The Sylvester-Gallai theorem asserts that for every collection of points in the plane, not all on a line, there is a line containing ** exactly** two of the points.

One high dimensional extension asserts that for every collection of points not all on a hyperplane in a d-dimensional space there is a [d/2]-space L whose intersection with the collection is a spanning set of cardinality [d/2]+1

**My question is:** Given a k-dimensional real algebraic variety V [perhaps of a certain kind] embedded in n-dimensional affine space (whose image spans this space) can one find an affine r-dimensional space L so that $V \cap L$ spans L and is topologically "simple".

**Remarks:** 1) For smooth complex varieties the Lefschetz hyperplane theorem describes sort of the opposite phenomenon: the homology of the hyperplane section is (more or less) as complicated as the homology of the original manifold all the way to half the dimension.

2) There is an analog of the Sylvester-Gallai Theorem over the complex numbers (there, if the points are not colinear you can always find a line containing 2 or 3 points, and if the points are not coplanar you can find a line containing precisely two points. The later statement was a conjecture by Serre first proved by Kelly. See this post in Konard Swanepoel's Blog).

So the difference between real AG and complex AG is not the only issue at hand (another issue seems to be how reducible the manifold is), and we can ask for sections with simple topology for complex varieties as well. Are there any results known in this direction?

3) Note that the whole point in the Sylvester-Gallai theorem and the proposed generalization is about non-generic embeddings and about intersection with non-generic flats.

However, the behavior of "generic embedding" (regarding intersection with non-generic flats) is sort of a role for what we expect for non-generic embedding. One difficulty is that *I am not aware of a definition of "generic embedding" of a real algebraic semi variety in a large Euclidean space.* Is there such a definition?

Let me formulate an open problem (and a few variations) which takes into account the discussion so far. We say that an embedding of a real semi algebraic variety V into an Euclidean space is genuine if its image span the space.

**Problem** (special case): Let V be a 2-dimensional variety genuinly embedded into n-space. (n cannot be too small, but maybe n=4 will suffice.) Then there is a plane L whose intersection with V is 1-dimensional, $V \cap L$ affinely span V, and $V \cap L$ is of very restricted topological type. (Maybe belong to a finite list of homotopy types.)

**Problem** (general case): The same conclusion (V \cap L belongs to a small set of homotopy types) when V is k dimensional, L is r dimensional, the intersection between V and L is j-dimensional and the dimension of the ambient space n is sufficiently large (but perhaps being moderately large suffices.) Greg's example shows that n cannot be too small. (For k=1,r=3,j=1 we cannot take n=4.) We know also that for k=j=0 we need n>=2r.

**Problem** (special case) Consider the case where V is an arrangement of subspaces.

**Problem** (complex analog) Consider the case that V is a complex variety; (and the special case of an arrangement of subspaces).