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 POINCARÉ SERIES OF COLLECTIONS OF PLANE VALUATIONS

2010 ◽  
Vol 21 (11) ◽  
pp. 1461-1473 ◽  
Author(s):  
A. CAMPILLO ◽  
F. DELGADO ◽  
S. M. GUSEIN-ZADE ◽  
F. HERNANDO
Keyword(s):  
Plane Curve ◽  
Poincaré Series ◽  
Poincare Series ◽  
Multi Index ◽  
Plane Curve Singularities ◽  
Functions Of Two Variables ◽  
Curve Singularities ◽  
Divisorial Valuations ◽  
General Collection ◽  
Reducible Plane

In earlier papers there were given formulae for the Poincaré series of multi-index filtrations on the ring [Formula: see text] of germs of functions of two variables defined by collections of valuations corresponding to (reducible) plane curve singularities and by collections of divisorial ones. It was shown that the Poincaré series of a collection of divisorial valuations determines the topology of the collection of divisors. Here we give a formula for the Poincaré series of a general collection of valuations on the ring [Formula: see text] centered at the origin and prove a generalization of the statement that the Poincaré series determines the topology of the collection.

THE ALEXANDER POLYNOMIAL OF A PLANE CURVE SINGULARITY AND INTEGRALS WITH RESPECT TO THE EULER CHARACTERISTIC

2003 ◽  
Vol 14 (01) ◽  
pp. 47-54 ◽  
Author(s):  
A. CAMPILLO ◽  
F. DELGADO ◽  
S. M. GUSEIN-ZADE
Keyword(s):  
Euler Characteristic ◽  
Plane Curve ◽  
Alexander Polynomial ◽  
Curve Singularity ◽  
Motivic Integration ◽  
Poincaré Polynomial ◽  
Plane Curve Singularity ◽  
Functions Of Two Variables ◽  
Several Variables ◽  
Reducible Plane

It was shown that the Alexander polynomial (of several variables) of a (reducible) plane curve singularity coincides with the (generalized) Poincaré polynomial of the multi-indexed filtration defined by the curve on the ring [Formula: see text] of germs of functions of two variables. The initial proof of the result was rather complicated (it used analytical, topological and combinatorial arguments). Here we give a new proof based on the notion of the integral with respect to the Euler characteristic over the projectivization of the space [Formula: see text] — the notion similar to (and inspired by) the notion of the motivic integration.

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.

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