Equivariant Poincaré Series

and Topology of Valuations

A. Campillo, F. Delgado^{1}, and S.M. Gusein-Zade ^{2}

Received: September 7, 2015 Revised: February 10, 2016

Communicated by Thomas Peternell

Abstract. The equivariant with respect to a finite group action Poincaré series of a collection of r valuations was defined earlier as a power series in r variables with the coefficients from a modification of the Burnside ring of the group. Here we show that (modulo simple exceptions) the equivariant Poincaré series determines the equivariant topology of the collection of valuations.

2010 Mathematics Subject Classification: 14B05, 13A18, 14R20, 16W70. Keywords and Phrases: finite group actions, Poincaré series, plane valuations, equivariant topology

1 Introduction

A definition of the Poincaré series of a multi-index filtration was first given in [3] (for filtrations defined by collections of valuations). It is a formal power series in several variables with integer coefficients, i.e., an element of the ring ℤ[[t_{1}, …, t_{r}]]. In [1] it was shown that, for the filtration defined by the curve valuations corresponding to the irreducible components of a plane curve singularity, the Poincaré series coincides with the Alexander polynomial in several variables of the corresponding algebraic link: the intersection of the curve with a small sphere in ℂ^{2} centred at the origin. This relation was obtained by a direct computation of the both sides in the same terms. Up to now there exist no conceptual proof of it. The Alexander polynomial in several variables of an algebraic link (and therefore the Poincaré series of the corresponding collection of valuations) determines the topological type of the corresponding plane curve singularity. In [2] the definition of the Poincaré series was reformulated in terms of an integral with respect to the Euler characteristics (over an infinite dimensional space). The desire to understand deeper this relation led to attempts to find an equivariant version of it (for actions of a finite group G) and thus to define equivariant versions of the Poincaré series and of the Alexander polynomial. Some equivariant versions of the monodromy zeta-function (that is of the Alexander polynomial in one variable) were defined in [9] and [10]. Equivariant versions of the Poincaré series were defined in [4], [5] and [7]. In some constructions of equivariant analogues of invariants (especially those related to the Euler characteristic) the role of the ring of integers ℤ (where the Euler characteristic takes values) is played by the Burnside ring A(G) of the group G. Therefore it would be attractive to define equivariant versions of the Poincaré series as elements of the ring A(G)[[t_{1}, …, t_{r}]] (or of a similar one). The equivariant versions of the monodromy zeta functions defined in [9] and [10] are formal power series with the coefficients from A(G) ⊗ ℚ and A(G) respectively. In [4] the equivariant Poincaré series was defined as an element of the ring R_{1}(G)[[t_{1}, …, t_{r}]] of formal power series in t_{1}, …, t_{r} with the coefficients from the subring R_{1}(G) of the ring R(G) of complex representations of the group G generated by the one-dimensional representations. This Poincaré series turned out to be useful for some problems: see, e.g., [8], [11]. However, it seems to be rather “degenerate”, especially for non-abelian groups. In [5] the G-equivariant Poincaré series P_{{ν_{i}}}^{G} of a collection of valuations (or order functions) {ν_{i}} was not in fact a series, but an element of the Grothendieck ring of so called locally finite (G, r)-sets. This Grothendieck ring was rather big and complicated, the Poincaré series P_{{ν_{i}}}^{G} was rather complicated as well and contained a lot of information about the valuations and the G-action. In particular, for curve and divisorial valuations on the ring O_{ℂ^{2},0} of functions in two variables the information contained in this Poincaré series was (almost) sufficient to restore the action of G on ℂ^{2} and the G-equivariant topology of the set of valuations: [6]. In [7] the equivariant Poincaré series P_{{ν_{i}}}^{G}(t_{1}, …, t_{r}) was defined as an element

1Supported by the grants MTM2012-36917-C03-01 / 02 (first grant with the help of FEDER Program).

2Supported by the grants RFBR–13-01-00755, NSh–5138.2014.1.

