A topological interpretation for the polar quotients of an algebraic plane curve singularity

1988 ◽  
pp. 122-143 ◽  
Author(s):  
Claude Weber
Keyword(s):  
Plane Curve ◽  
Curve Singularity ◽  
Plane Curve Singularity ◽  
Topological Interpretation

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

scholarly journals Monotonic invariants under blowups

2020 ◽  
Vol 31 (12) ◽  
pp. 2050093
Author(s):  
Zhenjian Wang
Keyword(s):  
General Framework ◽  
Plane Curve ◽  
Curve Singularity ◽  
Numerical Invariant ◽  
Plane Curve Singularity ◽  
New Perspective

We prove that the numerical invariant [Formula: see text] of a reduced irreducible plane curve singularity germ is non-negative, non-decreasing under blowups and strictly increasing unless the curve is non-singular. This provides a new perspective to understand the question posed by Dimca and Greuel. Moreover, our work can be put in the general framework of discovering monotonic invariants under blowups.

scholarly journals Improving the computation of invariants of plane curve singularities

2013 ◽  
Vol 21 (1) ◽  
pp. 51-57
Author(s):  
Muhammad Ahsan Binyamin
Keyword(s):  
Incidence Matrix ◽  
Plane Curve ◽  
Milnor Number ◽  
Curve Singularity ◽  
Plane Curve Singularity ◽  
Plane Curve Singularities ◽  
Resolution Graph ◽  
Curve Singularities

Abstract In this article we present an algorithm to compute the incidence matrix of the resolution graph, the total multiplicities, the strict multiplicities and the Milnor number of a reduced plane curve singularity and its implemetation in Singular

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