Cremona Maps of de Jonquières Type
01 Aug 2015-Canadian Journal of Mathematics (Canadian Mathematical Society)-Vol. 67, Iss: 4, pp 923-941
TL;DR: In this paper, a generalization of a plane de Jonquimap to higher dimensional space P n with n 3 is presented. But the generalization is restricted to the case where n = 3.
Abstract: This paper is concerned with suitable generalizations of a plane de Jonquimap to higher dimensional space P n with n 3. For each given point of P n there is a subgroup of the entire Cre- mona group of dimension n consisting of such maps. We study both geometric and group-theoretical properties of this notion. In the case where n = 3 we describe an explicit set of generators of the group and give a homological characterization of a basic subgroup thereof.
Citations
More filters
TL;DR: In this paper, the authors give formulas and effective sharp bounds for the degree of multi-graded rational maps and provide some effective and computable criteria for birationality in terms of their algebraic and geometric properties.
Abstract: We give formulas and effective sharp bounds for the degree of multi-graded rational maps and provide some effective and computable criteria for birationality in terms of their algebraic and geometric properties. We also extend the Jacobian dual criterion to the multi-graded setting. Our approach is based on the study of blow-up algebras, including syzygies, of the ideal generated by the defining polynomials of the rational map. A key ingredient is a new algebra that we call the saturated special fiber ring, which turns out to be a fundamental tool to analyze the degree of a rational map. We also provide a very effective birationality criterion and a complete description of the equations of the associated Rees algebra of a particular class of plane rational maps.
14 citations
TL;DR: In this article, the authors consider the behavior of a rational map under specialization of the coefficients of the defining linear system and develop the details of rational maps and their graphs when the ground ring of coefficients is a Noetherian domain.
Abstract:
One considers the behavior of the degree of a rational map under specialization of the coefficients of the defining linear system. The method rests on the classical idea of Kronecker as applied to the context of projective schemes and their specializations. For the theory to work, one is led to develop the details of rational maps and their graphs when the ground ring of coefficients is a Noetherian domain.
12 citations
10 Jul 2019
TL;DR: This formula is equal to an elementary symmetric polynomial in terms of the degrees of the syzygies of .
Abstract: Let R be a polynomial ring and let I subset of R be a perfect ideal of height two minimally generated by forms of the same degree. We provide a formula for the multiplicity of the saturated special fiber ring of I. Interestingly, this formula is equal to an elementary symmetric polynomial in terms of the degrees of the syzygies of I. Applying ideas introduced by Buse, D'Andrea, and the author, we obtain the value of the j-multiplicity of I and an effective method for determining the degree and birationality of rational maps defined by homogeneous generators of I.
10 citations
TL;DR: In this article, the authors give formulas and effective sharp bounds for the degree of multi-graded rational maps and provide some effective and computable criteria for birationality in terms of their algebraic and geometric properties.
Abstract: We give formulas and effective sharp bounds for the degree of multi-graded rational maps and provide some effective and computable criteria for birationality in terms of their algebraic and geometric properties. We also extend the Jacobian dual criterion to the multi-graded setting. Our approach is based on the study of blow-up algebras, including syzygies, of the ideal generated by the defining polynomials of the rational map. A key ingredient is a new algebra that we call the "saturated special fiber ring", which turns out to be a fundamental tool to analyze the degree of a rational map. We also provide a very effective birationality criterion and a complete description of the equations of the associated Rees algebra of a particular class of plane rational maps.
9 citations
TL;DR: In this paper, a conceptual revision of the main steps of the notion of complementary duality is presented, based on the work of the second author and B. Costa, and a conceptual interpretation of the three main steps is presented.
Abstract: This work deals with the notion of Newton complementary duality as raised originally in the work of the second author and B. Costa. A conceptual revision of the main steps of the notion is accompli...
8 citations
References
More filters
Book•
30 Mar 1995
TL;DR: In this article, the authors define basic constructions and dimension theory, and apply them to the problem of homological methods for combinatorial problem solving in the context of homology.
Abstract: Introduction.- Elementary Definitions.- I Basic Constructions.- II Dimension Theory.- III Homological Methods.- Appendices.- Hints and Solutions for Selected Exercises.- References.- Index of Notation.- Index.
5,674 citations
Book•
[...]
01 Jan 1993
TL;DR: In this article, the authors present a self-contained introduction to the homological and combinatorial aspects of the theory of Cohen-Macaulay rings, Gorenstein rings, local cohomology, and canonical modules.
Abstract: In the last two decades Cohen-Macaulay rings and modules have been central topics in commutative algebra. This book meets the need for a thorough, self-contained introduction to the homological and combinatorial aspects of the theory of Cohen-Macaulay rings, Gorenstein rings, local cohomology, and canonical modules. A separate chapter is devoted to Hilbert functions (including Macaulay's theorem) and numerical invariants derived from them. The authors emphasize the study of explicit, specific rings, making the presentation as concrete as possible. So the general theory is applied to Stanley-Reisner rings, semigroup rings, determinantal rings, and rings of invariants. Their connections with combinatorics are highlighted, e.g. Stanley's upper bound theorem or Ehrhart's reciprocity law for rational polytopes. The final chapters are devoted to Hochster's theorem on big Cohen-Macaulay modules and its applications, including Peskine-Szpiro's intersection theorem, the Evans-Griffith syzygy theorem, bounds for Bass numbers, and tight closure. Throughout each chapter the authors have supplied many examples and exercises which, combined with the expository style, will make the book very useful for graduate courses in algebra. As the only modern, broad account of the subject it will be essential reading for researchers in commutative algebra.
2,760 citations
Journal Article•
TL;DR: In this paper, the conditions générales d'utilisation (http://www.numdam.org/conditions) of the agreement with the Scuola Normale Superiore di Pisa are described.
Abstract: © Scuola Normale Superiore, Pisa, 1981, tous droits réservés. L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
190 citations
Book•
04 Dec 2001
TL;DR: Inverse Cremona maps and Noether's factorization theorem as mentioned in this paper have also been used to derive characteristic matrices with total principal and homaloidal curves, respectively.
Abstract: Preliminaries.- Plane Cremona maps.- Clebsch's theorems and jacobian.- Composition.- Characteristic matrices.- Total principal and special homaloidal curves.- Inverse Cremona map.- Noether's factorization theorem.
119 citations
TL;DR: In this paper, the existence of irreducible homaloidal hypersurfaces in projective space was studied and an infinite family of determinantal hypersurface based on a certain degeneration of a generic Hankel matrix was introduced.
110 citations