Documenta Mathematica 21 (2016) 271–286

Equivariant Poincaré Series and Topology of Valuations 273

of the ring Ã(G)[[t_{1}, …, t_{r}]] of formal power series in the variables t_{1}, …, t_{r} with the coefficients from a certain modification Ã(G) of the Burnside ring A(G) of the group G. A simple reduction of this Poincaré series is an element of the ring A(G)[[t_{1}, …, t_{r}]]. Thus it is somewhat close to the (“idealistic”) model discussed above. However, in order to define the equivariant Poincaré series of this form, it was necessary to lose quite a lot of information about the individual valuations from the collection. (It is possible to say that one used averaging of the information over the group.) Thus it was not clear how much information does it keep. Equivariant topology of germs of plane curves seems to be much more involved than the usual (non-equivariant) one. For example, it is unclear whether the equivariant topology of a collection of curves always determines the equivariant Poincaré series of the collection. Here we discuss to which extent the G-equivariant Poincaré series from [7] determines the topology of a set of plane valuations. We show that the G-equivariant Poincaré series of a collection of divisorial valuations determines the equivariant topology of this collection (in a natural “weak” sense: see below). We also show that with some minor exceptions the equivariant Poincaré series of a collection of curve valuations determines the weak equivariant topology of the collection. (This answer resembles the one from [6]. However reasons for that (and thus the proofs) are quite different. The version of the equivariant Poincaré series considered in [6] is apriori a much more fine invariant than that considered here.) The G-equivariant Poincaré series P_{{ν_{i}}}^{G} considered in [5] depends essentially on the set of valuations defining the filtration. In particular, the substitution of one of them (say, ν_{i}) by its shift a^{∗}ν_{i}, a ∈ G, changes the G-equivariant Poincaré series P_{\{ν_{i}\}}^{G}. The Poincaré series P_{\{ν_{i}\}}^{G}(\underline{t}) considered in [7] depends not on the valuations ν_{i} themselves, but on their G-orbits. The substitution of one of them by its shift does not change the G-equivariant Poincaré series P_{\{ν_{i}\}}^{G}(\underline{t}). Therefore this series cannot determine the G-topology of a collection of divisorial and/or of curve valuations on O_{ℂ^{2},0} in the form defined in [6]. One has to modify this notion a little bit. Assume first that we consider sets of curve valuations. Let {C_{i}}_{i=1}^{r} and {C′_{i}}_{i=1}^{r} be two collections of branches (that is of irreducible plane curve singularities) in the complex plane (ℂ^{2}, 0) with an action of a finite group G. We shall say that these collections are weakly G-topological equivalent if there exists a G-invariant germ of a homeomorphism ψ : (ℂ^{2}, 0) → (ℂ^{2}, 0) such that for each i = 1, …, r one has ψ(C_{i}) = a_{i}C′_{i} with an element a_{i} ∈ G (i.e if the image of the G-orbit of the branch C_{i} coincides with the G-orbit of the branch C′_{i}). To formulate an analogue of this definition for collections of divisorial valuations, one can describe a divisorial valuation ν on O_{ℂ^{2},0} by a pair of curvettes intersecting the corresponding divisor (transversally) at different points. The corresponding pair of curvettes allows to determine the divisor as the last one (and so the unique one with self-intersection equal to −1) appearing in the minimal embedded resolution of them. Two collections of divisorial valuations {ν_{i}}_{i=1}^{r} and {ν′_{i}}_{i=1}^{r} described by the corresponding collections of curvettes {L_{ij}}_{i=1,j=1,2}^{r} and {L′_{ij}}_{i=1,j=1,2}^{r} respectively are weakly G-topologically equivalent if there exists a G-invariant germ of a homeomorphism ψ : (ℂ^{2}, 0) → (ℂ^{2}, 0) such that for each i = 1, …, r one has ψ(L_{ij}) = a_{i}L′_{ij} for j = 1, 2 and an element a_{i} ∈ G. One has an obvious analogue of Theorem 2.9 from [6]. This means that, for a fixed representation of the group G on ℂ^{2}, the weak topology of a collection of curve or/and divisorial valuations on O_{ℂ^{2},0} is determined by the G-resolution graph Γ^{G} of the collection (where not individual branches or/and divisors, but their orbits are indicated) plus the correspondence between the tails of this graph emerging from special points of the first component of the exceptional divisor with these special points (see below).

2 Equivariant Poincaré series

Let us briefly recall the definition of the G-equivariant Poincaré series P_{{ν_{i}}}^{G}(t_{1}, …, t_{r}) of a collection of order functions on the ring O_{V,0} of germs of functions on (V, 0) and the equation for it in terms of a G-equivariant resolution of curve or/and divisorial plane valuations which will be used here.

Definition: A finite equipped G-set is a pair X̃ = (X, α) where:

• X is a finite G-set;

• α associates to each point x ∈ X a one-dimensional representation α_{x} of the isotropy subgroup G_{x} = {a ∈ G : ax = x} of the point x so that, for a ∈ G, one has α_{ax}(b) = α_{x}(a^{−1}ba), where b ∈ G_{ax} = aG_{x}a^{−1}.

Let Ã(G) be the Grothendieck group of finite equipped G-sets. The cartesian product defines a ring structure on it. The class of an equipped G-set X̃ in the Grothendieck ring Ã(G) will be denoted by [X̃]. As an abelian group Ã(G) is freely generated by the classes of the irreducible equipped G-sets [G/H]_{α} for all the conjugacy classes [H] of subgroups of G and for all one-dimensional representations α of H (a representative of the conjugacy class [H] ∈ Conjsub G). There is a natural homomorphism ρ from the ring Ã(G) to the Burnside rings A(G) of the group G defined by forgetting the one-dimensional representation corresponding to the points. The reduction ρ̂: Ã(G) → ℤ is defined by forgetting the representations and the G-action. There are natural pre-λ-structure on a rings A(G) and Ã(G) which give sense for the expressions of the form (1−t)^{−[X]}, [X] ∈ A(G), and (1−t)^{−[X̃]}, [X̃] ∈ Ã(G) respectively: see [7]. Both ρ and ρ̂ are homomorphisms of pre-λ-rings. Let (V, 0) be a germ of a complex analytic space with an action of a finite group G and let O_{V,0} be the ring of germs of functions on it. Without loss of generality we assume that the G-action on (V, 0) is faithful. The group G acts on O_{V,0} by a^{∗}f(z) = f(a^{−1}z) (z ∈ V, a ∈ G). A valuation ν on the ring O_{V,0} is a function ν : O_{V,0} → ℤ_{≥0} ∪ {+∞} such that:

1) ν(λf) = ν(f) for λ ∈ ℂ^{∗};

2) ν(f + g) ≥ min{ν(f), ν(g)};

3) ν(fg) = ν(f) + ν(g).

A function ν : O_{V,0} → ℤ_{≥0} ∪ {+∞} which possesses the properties 1) and 2) is called an order function. Let ν_{1}, …, ν_{r} be a collection of order functions on O_{V,0}. It defines an r-index filtration on O_{V,0}:

J(\underline{v}) = \{h ∈ O_{V,0} : \underline{ν}(h) ≥ \underline{v}\} ,

where \underline{v} = (v_{1}, …, v_{r}) ∈ ℤ_{≥0}^{r}, \underline{ν}(h) = (ν_{1}(h), …, ν_{r}(h)) and \underline{v}′ = (v′_{1}, …, v′_{r}) ≥ \underline{v}′′ = (v′′_{1}, …, v′′_{r}) if and only if v′_{i} ≥ v′′_{i} for all i. Let ω_{i} : O_{V,0} → ℤ_{≥0} ∪ {+∞} be defined by ω_{i} = ∑_{a∈G} a^{∗}ν_{i}. The functions ω_{i} are G-invariant (they are not, in general, order functions). For an element h ∈ ℙO_{V,0}, that is for a function germ considered up to a constant factor, let G_{h} be the isotropy subgroup G_{h} = {a ∈ G : a^{∗}h = α_{h}(a)h} and let Gh ≅ G/G_{h} be the orbit of h in ℙO_{V,0}. The correspondence a ↦ α_{h}(a) ∈ ℂ^{∗} determines a onedimensional representation α_{h} of the subgroup G_{h}. Let X̃_{h} = [G/G_{h}]_{α_{h}} be the element of the ring Ã(G) represented by the G-set Gh with the representation α_{a^{∗}h} associated to the point a^{∗}h ∈ Gh (a ∈ G). The correspondence h ↦ X̃_{h} defines a function (X̃) on ℙO_{V,0}/G with values in Ã(G). The equivariant Poincaré series P_{\{ν_{i}\}}^{G}(\underline{t}) of the collection \{ν_{i}\} is defined by the equation

