On Birational Maps and Jacobian Matrices
One is concerned with Cremona-like transformations, i.e., rational maps from $ P$n to $ P$m that are birational onto the image Y [subset or is implied by] $ P$m and, moreover, the inverse map from Y to $ P$n lifts to $ P$m. We establish a handy criterion of birationality in terms of certain syzygies and ranks of appropriate matrices and, moreover, give an effective method to explicitly obtaining the inverse map. A handful of classes of Cremona and Cremona-like transformations follow as applications.
Key Words: birational maps; Cremona transformations; algebraic geometry; Jacobian matrices.