When the Jacobian algebra of a curve singularity is Gorenstein

1995 ◽  
Vol 64 (3) ◽  
pp. 203-208
Author(s):  
A. Oneto ◽  
E. Zatini
Keyword(s):  
Curve Singularity ◽  
Jacobian Algebra

The semigroup of values of a curve singularity with several branches

1987 ◽  
Vol 59 (3) ◽  
pp. 347-374 ◽  
Author(s):  
Félix Delgado de la Mata
Keyword(s):  
Curve Singularity

scholarly journals Higher Nash blowups

2007 ◽  
Vol 143 (6) ◽  
pp. 1493-1510 ◽  
Author(s):  
Takehiko Yasuda
Keyword(s):  
Algebraic Variety ◽  
Characteristic Zero ◽  
Negative Integer ◽  
Curve Singularity ◽  
Irreducible Curve ◽  
Numerical Monoid

AbstractFor each non-negative integer n we define the nth Nash blowup of an algebraic variety, and call them all higher Nash blowups. When n=1, it coincides with the classical Nash blowup. We study higher Nash blowups of curves in detail and prove that any curve in characteristic zero can be desingularized by its nth Nash blowup with n large enough. Moreover, we completely determine for which n the nth Nash blowup of an analytically irreducible curve singularity in characteristic zero is normal, in terms of the associated numerical monoid.

On the deep structure of the blowing-up of curve singularities

2001 ◽  
Vol 131 (2) ◽  
pp. 227-240 ◽  
Author(s):  
JUAN ELIAS
Keyword(s):  
Tangent Cone ◽  
Hilbert Function ◽  
Deep Structure ◽  
Main Idea ◽  
Main Property ◽  
Finite Union ◽  
Curve Singularity ◽  
Blowing Up ◽  
Curve Singularities ◽  
Invariant Factors

Let C be a germ of curve singularity embedded in (kn, 0). It is well known that the blowing-up of C centred on its closed ring, Bl(C), is a finite union of curve singularities. If C is reduced we can iterate this process and, after a finite number of steps, we find only non-singular curves. This is the desingularization process. The main idea of this paper is to linearize the blowing-up of curve singularities Bl(C) → C. We perform this by studying the structure of [Oscr ]Bl(C)/[Oscr ]C as W-module, where W is a discrete valuation ring contained in [Oscr ]C. Since [Oscr ]Bl(C)/[Oscr ]C is a torsion W-module, its structure is determined by the invariant factors of [Oscr ]C in [Oscr ]Bl(C). The set of invariant factors is called in this paper as the set of micro-invariants of C (see Definition 1·2).In the first section we relate the micro-invariants of C to the Hilbert function of C (Proposition 1·3), and we show how to compute them from the Hilbert function of some quotient of [Oscr ]C (see Proposition 1·4).The main result of this paper is Theorem 3·3 where we give upper bounds of the micro-invariants in terms of the regularity, multiplicity and embedding dimension. As a corollary we improve and we recover some results of [6]. These bounds can be established as a consequence of the study of the Hilbert function of a filtration of ideals g = {g[r,i+1]}i [ges ] 0 of the tangent cone of [Oscr ]C (see Section 2). The main property of g is that the ideals g[r,i+1] have initial degree bigger than the Castelnuovo–Mumford regularity of the tangent cone of [Oscr ]C.Section 4 is devoted to computation the micro-invariants of branches; we show how to compute them from the semigroup of values of C and Bl(C) (Proposition 4·3). The case of monomial curve singularities is especially studied; we end Section 4 with some explicit computations.In the last section we study some geometric properties of C that can be deduced from special values of the micro-invariants, and we specially study the relationship of the micro-invariants with the Hilbert function of [Oscr ]Bl(C). We end the paper studying the natural equisingularity criteria that can be defined from the micro-invariants and its relationship with some of the known equisingularity criteria.

scholarly journals POINCARÉ SERIES FOR SEVERAL PLANE DIVISORIAL VALUATIONS

2003 ◽  
Vol 46 (2) ◽  
pp. 501-509 ◽  
Author(s):  
F. Delgado ◽  
S. M. Gusein-Zade
Keyword(s):  
Plane Curve ◽  
Poincaré Series ◽  
Finite Collection ◽  
Curve Singularity ◽  
Motivic Integration ◽  
Poincare Series ◽  
Plane Curve Singularity ◽  
Functions Of Two Variables ◽  
Irreducible Components ◽  
Divisorial Valuations