P_{\{ν_{i}\}}^{G}(\underline{t}) = ∫_{ℙO_{V,0}/G} X̃_{h}\underline{t}^{\underline{ω}(h)}dχ ∈ Ã(G)[[t_{1}, …, t_{r}]] ,  (1)

where \underline{t} := (t_{1}, …, t_{r}), \underline{t}^{\underline{ω}(h)} = t_{1}^{ω_{1}(h)} · . . . · t_{r}^{ω_{r}(h)}, t_{i}^{+∞} should be regarded as 0. The precise meaning of this integral see in [7]. Applying the reduction homomorphism ρ : Ã(G) → A(G) to the Poincaré series P_{\{ν_{i}\}}^{G}(\underline{t}), i.e. to its coefficients, one gets the series ρP_{\{ν_{i}\}}^{G}(\underline{t}) ∈ A(G)[[t_{1}, …, t_{r}]], i.e. a power series with the coefficients from the (usual) Burnside ring. Applying the homomorphism ρ̂ : Ã(G) → ℤ one gets the series ρ̂P_{\{ν_{i}\}}^{G}(\underline{t}) ∈ ℤ[[t_{1}, …, t_{r}]]. One has

ρ̂P_{\{ν_{i}\}}^{G}(\underline{t}) = P_{\{a^{∗}ν_{i}\}}(t_{1}, …, t_{1}, t_{2}, …, t_{2}, …, t_{r}, …, t_{r}) ,

where P_{{a^{∗}ν_{i}}}(•) is the usual (non-equivariant) Poincaré series of the collection of |G|r order functions {a^{∗}ν_{1}, a^{∗}ν_{2}, …, a^{∗}ν_{r}|a ∈ G} (each group of equal variables in P_{{a^{∗}ν_{i}}} consists of |G| of them). Now assume that a finite group G acts linearly on (ℂ^{2}, 0) and let ν_{i}, i = 1, …, r, be either a curve or a divisorial valuation on O_{ℂ^{2},0}. We shall write I_{0} = {1, 2, …, r} = I′ ⊔ I′′, where i ∈ I′ if and only if the corresponding valuation ν_{i} is a curve one. For i ∈ I′, let (C_{i}, 0) be the plane curve defining the valuation ν_{i}. A G-equivariant resolution (or a G-resolution for short) of the collection {ν_{i}} of valuations is a proper complex analytic map π : (X , D) → (ℂ^{2}, 0) from a smooth surface X with a G-action such that:

1) π is an isomorphism outside of the origin in ℂ^{2};

2) π commutes with the G-actions on X and on ℂ^{2};

3) the total transform π^{−1}( ⋃_{i∈I′, a∈G} aC_{i}) of the curve GC = G( ⋃_{i∈I′} C_{i}) is

a normal crossing divisor on X (in particular, the exceptional divisor D = π^{−1}(0) is a normal crossing divisor as well); 4) for each branch C_{i}, i ∈ I′, its strict transform C̃_{i} is a germ of a smooth

curve transversal to the exceptional divisor D at a smooth point x of it and is invariant with respect to the isotropy subgroup G_{x} = {g ∈ G : gx = x} of the point x;

5) for each i ∈ I′′, the exceptional divisor D = π^{−1}(0) contains the divisor defining the divisorial valuation ν_{i}.

A G-resolution can be obtained by a G-invariant sequence of blow-ups of points. The action of the group G on the first component of the exceptional divisor can either be trivial (this may happen only if G is cyclic) or have fixed points of (proper) subgroups of G. (If G is abelian, these are the fixed points of G itself.) These points are called special.

Let D^{∘} be the “smooth part” of the exceptional divisor D in the total transform π^{−1}(GC) of the curve GC, i.e., D itself minus all the intersection points of its components and all the intersection points with the components of the strict transform of the curve GC. For x ∈ D^{∘}, let L̃_{x} be a germ of a smooth curve on X transversal to D^{∘} at the point x and invariant with respect to the isotropy subgroup G_{x} of the point x. The image L_{x} = π(L̃_{x}) ⊂ (ℂ^{2}, 0) is called a curvette at the point x. Let the curvette L_{x} be given by an equation h_{x} = 0, h_{x} ∈ O_{ℂ^{2},0}. Without loss of generality one can assume that the function germ h_{x} is G_{x}-equivariant. Moreover we shall assume that the germs h_{x} associated to different points x ∈ D^{∘} are choosen so that h_{ax}(a^{−1}z)/h_{x}(z) is a constant (depending on a and x). Let E_{σ}, σ ∈ Γ, be the set of all irreducible components of the exceptional divisor D (Γ is a G-set itself). For σ and δ from Γ, let m_{σδ} := ν_{σ}(h_{x}), where ν_{σ} is the corresponding divisorial valuation, h_{x} is the germ defining the curvette at a point x ∈ E_{δ} ∩D^{∘}. One can show that the matrix (m_{σδ}) is minus the inverse matrix to the intersection matrix (E_{σ} ∘ E_{δ}) of the irreducible components of the exceptional divisor D. For i = 1, …, r, let m_{σi} := m_{σδ}, where E_{δ} is the component of D corresponding to the valuation ν_{i}, i.e. either the component defining the valuation ν_{i} if ν_{i} is a divisorial valuation (i.e. if i ∈ I′′), or the component intersecting the strict transform of the corresponding irreducible curve C_{i} if ν_{i} is a curve valuation (i.e. if i ∈ I′). Let \underline{m}_{σ} := (m_{σ1}, …, m_{σr}) ∈ ℤ_{≥0}^{r}, M_{σi} := ∑_{a∈G} m_{(aσ)i}, \underline{M}_{σ} := (M_{σ1}, …, M_{σr}) = ∑_{a∈G} \underline{m}_{aσ}.

