  In this paper, we consider rational maps whose source is a product of two
subvarieties, each one being embedded in a projective space. Our main objective
is to investigate birationality criteria for such maps. First, a general
criterion is given in terms of the rank of a couple of matrices that became to
be known as Jacobian dual matrices. Then, we focus on rational maps from the
product of two projectine lines to the projective plane in very low bidegrees
and provide new matrix-based birationality criteria by analyzing the syzygies
of the defining equations of the map, in particular by looking at the dimension
of certain bigraded parts of the syzygy module. Finally, applications of our
results to the context of geometric modeling are discussed at the end of the
paper.
