Consider a rational map from a projective space to a product of projective spaces, induced by a collection of linear projections. Motivated by the the theory of limit linear series and Abel-Jacobi maps, we study the basic properties of the closure of the image of the rational map using a combination of techniques of moduli functors and initial degenerations. We first give a formula of multi-degree in terms of the dimensions of intersections of linear subspaces and then prove that it is Cohen-Macaulay. Finally, we compute its Hilbert polynomials.

Inside a product of projective spaces, we try to understand which Chow classes come from irreducible subvarieties. The answer is closely related to the theory of integer polymatroids. The support of a representable class can be (partially) characterized as some integer point inside a particular polymatroid. If the class is multiplicity-free, we obtain a complete characterization in terms of representable polymatroids. We also generalize some of the results to the case of products of Grassmannians.