Let D̂ be the quotient D^{∘}/G and let p : D^{∘} → D̂ be the factorization map. Let {Ξ} be a stratification of the smooth curve D̂ such that:

1) each stratum Ξ is connected;

2) for each point x̂ ∈ Ξ and for each point x from its pre-image p^{−1}(x̂), the conjugacy class of the isotropy subgroup G_{x} of the point x is the same, i.e., depends only on the stratum Ξ.

The condition 2) is equivalent to say that the factorization map p : D^{∘} → D̂ is a (non-ramified) covering over each stratum Ξ. The condition 1) implies that the inverse image in D^{∘} of each stratum Ξ lies in the orbit of one component E_{σ} of the exceptional divisor. The element \underline{M}_{σ} ∈ ℤ_{≥0}^{r} depends only on the stratum Ξ and will be denoted by \underline{M}_{Ξ}.

For a point x ∈ D^{∘}, let X̃_{x} = [G/G_{x}]_{α_{hx}} ∈ Ã(G). The equipped G-set X̃_{x} is one and the same for all points x from the preimage of a stratum Ξ and therefore it defines an element of Ã(G) which we shall denote by [G/G_{Ξ}]_{α_{Ξ}}. In [7, Theorem 1] it was shown that

P_{\{ν_{i}\}}^{G}(\underline{t}) = ∏_{Ξ} (1 − \underline{t}^{\underline{M}^{Ξ}})^{−χ(Ξ)[G/G_{Ξ}]_{α_{Ξ}}} .  (2)

3 Topology of plane valuations

Let the complex plane (ℂ^{2}, 0) be endowed by a faithful linear G-action and let {ν_{i}}_{i=1}^{r} be a collection of divisorial valuations on O_{ℂ^{2},0}.

Theorem 1 The G-equivariant Poincaré series P_{\{ν_{i}\}}^{G}(\underline{t}) of the collection \{ν_{i}\} of divisorial valuations determines the weak G-equivariant topology of this collection.

Proof. One has to use the following “projection formula”. Let I = {i_{1}, …, i_{s}} be a subset of the set {1, …, r} of the indices numbering the valuations. Then one has

P_{{ν_{i}}_{i∈I}}^{G} (t_{i_{1}}, …, t_{i_{s}}) = P_{{ν_{i}}_{i=1}^{r}}^{G}(t_{1}, …, t_{r})_{|t_{i}=1 for i∉I} ,

i.e. the (G-equivariant) Poincaré series for a subcollection of valuations is obtained from the one for the whole collection by substituting t_{i} by 1 for all i numbering the valuations which do not participate in the subcollection. (This equation is not valid for other types of valuations, say, for curve ones: see the proof of Theorem 2). The projection formula implies, in particular, that the G-equivariant Poincaré series P_{\{ν_{i}\}}^{G}(\underline{t}) of a collection of divisorial valuations determines the G-equivariant Poincaré series (in one variable) of each individual valuation from it. First we shall show that the Poincaré series P_{\{ν_{i}\}}^{G}(\underline{t}) determines the G-resolution graph of the collection of valuations. It turns out that the necessary information about the G-equivariant resolution graph can be restored from the ρ-reduction ρP_{ν}^{G}(t) of the G-equivariant Poincaré series P_{ν}^{G}(t) (i.e. the series from A(G)[[t]] obtained by forgetting the one-dimensional representations associated with the G-orbits). Therefore we shall start with considering it. First let us prove the statement for one divisorial valuation. The dual graph Γ^{G} of the minimal G-equivariant resolution of a divisorial valuation ν looks like in Fig. 1. This means the following.

[figure: τ_{4}; σ_{4}; ρ_{2}; ν; τ_{5}; σ_{5}; 1 = σ_{0}; σ_{1}; σ_{2}; σ_{3}; τ_{1}; τ_{2}; τ_{3} = ρ_{1}]

Figure 1: The dual equivariant resolution graph Γ^{G} of the valuation ν.

[figure: τ_{4}; σ_{4}; ρ_{2}; ν; τ_{5}; σ_{5}; 1 = σ_{0}; σ_{1}; σ_{2}; σ_{3}; τ_{1}; τ_{2}; τ_{3} = ρ_{1}]

Figure 2: The dual resolution graph Γ of the valuation ν.

The standard (non-equivariant, minimal) dual resolution graph Γ of the valuation ν looks like in Fig. 2. The vertices σ_{q}, q = 0, 1, …, g, are the dead ends of the graph (g is the number of the Puiseux pairs of a curvette corresponding to the valuation, σ_{0} = 1 is the first component of the exceptional divisor), the vertices τ_{i}, q = 1, …, g, are the rupture points, the vertex ν corresponds to the divisorial valuation under consideration. (The vertex ν may coincide with τ_{g}.) The set of vertices of the graph Γ is ordered according to the order of the birth of the corresponding components of the exceptional divisor. On [σ_{0}, ν] (the geodesic from σ_{0} = 1 to ν) this order is the natural one: δ_{1} < δ_{2} if and only if the vertex δ_{1} lies on [σ_{0}, δ_{2}]. The integers m_{σ_{q}}, q = 0, 1, …, g, form the minimal set of generators of the semigroup of values of ν and are traditionally denoted by \overline{β}_{q}. One also uses the following notations. e_{q} := gcd(\overline{β}_{0}, \overline{β}_{1}, …, \overline{β}_{q}),

N_{q} := \frac{e_{q−1}}{e_{q}} (= \frac{m_{τ_{q}}}{m_{σ_{q}}} ) .

