A characteristic-free criterion of birationality
TL;DR: In this article, the Jacobian dual rank of a rational map is defined and attains its maximal possible value, and it is shown that a rational/birational map is irreducible if and only if it attains this value.
About: This article is published in Advances in Mathematics.The article was published on 2012-05-01 and is currently open access. It has received 82 citations till now. The article focuses on the topics: Invariant (mathematics) & Jacobian matrix and determinant.
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TL;DR: In this paper, the authors deal with arbitrary reduced free divisors in a polynomial ring over a field of characteristic zero, by stressing the ideal theoretic and homological behavior of the corresponding singular locus.
Abstract: This work deals with arbitrary reduced free divisors in a polynomial ring over a field of characteristic zero, by stressing the ideal theoretic and homological behavior of the corresponding singular locus. A particular emphasis is given to both weighted homogeneous and homogeneous polynomials, allowing to introduce new families of free divisors not coming from either hyperplane arrangements or discriminants in singularity theory.
56 citations
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TL;DR: In this paper, the authors consider reduced free divisors in a polynomial ring over a field of characteristic zero, by stressing the ideal theoretic and homological behavior of the corresponding singular locus.
Abstract: One deals with arbitrary reduced free divisors in a polynomial ring over a field of characteristic zero, by stressing the ideal theoretic and homological behavior of the corresponding singular locus. A particular emphasis is given to both weighted homogeneous and homogeneous polynomials, allowing to introduce new families of free divisors which do not come from hyperplane arrangements nor as explicit discriminants from singularity theory.
49 citations
TL;DR: In this paper, the authors focus on the ideal theoretic and homological properties of a plane Cremona map by focusing on its homogeneous base ideal (indeterminacy locus).
47 citations
TL;DR: In this article, a rational map defined by homogeneous forms, of the same degree, in the homogeneous coordinate ring of the graph of a variety parameterized by the Rees algebra is studied.
Abstract: Our object of study is a rational map defined by homogeneous forms , of the same degree , in the homogeneous coordinate ring of . Our goal is to relate properties of , of the homogeneous coordinate ring of the variety parameterized by , and of the Rees algebra , the bihomogeneous coordinate ring of the graph of . For a regular map , for instance, we prove that satisfies Serre’s condition , for some , if and only if satisfies and is birational onto its image. Thus, in particular, is birational onto its image if and only if satisfies . Either condition has implications for the shape of the core, namely, is the multiplier ideal of and Conversely, for , either equality for the core implies birationality. In addition, by means of the generalized rows of the syzygy matrix of , we give an explicit method to reduce the nonbirational case to the birational one when .
30 citations
Cites methods from "A characteristic-free criterion of ..."
...In [41, 16] the method has been advanced by emphasizing the role of th Rees algebra associated to the idealI = (g1, ....
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TL;DR: In this paper, it was shown that the Orlik-Terao algebra is graded isomorphic to the special fiber of the ideal $I$ generated by the $(n-1)$-fold products of the members of a central arrangement of size $n$.
Abstract: It is shown that the Orlik-Terao algebra is graded isomorphic to the special fiber of the ideal $I$ generated by the $(n-1)$-fold products of the members of a central arrangement of size $n$. This momentum is carried over to the Rees algebra (blowup) of $I$ and it is shown that this algebra is of fiber-type and Cohen-Macaulay. It follows by a result of Simis-Vasconcelos that the special fiber of $I$ is Cohen-Macaulay, thus giving another proof of a result of Proudfoot-Speyer about the Cohen-Macauleyness of the Orlik-Terao algebra.
25 citations
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Book•
25 Feb 1994TL;DR: In this paper, the authors provide an introduction to recent developments in the theory of blow up algebras - Rees algesbras, associated graded rings, Hilbert functions, and birational morphisms.
Abstract: This book provides an introduction to recent developments in the theory of blow up algebras - Rees algebras, associated graded rings, Hilbert functions, and birational morphisms. The emphasis is on deriving properties of rings from their specifications in terms of generators and relations. While this limits the generality of many results, it opens the way for the application of computational methods. A highlight of the book is the chapter on advanced computational methods in algebra using Grobner basis theory and advanced commutative algebra. The author presents the Grobner basis algorithm and shows how it can be used to resolve computational questions in algebra. This volume is intended for advanced students in commutative algebra, algebraic geometry and computational algebra, and homological algebra. It can be used as a reference for the theory of Rees algebras and related topics.
301 citations
"A characteristic-free criterion of ..." refers background in this paper
...The literature on this is quite extensive (see [10,11,13] and the book [18])....
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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
108 citations
TL;DR: In this article, the authors introduce the notion of Bourbaki ideals that allow the use of deformation theory to study Rees algebras of modules within a fairly general framework.
Abstract: 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.
93 citations
"A characteristic-free criterion of ..." refers background or methods in this paper
...To conclude we count dimensions according to [13], thereby finding...
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...According to [13], RS(E) SS(E)/(0 :SS(E) s), the symmetric algebra modulo torsion, where s is a suitable regular homogeneous element of S....
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...where RS(E) stands for the Rees algebra of E in the sense of [13]....
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...The literature on this is quite extensive (see [10,11,13] and the book [18])....
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TL;DR: In this article, a general characteristic-free criterion for a rational map between projective varieties to be birational in terms of ideal-theoretic and modulo-theory conditions is presented.
80 citations
"A characteristic-free criterion of ..." refers background or methods in this paper
...Still more recently, the present third author pushed the method further to the study of general rational maps between two integral projective schemes in arbitrary characteristic by an extended ideal-theoretic method emphasizing the role of the Rees algebra associated to the ideal generated by f [9]....
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...Some of the applications in this section envisage solving the questions posed in [9], in that they are characteristic-free....
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...1, (i) ⇒ (ii)], observing that the hypothesis of R (hence S) being a domain is superfluous because the two maps constructed in [9] were inverse...
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...This concept has evolved continuously from previous notions, of which the cradle is the Jacobian dual fibration of [10] (see the subsequent [8] and [9])....
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...The rank property was taken in [9] as definition of equivalent representatives of the same rational map....
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