Realization space of matroids - MathOverflow

Realization space of matroids

2010-08-04T21:36:20Z

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.'

Answer by David Speyer for Realization space of matroids

2010-08-04T22:31:51Z

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.

Answer by Michael Falk for Realization space of matroids

2011-09-05T19:39:29Z

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.)