The graph Γ^{G} of the minimal G-equivariant resolution consists of |G| copies of graph Γ (numbered by the elements of G) glued together. The gluing is defined by a sequence

G = H_{0} ⊃ H_{1} ⊃ H_{2} ⊃ … ⊃ H_{k}

of subgroups of the group G such that all H_{i} with i > 0 are abelian and H_{k} is the isotropy group of the valuation ν ({a ∈ G : a^{∗}ν = ν}) and by a sequence by vertices ρ_{1}, …, ρ_{k} of the graph Γ such that all of them lie on the geodesic from σ_{0} to ν, ρ_{1} < ρ_{2} < … < ρ_{k}. (Some of the vertices ρ_{i} may coincide with some of the vertices τ_{j}; the vertex ρ_{1} may coincide with the initial vertex σ_{0} = 1.) The copies of Γ numbered by the elements a_{1} and a_{2} from G are glued along the part preceeding ρ_{ℓ} (i.e., by identifying all the vertices smaller or equal to ρ_{ℓ}) if a_{1}a_{2}^{−1} ∈ H_{ℓ−1}. (In particular the initial vertices σ_{0} = 1 of all the copies are identified.) Pay attention that some of the vertices ρ_{i} can be inbetween the vertices τ_{g} and ν. For q = 1, 2, …, g, let j(q) be defined by the condition ρ_{j(q)} < τ_{q} ≤ ρ_{j(q)+1}. For δ ∈ Γ^{G} (or for the corresponding δ ∈ Γ), let M_{δ} := ∑_{a∈G} m_{aδ}. One can easily see that all the integers M_{δ}, δ ∈ Γ, are different. (One has M_{δ_{1}} = M_{δ_{2}} for δ_{1} and δ_{2} from Γ^{G} if and only if there exists a ∈ G such that δ_{2} = aδ_{1}.) One has M_{τ_{q}} = N_{q}M_{σ_{q}}. The series ρP_{ν}^{G}(t) is given by the equation

ρP_{ν}^{G}(t) = ∏_{q=0}^{g} (1 − t^{M_{σq}} )^{−[G/H_{j(q)}]} · ∏_{q=1}^{g} (1 − t^{N_{q}M_{σq}} )^{[G/H_{j(q)}]} ×

× ∏_{j=1}^{ℓ} (1 − t^{M_{ρ_{j}}} )^{[G/H_{j}]−[G/H_{j−1}]} · (1 − t^{M_{ν}} )^{−[G/H_{k}]} .

The fact that all the integers M_{δ} are different implies that the exponents M_{σ_{q}}, q = 1, …, g, are among those which participate in the decomposition of the series ρP_{ν}^{G}(t) with negative cardinalities of the multiplicities. (The multiplicity of a binomial (1 − t^{m})^{s_{m}}, s_{m} ∈ A(G), is s_{m}. Its cardinality is the (virtual) number of the points of it.) It is possible that the exponents of this sort include also M_{ν} corresponding to the divisorial valuation itself. The subgroups H_{1} ⊃ H_{2} ⊃ … ⊃ H_{k} are defined by the multiplicities of all the factors in the decomposition of the series ρP_{ν}^{G}(t) into the product of the binomials. The vertex σ_{0} = 1 coincides with ρ_{1} if and only if the binomial with the smallest exponent in the decomposition of the series ρP_{ν}^{G}(t) has a non-negative cardinality of the multiplicity. For σ_{q} ≤ ρ_{1} one has M_{σ_{q}} = |G|m_{σ_{q}} and M_{ρ_{1}} = |G|m_{ρ_{1}}. These equations give all the generators \overline{β}_{q} of the semigroup of values with σ_{q} ≤ ρ_{1} and also m_{ρ_{1}}. For ℓ ≥ 1, let σ_{q(ℓ)} be the minimal dead end greater than ρ_{ℓ} (i.e. there are the dead ends σ_{q(ℓ)}, …, σ_{q(ℓ+1)−1} inbetween ρ_{ℓ} and ρ_{ℓ+1}). Let us consider the dead ends σ_{q} such that ρ_{1} < σ_{q} < ρ_{2}. One has

M_{σ_{q(1)}} = |H_{1}|m_{σ_{q(1)}} + (|G| − |H_{1}|)m_{ρ_{1}} = |H_{1}|m_{σ_{q(1)}} + (M_{ρ_{1}} − |H_{1}|m_{ρ_{1}}) .

The smallest multiple of the exponent M_{σ_{q(1)}} in a binomial participating in the decomposition of the series ρP_{ν}^{G}(t) is M_{τ_{q(1)}} = N_{q(1)}M_{σ_{q(1)}}. Further, for ρ_{1} < σ_{q(1)} < σ_{q(1)+1} < σ_{q(1)+2} < ⋯ σ_{q(2)−1} < ρ_{2}, one has

M_{σ_{q(1)+1}} = |H_{1}|m_{σ_{q(1)+1}} + (M_{ρ_{1}} − |H_{1}|m_{ρ_{1}})N_{q(1)} ,
M_{σ_{q(1)+2}} = |H_{1}|m_{σ_{q(1)}+2} + (M_{ρ_{1}} − |H_{1}|m_{ρ_{1}})N_{q(1)}N_{q(1)+1} ,
…
M_{ρ_{2}} = |H_{1}|m_{ρ_{2}} + (M_{ρ_{1}} − |H_{1}|m_{ρ_{1}})N_{q(1)}N_{q(1)+1} · . . . · N_{q(2)−1} .