AbstractWe compute the (generalized) Poincaré series of the multi-index filtration defined by a finite collection of divisorial valuations on the ring $\mathcal{O}_{\mathbb{C}^2,0}$ of germs of functions of two variables. We use the method initially elaborated by the authors and Campillo for computing the similar Poincaré series for the valuations defined by the irreducible components of a plane curve singularity. The method is essentially based on the notions of the so-called extended semigroup and of the integral with respect to the Euler characteristic over the projectivization of the space of germs of functions of two variables. The last notion is similar to (and inspired by) the notion of the motivic integration.AMS 2000 Mathematics subject classification: Primary 14B05; 16W70

Algorithm for the Semigroup of a Space Curve Singularity

2004 ◽  
Vol 70 (1) ◽  
pp. 44-60 ◽  
Author(s):  
Angel Castellanos ◽  
Julio Castellanos
Keyword(s):  
Space Curve ◽  
Curve Singularity

scholarly journals ON RATIONAL CUSPIDAL PROJECTIVE PLANE CURVES

2005 ◽  
Vol 92 (1) ◽  
pp. 99-138 ◽  
Author(s):  
J. FERNÁNDEZ DE BOBADILLA ◽  
I. LUENGO-VELASCO ◽  
A. MELLE-HERNÁNDEZ ◽  
A. NÉMETHI
Keyword(s):  
Singular Point ◽  
Projective Plane ◽  
Plane Curve ◽  
Topological Type ◽  
Homology Sphere ◽  
Plane Curves ◽  
Curve Singularity ◽  
Witten Invariant ◽  
Rational Homology ◽  
Rational Homology Sphere

In 2002, L. Nicolaescu and the fourth author formulated a very general conjecture which relates the geometric genus of a Gorenstein surface singularity with rational homology sphere link with the Seiberg--Witten invariant (or one of its candidates) of the link. Recently, the last three authors found some counterexamples using superisolated singularities. The theory of superisolated hypersurface singularities with rational homology sphere link is equivalent with the theory of rational cuspidal projective plane curves. In the case when the corresponding curve has only one singular point one knows no counterexample. In fact, in this case the above Seiberg--Witten conjecture led us to a very interesting and deep set of `compatibility properties' of these curves (generalising the Seiberg--Witten invariant conjecture, but sitting deeply in algebraic geometry) which seems to generalise some other famous conjectures and properties as well (for example, the Noether--Nagata or the log Bogomolov--Miyaoka--Yau inequalities). Namely, we provide a set of `compatibility conditions' which conjecturally is satisfied by a local embedded topological type of a germ of plane curve singularity and an integer $d$ if and only if the germ can be realized as the unique singular point of a rational unicuspidal projective plane curve of degree $d$. The conjectured compatibility properties have a weaker version too, valid for any rational cuspidal curve with more than one singular point. The goal of the present article is to formulate these conjectured properties, and to verify them in all the situations when the logarithmic Kodaira dimension of the complement of the corresponding plane curves is strictly less than 2.

NON-NEGATIVE DEFORMATIONS OF WEIGHTED HOMOGENEOUS SINGULARITIES

2017 ◽  
Vol 60 (1) ◽  
pp. 175-185 ◽  
Author(s):  
J. J. NUÑO-BALLESTEROS ◽  
B. ORÉFICE-OKAMOTO ◽  
J. N. TOMAZELLA
Keyword(s):  
Sufficient Conditions ◽  
Additional Term ◽  
Milnor Number ◽  
Curve Singularity ◽  
Hypersurface Singularity ◽  
Weighted Degree ◽  
Function Germ ◽  
Analytic Variety ◽  
Complex Analytic Variety ◽  
Necessary And Sufficient

AbstractWe consider a weighted homogeneous germ of complex analytic variety (X, 0) ⊂ (ℂn, 0) and a function germ f : (ℂn, 0) → (ℂ, 0). We derive necessary and sufficient conditions for some deformations to have non-negative degree (i.e., for any additional term in the deformation, the weighted degree is not smaller) in terms of an adapted version of the relative Milnor number. We study the cases where (X, 0) is an isolated hypersurface singularity and the invariant is the Bruce-Roberts number of f with respect to (X, 0), and where (X, 0) is an isolated complete intersection or a curve singularity and the invariant is the Milnor number of the germ f: (X, 0) → ℂ. In the last part, we give some formulas for the invariants in terms of the weights and the degrees of the polynomials.

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