A characteristic free criterion of birationality

TL;DR

This paper introduces the Jacobian dual rank as a new, characteristic-free invariant to determine birationality of rational maps on reduced projective varieties, providing a universal criterion applicable in arbitrary characteristic.

Contribution

It develops a characteristic-free theory of rational maps, introducing the Jacobian dual rank as a new invariant for birationality, extending classical results to positive characteristic.

Findings

Jacobian dual rank characterizes birationality in general settings

The invariant is related to commutative algebra constructions

Application to characteristic zero results and Cremona maps

Abstract

One develops {\em ab initio} the theory of rational/birational maps over reduced, but not necessarily irreducible, projective varieties in arbitrary characteristic. A 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 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 as it is, on one hand, intrinsically related to natural constructions in commutative algebra and, on the other hand, is effectively straightforwardly computable. Applications are given to results so far only known in characteristic zero. In particular, the surprising result of Dolgachev concerning the degree of a plane polar Cremona map is given an alternative conceptual angle.