No abstract

We introduce various families of irreducible homaloidal hypersurfaces in projective space P r , for all r ≥ 3. Some of these are families of homaloidal hypersurfaces whose degrees are arbitrarily large as compared to the dimension of the ambient projective space. The existence of such a family solves a question that has naturally arisen from the consideration of the classes of homaloidal hypersurfaces known so far. The result relies on a fine analysis of hypersurfaces that are dual to certain scroll surfaces. We also introduce an infinite family of determinantal homaloidal hypersurfaces based on a certain degeneration of a generic Hankel matrix. The latter family fit non-classical versions of de Jonquières transformations. As a natural counterpoint, we broaden up aspects of the theory of Gordan-Noether hypersurfaces with vanishing Hessian determinant, bringing over some more precision into the present knowledge. * This author thanks CNPq for support and the Departamento de Matemática at Recife for hospitality during the preparation of this paper.

We study Rees algebras of modules within a fairly general framework. We introduce an approach through the notion of Bourbaki ideals that allows the use of deformation theory. One can talk about the (essentially unique) generic Bourbaki ideal I(E) of a module E which, in many situations, allows one to reduce the nature of the Rees algebra of E to that of its Bourbaki ideal I(E). Properties such as Cohen–Macaulayness, normality and being of linear type are viewed from this perspective. The known numerical invariants, such as the analytic spread, the reduction number and the analytic deviation, of an ideal and its associated algebras are considered in the case of modules. Corresponding notions of complete intersection, almost complete intersection and equimultiple modules are examined in some detail. Special consideration is given to certain modules which are fairly ubiquitous because interesting vector bundles appear in this way. For these modules one is able to estimate the reduction number and other invariants in terms of the Buchsbaum–Rim multiplicity. 2000 Mathematics Subject Classification 13A30 (primary), 13H10, 13B21 (secondary)

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...

No abstract

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.

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.

No abstract