One develops 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.The simplification comes about by showing that birationality is controlled by the behavior of a unique numerical invariant -called the Jacobian dual rank of a rational map. Alas, this sounds like old knowledge because the classical theory also depends only on the degree of the rational map. However, the latter is in full control only in the integral case. Habitually, in positive characteristic one treats the inseparability degree apart from the main stream of the natural ideas in birational theory. The new invariant introduced here looks more intrinsic and makes no explicit reference to inseparability, so the criterion itself and the applications will be characteristic-free. Finally, the Jacobian dual rank is straightforwardly effectively computable in the usual implementation of the Gröbner basis algorithm, an appreciable advantage over the field degree.In addition, the Jacobian dual rank calls attention to several aspects of the theory of Rees algebras and base ideals of maps, a trend sufficiently shown in many modern accounts (see, e.g., [2], [5], [6], [7]).The paper is divided in two sections. The first section hinges on the needed background to state the general criterion of birationality.In the initial subsections we develop the ground material on rational and birational maps on a reduced source. Our approach is entirely algebraic, but we mention the transcription to the geometric side. A degree of care is required to show that the present notion is stable under the expected manipulations from the "classical" case. One main result in this part is Proposition 1.11 which drives us back to an analogue of the field extension version.The main core is the subsequent subsection, where we introduce the Jacobian dual rank and prove the basic characteristic-free criterion of birationality in terms of this rank. The criterion also holds component-wise as possibly predictable (but not obviously proved!). We took pains to transcribe the criterion into purely geometric terms, except for the Jacobian dual rank itself, whose geometric meaning is not entirely apparent at this stage. This concept has evolved continuously from previous notio...

One studies plane Cremona maps by focusing on the ideal theoretic and homological properties of its homogeneous base ideal ("indeterminacy locus"). The leitmotiv driving a good deal of the work is the relation between the base ideal and its saturation. As a preliminary one deals with the homological features of arbitrary codimension 2 homogeneous ideals in a polynomial ring in three variables over a field which are generated by three forms of the same degree. The results become sharp when the saturation is not generated in low degrees, a condition to be given a precise meaning. An implicit goal, illustrated in low degrees, is a homological classification of plane Cremona maps according to the respective homaloidal types. An additional piece of this work relates the base ideal of a rational map to a few additional homogeneous "companion" ideals, such as the integral closure, the µ-fat ideal and a seemingly novel ideal defined in terms of valuations.

Cohen-Macaulayness, unmixedness, the structure of the canonical module and the stability of the Hilbert function of algebraic residual intersections are studied in this paper. Some conjectures about these properties are established for large classes of residual intersections without restricting local number of generators of the ideals involved. A family of approximation complexes for residual intersections is constructed to determine the above properties. Moreover some general properties of the symmetric powers of quotient ideals are determined which were not known even for special ideals with a small number of generators. Acyclicity of a prime case of these complexes is shown to be equivalent to find a common annihilator for higher Koszul homologies. So that, a tight relation between residual intersections and the uniform annihilator of positive Koszul homologies is unveiled that sheds some light on their structure. a residual intersection of X if the number of equations needed to define X ∪ Y as a subscheme of Z is the smallest possible that is s. Precisely, if R is a Noetherian ring, I an ideal of height g and s ≥ g an integer, then • An (algebraic) s-residual intersection of I is a proper ideal J of R such that ht(J) ≥ s and J = (a : R I) for some ideal a ⊂ I generated by s elements. • A geometric s-residual intersection of I is an algebraic s-residual intersection J of I such that ht(I + J) ≥ s + 1.Based on a construction of Laksov for residual intersection, Fulton [Fu, Definition 9.2.2] presents a formulation for residual intersection that, locally, can be expressed as follows: Suppose that X = Spec(R)

Abstract. In this article we study the structure of residual intersections via constructing a finite complex which is acyclic under some sliding depth conditions on the cycles of the The complex we construct also provides a bound for the Castelnuovo-Mumford regularity of a residual intersection in term of the degrees of the minimal generators. More precisely, in a positively graded Cohen-Macaulay *local ring R = n≥0 R n , if J = a : I is a (geometric) s-residual intersection of the ideal I such that ht(I) = g > 0 and satisfies a sliding depth condition, then reg(R/J) ≤ reg(R)+dim(R 0 )+σ(a)−(s−g+1) indeg(I/a)− s, where σ(a) is the sum of the degrees of elements of a minimal generating set of a. It is also shown that the equality holds whenever I is a perfect ideal of height 2, and the base ring R 0 is a field.