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View article: Quasihomogeneous isolated singularities in terms of syzygies and foliations
Quasihomogeneous isolated singularities in terms of syzygies and foliations Open
One considers quasihomogeneous isolated singularities of hypersurfaces in arbitrary dimensions through the lenses of three apparently quite apart themes: syzygies, singularity invariants, and foliations. In the first of these, one adds to …
View article: Additions to a theorem of Morey-Ulrich
Additions to a theorem of Morey-Ulrich Open
Let $R = k[x_1,\ldots, x_d]$ denote a standard graded polynomial ring over an algebraically closed field $k$, and let $I \subset R$ be a perfect ideal of codimension $2$ with an $n\times (n-1)$ linear presentation matrix $ϕ$. We prove an e…
View article: Cohen--Macaulay ideals of codimension two and the geometry of plane points
Cohen--Macaulay ideals of codimension two and the geometry of plane points Open
We consider classes of codimension two Cohen--Macaulay ideals over a standard graded polynomial ring over a field. We revisit Vasconcelos' problem on $3\times 2$ matrices with homogeneous entries and describe the homological details of Ger…
View article: Rose-Terao-Yuzvinsky theorem for reduced forms
Rose-Terao-Yuzvinsky theorem for reduced forms Open
Yuzvinsky and Rose-Terao have shown that the homological dimension of the gradient ideal of the defining polynomial of a generic hyperplane arrangement is maximum possible. In this work one provides yet another proof of this result, which …
View article: Matrices over polynomial rings approached by commutative algebra
Matrices over polynomial rings approached by commutative algebra Open
The main goal of the paper is the discussion of a deeper interaction between matrix theory over polynomial rings over a field and typical methods of commutative algebra and related algebraic geometry. This is intended in the sense of bring…
View article: Bounds on the degrees of vector fields
Bounds on the degrees of vector fields Open
In this article, we study the generalized Poincare problem from the opposite perspective, by establishing lower bounds on the degree of the vector field in terms of invariants of the variety.
View article: The Bourbaki Degree of Plane Projective Curves
The Bourbaki Degree of Plane Projective Curves Open
Bourbaki sequences and Bourbaki ideals have been studied by several authors since its inception sixty years ago circa. Generic Bourbaki sequences have been thoroughly examined by the senior author with B. Ulrich and W. Vasconcelos, but due…
View article: De Jonquières transformations in arbitrary dimension. An ideal theoretic view
De Jonquières transformations in arbitrary dimension. An ideal theoretic view Open
A generalization of the plane de Jonquières transformation to arbitrary dimension is studied, with an eye for the ideal theoretic side. In particular, one considers structural properties of the corresponding base ideal and of its defining …
View article: Generic freeness of local cohomology and graded specialization
Generic freeness of local cohomology and graded specialization Open
The main focus is the generic freeness of local cohomology modules in a graded setting. The present approach takes place in a quite nonrestrictive setting, by solely assuming that the ground coefficient ring is Noetherian. Under additional…
View article: ON THE JACOBIAN IDEAL OF CENTRAL ARRANGEMENTS
ON THE JACOBIAN IDEAL OF CENTRAL ARRANGEMENTS Open
Let $\mathcal{A}$ denote a central hyperplane arrangement of rank $n$ in affine space $\mathbb{K}^n$ over an infinite field $\mathbb{K}$ and let $l_1,\ldots, l_m\in R:= \mathbb K[x_1,\ldots,x_n]$ denote the linear forms defining the corres…
View article: Equigenerated ideals of analytic deviation one
Equigenerated ideals of analytic deviation one Open
The overall goal is to approach the Cohen--Macaulay property of the special fiber $\mathcal{F}(I)$ of an equigenerated homogeneous ideal $I$ in a standard graded ring over an infinite field. When the ground ring is assumed to be local, the…
View article: Degree of Rational Maps versus Syzygies
Degree of Rational Maps versus Syzygies Open
One proves a far-reaching upper bound for the degree of a generically finite rational map between projective varieties over a base field of arbitrary characteristic. The bound is expressed as a product of certain degrees that appear natura…
View article: Degree of Rational Maps and Specialization
Degree of Rational Maps and Specialization Open
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 thei…
View article: Equigenerated Gorenstein ideals of codimension three
Equigenerated Gorenstein ideals of codimension three Open
We focus on the structure of a homogeneous Gorenstein ideal $I$ of codimension three in a standard polynomial ring $R=\kk[x_1,\ldots,x_n]$ over a field $\kk$, assuming that $I$ is generated in a fixed degree $d$. For such an ideal $I$ this…
View article: Coordinate sections of generic Hankel matrices
Coordinate sections of generic Hankel matrices Open
One deals with degenerations by coordinate sections of the square generic Hankel matrix over a field $k$ of characteristic zero, along with its main related structures, such as the determinant of the matrix, the ideal generated by its part…
View article: Generic freeness of local cohomology and graded specialization
Generic freeness of local cohomology and graded specialization Open
The main focus is the generic freeness of local cohomology modules in a graded setting. The present approach takes place in a quite nonrestrictive setting, by solely assuming that the ground coefficient ring is Noetherian. Under additional…
View article: Degree of rational maps via specialization
Degree of rational maps via specialization Open
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 thei…
View article: The depth of the Rees algebra of three general binary forms
The depth of the Rees algebra of three general binary forms Open
One proves that the Rees algebra of an ideal generated by three general binary forms of same degree $\geq 5$ has depth one. The proof hinges on the behavior of the Ratliff-Rush filtration for low powers of the ideal and on establishing tha…
View article: A blowup algebra for hyperplane arrangements
A blowup algebra for hyperplane arrangements Open
It is shown that the Orlik-Terao algebra is graded isomorphic to the special\nfiber of the ideal $I$ generated by the $(n-1)$-fold products of the members of\na central arrangement of size $n$. This momentum is carried over to the Rees\nal…
View article: Degenerations of the generic square matrix, the polar map and the determinantal structure
Degenerations of the generic square matrix, the polar map and the determinantal structure Open
One studies certain degenerations of the generic square matrix over a field [Formula: see text] along with its main related structures, such as the determinant of the matrix, the ideal generated by its partial derivatives, the polar map de…
View article: Linearly presented perfect ideals of codimension $2$ in three variables
Linearly presented perfect ideals of codimension $2$ in three variables Open
The goal of this paper is the fine structure of the ideals in the title, with emphasis on the properties of the associated Rees algebra and the special fiber. The watershed between the present approach and some of the previous work in the …
View article: Symbolic powers of monomial ideals and Cohen-Macaulay vertex-weighted digraphs
Symbolic powers of monomial ideals and Cohen-Macaulay vertex-weighted digraphs Open
In this paper we study irreducible representations and symbolic Rees algebras of monomial ideals. Then we examine edge ideals associated to vertex-weighted oriented graphs. These are digraphs having no oriented cycles of length two with we…
View article: Homaloidal nets and ideals of fat points II: subhomaloidal nets
Homaloidal nets and ideals of fat points II: subhomaloidal nets Open
This paper is a natural sequel to [22] in that it tackles problems of the same nature. Here one aims at the ideal theoretic and homological properties of a class of ideals of general plane fat points whose second symbolic powers hold virtu…