These equations give all the generators \overline{β}_{q} of the semigroup of values with σ_{q} < ρ_{2} and also m_{ρ_{2}}. Assume that we have determined all the exponents m_{σ_{q}} for q < q(ℓ) and also the exponent m_{ρ_{ℓ}}. Let us consider the dead ends σ_{q} such that ρ_{ℓ} < σ_{q} < ρ_{ℓ+1}. One has

M_{σ_{q(ℓ)}} = |H_{ℓ}|m_{σ_{q(ℓ)}} + (M_{ρ_{ℓ}} − |H_{ℓ}|m_{ρ_{ℓ}}) ,
M_{σ_{q(ℓ)+1}} = |H_{ℓ}|m_{σ_{q(ℓ)+1}} + (M_{ρ_{ℓ}} − |H_{ℓ}|m_{ρ_{ℓ}})N_{q(ℓ)} ,
M_{σ_{q(ℓ)+2}} = |H_{ℓ}|m_{σ_{q(ℓ)+2}} + (M_{ρ_{ℓ}} − |H_{ℓ}|m_{ρ_{ℓ}})N_{q(ℓ)}N_{q(ℓ)+1} ,
…
M_{ρ_{ℓ+1}} = |H_{ℓ}|m_{ρ_{ℓ}} + (M_{ρ_{ℓ}} − |H_{ℓ}|m_{ρ_{ℓ}})N_{q(ℓ)}N_{q(ℓ)+1} · . . . · N_{q(ℓ+1)−1} .

These equations give all the generators m_{σ_{q}} of the semigroup of values with q < q(ℓ + 1) and also m_{ρ_{ℓ+1}}. The described procedure recovers m_{σ_{q}} for all q ≤ g. If, in the binomials of the decomposition of the series ρP_{ν}^{G}(t), there are no exponents proportional to M_{σ_{g}}, one has ν = τ_{g} and the resolution graph Γ is determined by the semigroup ⟨\overline{β}_{0}\overline{β}_{1}, …, \overline{β}_{g}⟩. Otherwise the described above procedure permits to determine the exponents m_{ρ_{j}} with ρ_{j} ≥ τ_{g} and m_{ν}. This gives the Gequivariant resolution graph of one divisorial valuation. Assume that we have a collection {ν_{i}} of divisorial valuations, i = 1, 2, …, r. To restore the equivariant resolution graph Γ^{G} of the collection from the resolution graphs of each individual valuation ν_{i}, one has to determine the separation point δ_{ij} between each two valuations ν_{i} and ν_{j} (for simplicity let us assume that i = 1, j = 2). Let

ρP_{ν}^{G}(t_{1}, t_{2}, 1, …, 1) = ∏(1 − t_{1}^{M_{1}}t_{2}^{M_{2}})^{s_{M_{1}M_{2}}} ,  (3)

s_{M_{1}M_{2}} ∈ ℤ, be the decomposition into the product of the binomials. The separation point δ_{12} corresponds to the maximal exponent in the decomposition (3) with

\frac{M_{δ1}}{M_{δ2}} = \frac{M_{σ_{0}1}}{M_{σ_{0}2}} .

This proves that the reduction ρP_{\{ν_{i}\}}^{G}(\underline{t}) ∈ A(G)[[t_{1}, …, t_{r}]] of the Gequivariant Poincaré series P_{\{ν_{i}\}}^{G}(\underline{t}) determines the minimal G-resolution graph of the set {ν_{i}} of divisorial valuations.

In order to prove that one can also determine the weak G-topology of the collection of valuations, one has to show how is it possible to restore the representation of the group G on ℂ^{2} and the correspondence between (some) tails of the (minimal) G-resolution graph and the special points on the first component of the exceptional divisor. For that one should use the non-reduced Poincaré series P_{\{ν_{i}\}}^{G}(\underline{t}) ∈ Ã(G)[[t_{1}, …, t_{r}]] itself. (If there are no special points on the first component of the exceptional divisor (this can happen only if G is cyclic), only the representation of G on ℂ^{2} has to be determined.) We follow the scheme described in [6].

Let us consider the case of an abelian group G first. If there are no special points on the first component E_{1} of the exceptional divisor, all points of E_{1} are fixed with respect to the group G, the group G is cyclic and the representation is a scalar one. This (one dimensional) representation is dual to the representation of the group G on the one-dimensional space generated by any linear function. The case when there are no more components in D, i.e. if the resolution is achieved by the first blow-up, is trivial. Otherwise let us consider a maximal component E_{σ} among those components E_{τ} of the exceptional divisor for which G_{τ} = G and the corresponding curvette is smooth. (The last condition can be easily detected from the resolution graph.) The smooth

part E^{•}_{σ} of this component contains a special point x with G_{x} = G (or all the

points of E^{•}_{σ} are such that G_{x} = G). The point(s) from E^{•}_{σ} with G_{x} = G bring(s) into the decomposition of the Poincaré series P_{\{ν_{i}\}}^{G}(\underline{t}) a factor of the form (1 − \underline{t}\underline{M})^{−[G/G]_{α}}. The (G-equivariant) curvette L at the described special point of the divisor is smooth. Therefore the representation of G on the onedimensional space generated by a G-equivariant equation of L coincides with the representation on the space generated by a linear function. Let us take all factors of the form (1 − \underline{t}\underline{M})^{−[G/G]_{α}} in the decomposition of the Poincaré series P_{\{v_{i}\}}^{G}. For each of them, the exponent \underline{M} determines the corresponding component of the exceptional divisor and therefore the topological type of the corresponding curvettes. The factor which corresponds to a component with a smooth curvette gives us the representation α on the space generated by a linear function.

Now assume that there are two special points on the first component of the resolution. Without loss of generality we can assume that they correspond to the coordinate axis {x = 0} and {y = 0}. The representation of the group G on ℂ^{2} is defined by its action on the linear functions x and y. For each of them this action can be recovered from a factor of the form described above just in the same way. Moreover, a factor, which determines the action of the group G on the function x, corresponds to a component of the exceptional divisor from the tail emerging from the point {x = 0}.

