Hilbert function, generalized Poincar\'e series and topology of plane valuations

TL;DR

This paper explores the relationships between various invariants of multi-index filtrations on complex analytic varieties, demonstrating how they encode the topology of plane valuations and their resolutions.

Contribution

It establishes the equivalence of the Hilbert function, generalized Poincaré series, and semigroup Poincaré series for filtrations defined by plane valuations, linking these invariants to topology.

Findings

Hilbert function and generalized Poincaré series are equivalent

These invariants determine the topology of plane valuations

The invariants encode the topology of the minimal resolution

Abstract

To a multi-index filtration (say, on the ring of germs of functions on a germ of a complex analytic variety) one associates several invariants: the Hilbert function, the Poincar\'e series, the generalized Poincar\'e series, and the generalized semigroup Poincar\'e series. The Hilbert function and the generalized Poincar\'e series are equivalent in the sense that each of them determines the other one. We show that for a filtration on the ring of germs of holomorphic functions in two variables defined by a collection of plane valuations both of them are equivalent to the generalized semigroup Poincar\'e series and determine the topology of the collection of valuations, i.e. the topology of its minimal resolution.