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A pair of ordered collections of linear subspaces $\Lambda_1, \ldots, \Lambda_k$ and $\Lambda'_1, \ldots, \Lambda'_k$ of $\mathbb{P}^n$ are called projectively equivalent if there exists a regular automorphism $\phi : \mathbb{P}^n \to \mathbb{P}^n$ (equivalently, a member of $PGL_{n+1}$) such that $\phi(\Lambda_i) = \Lambda'_i$ for each $i$.

It is well-known that two ordered sets of $n+2$ points in $\mathbb{P}^n$ in general position are projectively equivalent, and that any two ordered sets of three pairwise disjoint lines in $\mathbb{P}^3$ are projectively equivalent. Likewise, any two pairs of a hyperplane in $\mathbb{P}^n$ and a point outside of it are projectively equivalent. These should all be examples of ordered collections of linear subspaces in general position.

I have two questions:

1) In what general sense does collections of linear subspaces lie in "general position"? It cannot simply be that any subset of them span a linear subspace of maximal dimension, as this does not leave out the case of three concurrent lines in the plane. It can also not simply be that any subset of them has a proper intersection, as this does not leave out three colinear points. I imagine both of these two conditions must be satisfied for a collection to be in "general position".

2) What kind of "classifying theorems" are there regarding which pairs of ordered collections of linear subspaces in $\mathbb{P}^n$ are projectively equivalent? I'm looking for a sharp relation between the number $k$ and the individual dimensions of each $\Lambda_i$ under which two such collections are projectively equivalent, relative to the strictness of the condition of their relative position.

What I mean by the latter is in reference to the first question: such a condition could be that they are in general position, but also other types of conditions: such as that each subset span a linear space of maximal dimension, or that each subset has a proper intersection. These three types of conditions should all yield a different relation between $k$ and the individual dimensions of the $\Lambda_i$'s.

Note: I posted this question first on math.stackexchange.

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  • $\begingroup$ I'm not sure what you mean in (1). To be in general position means some unspecified condition holding in a dense open subset... if there is an open orbit on some product of Grassmanians, then this indeed exhibits one such open subset, which we can try to identify. $\endgroup$ – YCor Jan 3 '17 at 21:53
  • $\begingroup$ I expanded my first question a bit to clarify what I mean. Essentially, whatever "general position" could be, it should equate to the members of an open subset of the product of Grassmannians (parametrizing such collections) on which the optimal relation between $k$ and the the individual dimensions of each $\Lambda_i$ is achieved. I am also open to other suggestions. $\endgroup$ – Bubbles Jan 3 '17 at 22:05
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    $\begingroup$ In full generality I don't see any meaning to (1) other than: "enumerate all $PGL_{n+1}$-invariant Zariski open dense subsets of the product of Grassmanians". $\endgroup$ – YCor Jan 3 '17 at 22:14
  • $\begingroup$ I'm not looking for every possible definition of "general position", only a natural one (perhaps simply both maximal span and proper intersections?) for which an optimal sharp relation exists. $\endgroup$ – Bubbles Jan 3 '17 at 22:19
  • $\begingroup$ I may be wrong but I thought that the somewhat archaic way of saying "configurations in general position satisfy property $P$" meant that there is a Zariski open set $U_P$ whose elements satisfy $P$. Are you trying to have the same $U$ work for all $P$'s? $\endgroup$ – Abdelmalek Abdesselam Jan 4 '17 at 1:52
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1) There's an obvious notion of general position for subspaces: the span of any subset of the subspaces has the maximal possible dimension (that is, the minimum of the sum of the dimensions, and the dimension of the whole space). A set of lines "in general position" for this notion is precisely one with the uniform matroid structure (every subset of the correct size is a basis). This is obviously an open condition, and non-empty since it's an intersection of non-empty Zariski open subsets (obviously, each individual dimension condition can be satisfied) in an irreducible algebraic variety. Of course, I don't know for sure this is the best one: in particular, I don't know that if there is a dense orbit, then it must be precisely this subspace (though it sure wouldn't surprise me).

2) This seems to be a pretty rich area of study. I think this paper is probably state of the art at the moment. It doesn't completely solve the problem, but I think provides a good handle on what's going on.

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  • $\begingroup$ Thank you for the article. This is exactly what I was looking for. With regard to 1) - if I understand you correctly - it would seem that the condition that the span of any subset has the maximal possible dimension is not enough to characterize general position in this context. Three concurrent lines in $\mathbb{P}^2$ is an example of a set which satisfies that condition, but its orbit is not dense. $\endgroup$ – Bubbles Jan 4 '17 at 10:27
  • $\begingroup$ The condition "the span of any subset of the subspaces has the maximal possible dimension" is not even closed under taking orthogonals in the dual (e.g., it's satisfied by any family of distinct hyperplanes, while it's not satisfied by any family of distinct lines). Precisely in the case of, say triples of hyperplanes, there is a dense open orbit but this is not it. $\endgroup$ – YCor Jan 4 '17 at 15:10

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