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Let $M$ be a matroid admitting a coordinatization over a complex vector space. If we know that the complex coordinatization space for $M$ is connected, then may we conclude that the matroid admits a coordinatization over the real numbers?

The only examples that I am able to construct which do not have real coordinatizations have disconnected coordinatization spaces.

Note: Some texts refere to 'coordinatization over a vector space' as 'realizability over a vector space.'

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2 Answers 2

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The answer should be no. Here is the reason: It is perfectly possible to have a connected algebraic variety, defined over $\mathbb{R}$, which has no $\mathbb{R}$-points. For example, $\{ (x,y) : x^2+y^2=-1 \}$. Using Mnev's universality theorem, you should be able to build a matroid whose realization space is stably equivalent to this variety, and thus has the same property. I have not checked the details here.

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Does the proof of Mnëv's universality theorem preserve the structure of the matroid over $\mathbb{C}$? This should be checked, because all the proofs I've seen are only concerned about what happens over $\mathbb{R}$. Checking directly whether Mnëv's construction works for the example above shouldn't be too hard. –  Peter Shor Aug 4 '10 at 22:47
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I am certain the answer is yes. I believe Ravi's paper on Murphy's law in algebraic geometry front.math.ucdavis.edu/0411.5469 has a proof which is particularly careful about these field of definition issues. But this sort of uncertainty is why I wasn't willing to completely commit to my answer. –  David Speyer Aug 4 '10 at 23:35

Mnev wrote two papers in English about his theorem, as far as I know. (I don't read Russian.) A two-page paper (in Doklady I think) states the result for arbitrary "partially-oriented" matroids (including unoriented matroids), citing his dissertation, but with no proof. The paper in LNM that is usually cited has a sketch of the proof, but treats only the oriented case. Vakil's paper doesn't have a proof, but refers to the beginning of Lafforgue's book, where there is an algebraic argument (at the level of schemes) for the unoriented case, using the same method as Sturmfels used to prove a birational version of the universality theorem in the unoriented case, independently and simultaneously to Mnev. (Bernd doesn't get enough credit, I think.)

While that answers the question in principle, modeling the equation $x^2+y^2 = -1$ with a matroid requires a very large number of points. I once built the matroid for $x^2-1=0$, based on Sturmfels' construction, and needed 17 points (if I recall correctly), while there is a nine-point matroid with real, disconnected realization space.

So, consider the nine-point matroid of the line arrangement consisting of the irreducible components of $(x^3-y^3)(y^3-z^3)(z^3-x^3)=0$. (This is the matroid AG(2,3).) The matroid is (obviously) realizable over $\mathbb C$, but is projectively unique. (I'm pretty sure it is, but I don't have a reference.) So the realization space is a single point (connected), which, when put in a normal form, is not real. One can then argue that there is no linear change of variables that carries this matrix to a real matrix (up to scaling the rows). (What this means is that the conjugate of any realization is linearly equivalent to the original.)

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Can you mention the reference to Sturmfels' version of the universality theorem? –  Camilo Sarmiento May 4 '12 at 9:43
    
It is in Bulletin of the AMS, New Series, Vol. 17 (1987), pp. 121-124. –  Michael Falk May 9 '12 at 15:05

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