A characteristic-free criterion of birationality

One develops ab initio the theory of rational/birational maps over reduced, but not necessarily irreducible, projective varieties in arbitrary characteristic. A virtual numerical invariant of a rational map is introduced, called the Jacobian dual rank. It is proved that a rational map in this general setup is birational if and only if the Jacobian dual rank is well defined and attains its maximal possible value. Even in the “classical” case where the source variety is irreducible there is some gain for this invariant over the degree of the map because, on one hand, it relates naturally to constructs in commutative algebra and, on the other hand, is effectively computable. Applications are given to results only known so far in characteristic zero. One curious byproduct is an alternative approach to deal with the result of Dolgachev concerning the degree of a plane polar Cremona map.