On the monodromy of a plane curve singularity and the Poincaré series of the ring of functions on the curve

1999 ◽  
Vol 33 (1) ◽  
pp. 56-57 ◽  
Author(s):  
S. M. Gusein-Zade ◽  
F. Delgado ◽  
A. Campillo
Keyword(s):  
Plane Curve ◽  
Poincaré Series ◽  
Curve Singularity ◽  
Poincare Series ◽  
Plane Curve Singularity ◽  
Ring Of Functions

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.

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