Cremona transformations and some related algebras

One proves a general characteristic-free criterion for a rational map between projective varieties to be birational in terms of ideal-theoretic and modulo-theoretic conditions. This criterion is more inclusive than that of [F. Russo, A. Simis, Compositio Math. 126 (2001) 335–358] and, moreover, differs from previous criteria in its nature in that the syzygies of the base ideal of the map are not directly involved in its formulation. However, a great deal of the consequences are phrased by means of those very syzygies avoided in the formulation of the criterion! In any case, the criterion is stated in effective terms so it yields an efficient computable test of birationality. One also introduces a so-called linear obstruction principle for base ideals of linear type, thus raising a basic question concerning the structure of a certain related “bilinear” algebra.