The Poincaré series of divisorial valuations in the plane defines the topology of the set of divisorsстатья

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[1] Campillo A., Delgado F., Gusein-Zade S. M. The poincaré series of divisorial valuations in the plane defines the topology of the set of divisors // Functional Analysis and Other Mathematics. — 2010. — Vol. 3, no. 1. — P. 39–46. With a plane curve singularity one associates a multi-index filtration on the ring of germs of functions of two variables defined by the orders of a function on irreducible components of the curve. The Poincaré series of this filtration turns out to coincide with the Alexander polynomial of the curve germ. For a finite set of divisorial valuations on the ring corresponding to some components of the exceptional divisor of a modification of the plane, in a previous paper there was obtained a formula for the Poincaré series of the corresponding multi-index filtration similar to the one associated with plane germs. Here we show that the Poincaré series of a set of divisorial valuations on the ring of germs of functions of two variables defines “the topology of the set of the divisors” in the sense that it defines the minimal resolution of this set up to combinatorial equivalence. For the plane curve singularity case, we also give a somewhat simpler proof of the statement by Yamamoto which shows that the Alexander polynomial is equivalent to the embedded topology. [ DOI ]

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