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Multi-index filtrations and generalized Poincaré series. (English) Zbl 1111.14020

Summary: A multi-index filtration on the ring of germs of functions can be described by its Poincaré series. We consider a finer invariant (or rather two invariants) of a multi-index filtration than the Poincaré series generalizing the last one. The construction is based on the fact that the Poincaré series can be written as a certain integral with respect to the Euler characteristic over the projectivization of the ring of functions. The generalization of the Poincaré series is defined as a similar integral with respect to the generalized Euler characteristic with values in the Grothendieck ring of varieties. For the filtration defined by orders of functions on the components of a plane curve singularity \(C\) and for the so called divisorial filtration for a modification of \(({\mathbb C}^2,0)\) by a sequence of blowing-ups there are given formulae for this generalized Poincaré series in terms of an embedded resolution of the germ \(C\) or in terms of the modification respectively. The generalized Euler characteristic of the extended semigroup corresponding to the divisorial filtration is computed giving a curious “motivic version” of an A’Campo type formula.

MSC:

14H20 Singularities of curves, local rings
14B05 Singularities in algebraic geometry
32S99 Complex singularities
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[12] Looijenga E (2002) Motivic measures. In: Séminaire Bourbaki 1999/2000. Astérisque 276: 267–297
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