Now let G be an arbitrary (not necessarily abelian) group. For an element g ∈ G consider the action of the cyclic group ⟨g⟩ generated by g on ℂ^{2}. One can see that the G-equivariant Poincaré series P_{\{v_{i}\}}^{G}(\underline{t}) determines the ⟨g⟩- Poincaré series P_{\{v_{i}\}}^{⟨g⟩}(\underline{t}) just like in [5, Proposition 2]. This implies that the G-equivariant Poincaré series determines the representation of the subgroup ⟨g⟩. (Another way is to repeat the arguments above adjusting them to the subgroup ⟨g⟩.) Therefore the G-Poincaré series P_{\{v_{i}\}}^{G}(\underline{t}) determines the value of the character of the G-representation on ℂ^{2} for each element g ∈ G and thus the representation itself. Special points of the G-action on the first component E_{1} of the exceptional divisor correspond to some abelian subgroups H of G. For each such subgroup H there are two special points corresponding to different one-dimensional representations of H. Again the construction above for an abelian group permits to identify tails of the dual resolution graph with these two points. □

Let {C_{i}}, i = 1, …, r, be a collection of irreducible curve singularities in (ℂ^{2}, 0) such that it does not contain curves from the same G-orbit and it does not contain a smooth curve invariant with respect to a non-trivial element of G whose action on ℂ^{2} is not a scalar one. Let {ν_{i}} be the corresponding collection of valuations. Let G_{i} ⊂ G be the isotropy group of the branch C_{i}, 1 ≤ i ≤ r.

Theorem 2 The G-equivariant Poincaré series P_{\{ν_{i}\}}^{G}(\underline{t}) of the collection \{ν_{i}\} determines the weak G-equivariant topology of the collection {ν_{i}} of curve valuations.

Proof. The minimal resolution graph Γ of the plane curve singularity C =

⋃_{i=1}^{r} C_{i} is essentially the same as the graph of the divisorial valuations defined

by the set of irreducible components {E_{α_{i}}} of the exceptional divisor such that the strict transform of C_{i} intersects the component E_{α_{i}}. Instead of the mark used for the divisor E_{α_{i}} (like in Figures 1 and 2 for one valuation) one puts an arrow corresponding to C_{i} connected to the vertex α_{i}. Notice that there can be several arrows connected to the same vertex, i.e. α_{i} = α_{j} for different branches C_{i}, C_{j}. In the case of one branch the graph looks like the one in Figure 2 but the vertex marked by ν coincides with τ_{g} and there is an arrow connected with τ_{g}. The number g is equal to the number of Puiseux pairs of the curve and m_{σ_{i}} = β̄_{i}, 0 ≤ i ≤ g, are the elements of the minimal set of generators of the semigroup of the branch. (In particular they determine the minimal resolution graph of the curve.)

[figure: α_{i}; (a) Γ; ρ; C_{i}; aC_{i}; (b) Γ^{G}; ρ = α_{i}; C_{i}; (c) Γ enlarged]

Figure 3: The graphs Γ, Γ^{G} and Γ enlarged.

The same rules apply for the graph Γ^{G}. However Γ^{G} corresponds to the embedded resolution of the union of all the orbits of the branches of C. So, it is possible that, in order to achieve the minimal equivariant resolution (i.e. in order to separate all the conjugates of each one of the branches C_{i}), one has to add some additional blow-ups starting at the point α_{i}. Note that in this case some of the vertices ρ (see the notations in the proof of Theorem 1 and Figures 1 and 2) do not appear in Γ. In order to preserve the scheme and the notations from the proof of the case of divisorial valuations it is better to enlarge Γ in such a way that the new one (also denoted by Γ) is the minimal one in which all the vertices ρ are present (see Figure 3). Note that aE_{α_{i}} = E_{aα_{i}} for a ∈ G, so in this way the (new) resolution graph Γ is just the quotient of Γ^{G} by the obvious action of G on Γ^{G}. As in the case of divisorial valuations, for each δ ∈ Γ^{G} let h_{δ} = 0, h_{δ} ∈ O_{ℂ^{2},0}, be the equation of a curvette at the component E_{δ}, m_{δi} be the value ν_{i}(h_{δ}), M_{δi} = ∑_{a∈G} m_{(aδ)i} = ∑_{a∈G}(a^{∗}ν_{i})(h_{δ}) and \underline{M}_{δ} = (M_{δ1}, …, M_{δr}) ∈ ℤ_{≥0}^{r}. All the \underline{M}_{σ}, σ ∈ Γ, are different and for σ, τ ∈ Γ^{G} \underline{M}_{σ} = \underline{M}_{τ} if and only if E_{τ} = aE_{σ} for some a ∈ G. Let G_{i} ⊂ G be the isotropy group of the branch C_{i}, 1 ≤ i ≤ r. For i, j ∈ {1, …, r}, m_{α_{i}j} is just the intersection multiplicity between C_{i} and C_{j} and

M_{α_{i}j} =∑_{a∈G} m_{(aα_{i})j} =∑_{a∈G}(a^{∗}ν_{j})(h_{α_{i}}) = (C_{i}, ⋃_{a∈G} aC_{j}) = (C_{j}, ⋃_{a∈G} aC_{i}) = M_{α_{j}i} .

In contrast with the case of divisorial valuations the projection formula is different from the one for divisorial valuations formulated at the beginning of the proof of Theorem 1. Instead of it one has the following one: For i_{0} ∈ {1, …, r} one has

P_{\{ν_{i}\}}^{G}(\underline{t})_{|_{t_{i_{0}} =1}} = (1 − \underline{t}^{\underline{M}^{α_{i_{0}}}} )_{|_{t_{i_{0}} =1}}^{[G/G_{i_{0}} ]_{α_{i_{0}}}} P_{\{ν_{i}\}_{i≠i_{0}}}^{G} (t_{1}, …, t_{i_{0}−1}, t_{i_{0}+1}, …, t_{r}) . (4)

(This can be easily deduced from (2).) Using (4) repeatedly one also has:

P_{\{ν_{i}\}}^{G}(\underline{t})_{|_{t_{i}=1,i≠i_{0}}} = ∏_{i≠i_{0}}(1 − t_{i_{0}}^{M_{α_{i}i_{0}}} )^{[G/G_{i}]_{α_{i}}} P_{ν_{i_{0}}}^{G} (t_{i_{0}}) .  (5)

Equations (4) and (5) imply that in order to describe inductively the minimal G-resolution graph Γ^{G} one has to detect the binomial (1−\underline{t}^{\underline{M}^{α_{i_{0}}}} ) corresponding to some i_{0} from the G-equivariant Poincaré series and also the intersection multiplicities of C_{i_{0}} with the other branches of C. As in the divisorial case, the necessary information about the G-equivariant resolution graph can be restored from the ρ-reduction ρP_{\{ν_{i}\}}^{G}(\underline{t}) of the Poincaré series P_{\{ν_{i}\}}^{G}(\underline{t}) to the ring A(G)[[t_{1}, …, t_{r}]]. From the factorization given in (2) one can write ρP_{\{ν_{i}\}}^{G}(\underline{t}) = ∏_{σ∈Γ}(1−\underline{t}^{\underline{M}^{σ}})^{s_{σ}}, where s_{σ} ∈ A(G). Note that the multiplicity s_{σ} may be equal to zero, i.e. the binomial factor corresponding to σ may be absent.

