Premium
On the topological type of a set of plane valuations with symmetries
Author(s) -
Campillo A.,
Delgado F.,
GuseinZade S. M.
Publication year - 2017
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201600198
Subject(s) - mathematics , monodromy , homogeneous space , type (biology) , pure mathematics , plane curve , plane (geometry) , group (periodic table) , polynomial , gravitational singularity , function (biology) , mathematical analysis , geometry , ecology , chemistry , organic chemistry , evolutionary biology , biology
Let { C i : i = 1 , … , r } be a set of irreducible plane curve singularities. For an action of a finite group G , letΔ L ( { t a i } )be the Alexander polynomial in r | G | variables of the algebraic link( ⋃ i = 1 r ⋃ a ∈ Ga C i ) ∩ S ε 3and let ζ ( t 1 , … , t r ) = Δ L ( t 1 , … , t 1 , t 2 , … , t 2 , … , t r , … , t r )with | G | identical variables in each group. (If r = 1 , ζ ( t ) is the monodromy zeta function of the function germ∏ a ∈ Ga ∗ f , where f = 0 is an equation defining the curve C 1 .) We prove that ζ ( t 1 , … , t r ) determines the topological type of the link L . We prove an analogous statement for plane divisorial valuations formulated in terms of the Poincaré series of a set of valuations.