Arkiv för Matematik

Volume 60 (2022)

Number 2

Weil–Poincaré series and topology of collections of valuations on rational double points

Pages: 297 – 322

DOI: https://dx.doi.org/10.4310/ARKIV.2022.v60.n2.a4

Authors

A. Campillo (IMUVA (Instituto de Investigación en Matemáticas), Universidad de Valladolid, Paseo de Belén, Valladolid, Spain)

F. Delgado (IMUVA (Instituto de Investigación en Matemáticas), Universidad de Valladolid, Paseo de Belén, Valladolid, Spain)

S.M. Gusein-Zade (Faculty of Mathematics and Mechanics, Moscow Center for Fundamental and Applied Mathematics, Moscow State University, Moscow, Russia; and National Research University “Higher School of Economics”, Moscow, Russia)

Abstract

Earlier it was described to which extent the Alexander polynomial in several variables of an algebraic link in the Poincaré sphere determines the topology of the link. It was shown that, except some explicitly described cases, the Alexander polynomial of an algebraic link determines the combinatorial type of the minimal resolution of the curve and therefore the topology of the corresponding link. The Alexander polynomial of an algebraic link in the Poincaré sphere coincides with the Poincaré series of the corresponding set of curve valuations. The latter one can be defined as an integral over the space of divisors on the $\mathbb{E}_8$-singularity. Here, we consider a similar integral for rational double point surface singularities over the space of Weil divisors called the Weil–Poincaré series. We show that, except a few explicitly described cases, the Weil–Poincaré series of a collection of curve valuations on a rational double point surface singularity determines the topology of the corresponding link. We give analogous statements for collections of divisorial valuations.

Keywords

rational double points, algebraic links, Poincaré series, topological type

2010 Mathematics Subject Classification

14B05, 16W60, 32S25, 57M27

Full Text (PDF format)

The first two authors were partially supported by MCIN/AEI/10.13039/501100011033 and by “ERDF A Way of Making Europe”, grant PGC2018-096446-B-C21. The work of the third author (Sections 1, 3 and 5) was supported by the grant 21-11-00080 of the Russian Science Foundation.

Received 1 December 2021

Received revised 20 April 2022

Accepted 18 May 2022

Published 26 October 2022