In addition to the standard matroid axioms, let's assume:

  1. cl(emptyset) = emptyset;
  2. cl({a}) = {a} for any a in S;
  3. (nontriviality) There is a subset X of S such that cl(X) is not the union of {cl(a) : a is in X};
  4. (modularity) For any finite-dimensional closed subsets X and Y of S, dim(X union Y) = dim(X) + dim(Y) - dim(X intersect Y);
  5. dim(S) is at least 4, and any 2-dimensional subset contains at least 3 elements.

Then it turns out that (S,cl) is isomorphic to a projective space over some skew field.

I believe this was first proved in Emil Artin's book Geometric Algebra (1957).

In the context of model theory, there has been a lot of research recently on matroids that arise in the models of certain theories (only we call them "pregeometries") and how properties of these matroids are related to definable group actions and vector spaces, in analogy to Artin's result above. See, for instance, Hrushovski's "stable group configuration theorem," which says that there is an infinite definable group whenever you have a certain finite configuration in a model of your theory.

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Thanks! Has model theory been applied to arithmetic geometry? E.g. rational points, resp. the geometric constructions associated with them, on varieties viewed as configurations (sorry if such a spontaneaus question is stupid). – Thomas Riepe Oct 18 '09 at 13:06
Yes! The most striking application is Hrushovski's proof of the Mordell-Lang conjecture for function fields of any characteristic, which used a lot of model-theoretic machinery, in particular geometric stability theory (which studies matroids like the ones I mentionned in my answer). The original result is in Hrushovski's "Mordell-Lang conjecture for function fields" (Journal of the AMS vol. 9 (1996), 667-690), and a good beginner's introduction to the subject is the book Model theory and algebraic geometry (ed. by E. Bouscaren, Springer LNM 1696). – John Goodrick Oct 18 '09 at 22:30

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