
A fundamental problem in Algebraic Geometry for projective schemes X is postulation problem. In fact to understand the geometry of X as an embedded scheme, one of the first points of interest is considering the postulation problem, i.e. determining the number of conditions imposed by asking hypersurfaces of any degree to contain X. There is an expected number of conditions in this situation the scheme X is said to have good postulation. So a natural question to ask is: when does X have good postulation, and if it is not, why not?
In the case when X is a scheme supported on unions of generic linear subspaces of a projective space there is much interest in this question because of many applications, and not only among algebraic geometers. This problem can be traced back to classical projective geometry, but the first complete result was achieved in 1981 by R. Hartshorne and A. Hirschowitz on the postulation of generic lines. After that, despite all the progress made on this problem, surprisingly very little is known about it, and the problem is still very live and widely open.
In this talk we first introduce the postulation problem, then we discuss this problem for the scheme theoretic unions of linear spaces in projective space by giving main results and conjectures about them, and finally, we point out its connections with several research areas inside and outside of mathematics.

