The Hessian polynomial and the Jacobian ideal of a reduced hypersurface in Pn

2021 ◽  
Vol 392 ◽  
pp. 108035
Author(s):  
Laurent Busé ◽  
Alexandru Dimca ◽  
Hal Schenck ◽  
Gabriel Sticlaru
Keyword(s):  
Jacobian Ideal

scholarly journals On Plane Curves with Double and Triple Points

2016 ◽  
Vol 119 (1) ◽  
pp. 60 ◽  
Author(s):  
Nancy Abdallah
Keyword(s):  
Plane Curve ◽  
Plane Curves ◽  
Jacobian Ideal ◽  
Pole Order ◽  
Hodge Filtration ◽  
Triple Points

We describe in simple geometric terms the Hodge filtration on the cohomology $H^*(U)$ of the complement $U=\mathsf{P}^2 \setminus C$ of a plane curve $C$ with ordinary double and triple points. Relations to Milnor algebra, syzygies of the Jacobian ideal and pole order filtration on $H^2(U)$ are given.

scholarly journals Orbifolding Frobenius Algebras

2003 ◽  
Vol 14 (06) ◽  
pp. 573-617 ◽  
Author(s):  
Ralph M. Kaufmann
Keyword(s):  
General Theory ◽  
Gauge Group ◽  
Physical Theory ◽  
Group Actions ◽  
Vector Spaces ◽  
Algebraic Structures ◽  
Jacobian Ideal ◽  
Frobenius Algebras

We study the general theory of Frobenius algebras with group actions. These structures arise when one is studying the algebraic structures associated to a geometry stemming from a physical theory with a global finite gauge group, i.e. orbifold theories. In this context, we introduce and axiomatize these algebras. Furthermore, we define geometric cobordism categories whose functors to the category of vector spaces are parameterized by these algebras. The theory is also extended to the graded and super-graded cases. As an application, we consider Frobenius algebras having some additional properties making them more tractable. These properties are present in Frobenius algebras arising as quotients of Jacobian ideal, such as those having their origin in quasi-homogeneous singularities and their symmetries.

scholarly journals Hypertetrahedral arrangements

2021 ◽  
Author(s):  
Liena Colarte-Gómez ◽  
Laura Costa ◽  
Simone Marchesi ◽  
Rosa M. Miró-Roig ◽  
Marti Salat-Moltó
Keyword(s):  
Graph Theory ◽  
Jacobian Ideal ◽  
Initial Degree ◽  
Syzygy Module

AbstractIn this paper, we introduce the notion of a complete hypertetrahedral arrangement $${\mathcal {A}}$$ A in $${\mathbb {P}}^{n}$$ P n . We address two basic problems. First, we describe the local freeness of $${\mathcal {A}}$$ A in terms of smaller complete hypertetrahedral arrangements and graph theory properties, specializing the Mustaţă–Schenck criterion. As an application, we obtain that general complete hypertetrahedral arrangements are not locally free. In the second part of this paper, we bound the initial degree of the first syzygy module of the Jacobian ideal of $${\mathcal {A}}$$ A .

scholarly journals A comparison formula for residue currents

2019 ◽  
Vol 125 (1) ◽  
pp. 39-66
Author(s):  
Richard Lärkäng
Keyword(s):  
Holomorphic Mapping ◽  
Holomorphic Functions ◽  
Analytic Proof ◽  
Jacobian Ideal ◽  
Residue Currents

Given two ideals $\mathcal {I}$ and $\mathcal {J}$ of holomorphic functions such that $\mathcal {I} \subseteq \mathcal {J}$, we describe a comparison formula relating the Andersson-Wulcan currents of $\mathcal {I}$ and $\mathcal {J}$. More generally, this comparison formula holds for residue currents associated to two generically exact Hermitian complexes together with a morphism between the complexes. One application of the comparison formula is a generalization of the transformation law for Coleff-Herrera products to Andersson-Wulcan currents of Cohen-Macaulay ideals. We also use it to give an analytic proof by means of residue currents of theorems of Hickel, Vasconcelos and Wiebe related to the Jacobian ideal of a holomorphic mapping.

The Valabrega-Valla module of the Jacobian ideal of points in a projective plane

2020 ◽  
Vol 48 (5) ◽  
pp. 2110-2126
Author(s):  
Abbas Nasrollah Nejad ◽  
Zahra Shahidi
Keyword(s):  
Projective Plane ◽  
Jacobian Ideal

scholarly journals The Jacobian ideal of a hyperplane arrangement

2008 ◽  
Vol 15 (4) ◽  
pp. 795-799 ◽  
Author(s):  
Max Wakefield ◽  
Masahiko Yoshinaga
Keyword(s):  
Hyperplane Arrangement ◽  
Jacobian Ideal

scholarly journals Hypersurfaces with linear type singular loci

2019 ◽  
Vol 19 (09) ◽  
pp. 2050169
Author(s):  
Amir Behzad Farrahy ◽  
Abbas Nasrollah Nejad
Keyword(s):  
Projective Plane ◽  
Plane Curve ◽  
Isolated Singularity ◽  
Linear Type ◽  
Jacobian Ideal ◽  
Quartic Curve ◽  
Singular Loci ◽  
Necessary And Sufficient Criteria ◽  
Simple Singularities ◽  
Necessary And Sufficient

In this paper, necessary and sufficient criteria for the Jacobian ideal of a reduced hypersurface with isolated singularity to be of linear type are presented. We prove that the gradient ideal of a reduced projective plane curve with simple singularities ([Formula: see text]) is of linear type. We show that any reduced projective quartic curve is of gradient linear type.

scholarly journals Schemes Supported on the Singular Locus of a Hyperplane Arrangement in $\mathbb P^n$

2020 ◽  
Author(s):  
Juan Migliore ◽  
Uwe Nagel ◽  
Henry Schenck
Keyword(s):  
Positive Integer ◽  
Hyperplane Arrangement ◽  
Singular Locus ◽  
Jacobian Ideal ◽  
Liaison Class

Abstract A hyperplane arrangement in $\mathbb P^n$ is free if $R/J$ is Cohen–Macaulay (CM), where $R = k[x_0,\dots ,x_n]$ and $J$ is the Jacobian ideal. We study the CM-ness of two related unmixed ideals: $ J^{un}$, the intersection of height two primary components, and $\sqrt{J}$, the radical. Under a mild hypothesis, we show these ideals are CM. Suppose the hypothesis fails. For equidimensional curves in $\mathbb P^3$, the Hartshorne–Rao module measures the failure of CM-ness and determines the even liaison class of the curve. We show that for any positive integer $r$, there is an arrangement for which $R/J^{un}$ (resp. $R/\sqrt{J}$) fails to be CM in only one degree, and this failure is by $r$. We draw consequences for the even liaison class of $J^{un}$ or $\sqrt{J}$.

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