“…However, there seems to be very few results and no general theory available for multi-graded rational maps, that is, maps that are defined by a collection of multi-homogeneous polynomials over a subvariety of a product of projective spaces. At the same time, there is an increasing interest in those maps, for both theoretic and applied purposes (see, for example, [4,5,34,52,53]). This paper is the first step toward the development of a general theory of rational and birational multi-graded maps, providing also new insights in the single-graded case.…”