The determination of the G-equivariant resolution graph from the series ρP_{ν}^{G}(t) for one branch almost repeats the one described for one divisorial valuation, e.g. the semigroup is the same as the one of the divisorial valuation defined by the component E_{τ_{g}} of the exceptional divisor. So, let us assume r > 1 and let us fix j, k ∈ {1, …, r}. The separation point s(α_{j}, α_{k}) ∈ Γ^{G} of α_{j} and α_{k} is defined by the condition [1, α_{j}] ∩ [1, α_{k}] = [1, s(α_{j}, α_{k})]. Here [1, σ] is the geodesic in the dual graph Γ^{G} joining the first vertex 1 with the vertex σ. Now, let us define the separation vertex s(α_{j}, k) of C_{j} and GC_{k} as the maximun of s(α_{j}, aα_{k}) for a ∈ G. Note that, if a ∈ G then s(aα_{j}, k) = as(α_{j}, k) ∈ Γ^{G} so s(j, k) = s(α_{j}, k) is a well defined vertex of the graph Γ. We refer to it as the separation vertex of C_{i} and C_{j} in Γ. The ratio M_{σj}/M_{σk} is constant for σ in [1, s(j, k)] and is a strictly increasing function for σ ∈ [s(i, j), α_{j}] ⊂ Γ as well as in the geodesic [as(j, k), aα_{j}] ⊂ Γ^{G} for a ∈ G. Notice that for σ ∉ ⋃_{a∈G} ([1, aα_{j}] ∪ [1, aα_{k}]) the ratio M_{σj}/M_{σk} is equal to M_{σ′j}/M_{σ′k} where σ′ is the vertex such that

[1, σ′] = max_{a∈G} {([1, aα_{j}] ∪ [1, aα_{k}]) ∩ [1, σ]} .

Let σ ∈ Γ be such that the exponent \underline{M}_{σ} is a maximal one among the set of exponents \underline{M}_{τ} appearing in the factorization

ρP_{\{ν_{i}\}}^{G}(\underline{t}) = ∏_{τ∈Γ , s_{τ} ≠0}(1 − \underline{t}^{\underline{M}^{τ}})^{s_{τ}} .  (6)

(Here we use the partial order \underline{M} = (M_{1}, …, M_{r}) ≤ \underline{M}′ = (M′_{1}, …, M′_{r}) if and only if M_{i} ≤ M′_{i} for all i = 1, …, r.) Note that in this case the corresponding factor has positive cardinality and there exists an index j ∈ {1, …, r} such that α_{j} = σ. Let A ⊂ {1, …, r} be the set of indices j such that M_{σj}/M_{σk} ≥ M_{τj}/M_{τk} for all k ∈ \{1, …, r\} and all τ ∈ Γ^{G} such that the binomial (1 − \underline{t}^{\underline{M}^{τ}}) appears in (6), i.e. s_{τ} ≠ 0. From the comments above it is clear that all indices j such that α_{j} = σ belong to A, however A could contain some other indices ℓ such that α_{ℓ} ≠ σ. Let us assume that there exists ℓ ∈ A such that α_{ℓ} ≠ σ. The behaviour of the ratios M_{τℓ}/M_{τk} along [1, α_{ℓ}] described above implies that σ ∈ [1, α_{ℓ}]. By definition of the set A, for any τ ∈ [σ, α_{ℓ}], τ ≠ σ, the binomial (1 − \underline{t}^{\underline{M}^{τ}}) does

not appear in (6), i.e. s_{τ} = 0, in particular χ(E^{∘}_{τ}) = 0. As a consequence, α_{ℓ} < σ and α_{ℓ} is the end point σ_{g} on the dual graph of C_{j} (here j ∈ A such that α_{j} = σ). In this case one has M_{σℓ} < M_{σj} and one can distinguish ℓ by this condition. Note that if such an ℓ ∈ A exists then it is unique. Let i_{0} ∈ A be such that M_{σi_{0}} ≥ M_{σj} for all j ∈ A. Then α_{i_{0}} = σ and the factor (1 − \underline{t}^{\underline{M}^{α_{i_{0}}}} )^{[G/G_{i_{0}} ]} appears in the factorization (6). Thus, the projection formula permits to recover the G-equivariant resolution graph by induction. As in Theorem 1 one has to show that the Poincaré series P_{\{ν_{i}\}}^{G}(\underline{t}) determines the representation of G on ℂ^{2}, and the correspondence between “tails” of the resolution graph. The proof in this case does not differ from the one made in Theorem 1 for divisorial valuations since the collection {C_{i}} does not contains smooth curves invariant with respect to a non-trivial element of G whose action is not a scalar one. □

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A. Campillo IMUVA (Instituto de Investigación en Matemáticas) Universidad de Valladolid Paseo de Belén, 7 47011 Valladolid, Spain campillo@agt.uva.es F. Delgado IMUVA (Instituto de Investigación en Matemáticas) Universidad de Valladolid Paseo de Belén, 7 47011 Valladolid, Spain fdelgado@agt.uva.es

S. M. Gusein-Zade

Moscow State University

Faculty of Mathematics and Mechanics Moscow, GSP-1, 119991, Russia sabir@mccme.ru