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The "piercing subspace" problem may be stated as follows:

There are given several subspaces in a projective space, rather non-intersecting. Find an additional subspace of a prescribed dimension that has intersections of prescribed dimensions with each of the given subspaces. Determine how many solutions there are.

I believe this problem was popular many years ago. Can you suggest where to find the best results related to this problem?

The simplest example of this type of problems is the line piercing three skew lines in a 3d space. This problem does have infinite number of solutions. When the additional line is to intersect four lines then in real spaces there may be two, one double or no solutions while in complex spaces there are two solutions or one double. A theorem related to this is the so-called sixteen-point theorem.

Can you suggest some other theorems of this type and how to find them?

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Schubert calculus; a newer book would be 3264 & All That. Intersection Theory in Algebraic Geometry by David Eisenbud and Joe Harris

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    $\begingroup$ This book is an over-kill. It contains much more general theory of intersections of nonlinear variettes. The problem of intersections of linear variettes takes less than 1% of the contents of the book. Intersections of linear variettes serve only as introduction to more general methods that are applicable to non-linear problems. Are there any methods simpler and more specialized? In case of intersections of linear variettes one may construct also explicit solutions which are not paid any attention to in this book. But thanks anyway. $\endgroup$
    – peter smoczynski
    Nov 20, 2014 at 0:43

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