# Weil–Poincaré series and topology of collections of valuations on rational
double points 111Math. Subject Class. 2010: 14B05, 32S25, 57M27, 16W60.
Keywords: rational double points, algebraic links, Poincaré series,
topological type.

A. Campillo  F. Delgado The first two authors partially supported by
MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe”, grant
PGC2018-096446-B-C21. S.M. Gusein-Zade  The work of the third author was
supported by the grant 21-11-00080 of the Russian Science Foundation.

###### 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.

##  1 Introduction

An algebraic link in the three-dimensional sphere is the intersection
$K=C\cap\mathbb{S}_{\varepsilon}^{3}$ of a germ
$(C,0)\subset(\mathbb{C}^{2},0)$ of a complex analytic plane curve with the
sphere $\mathbb{S}_{\varepsilon}^{3}$ of radius $\varepsilon$ centred at the
origin in $\mathbb{C}^{2}$ with $\varepsilon$ small enough. The number $r$ of
the components of the link $K$ is equal to the number of the irreducible
components of the curve $(C,0)$. It is well-known that the Alexander
polynomial in $r$ variables determines the topological type of an algebraic
link (or, equivalently, the (local) topological type of the triple
$(\mathbb{C}^{2},C,0)$): [16]. This follows from the fact that the Alexander
polynomial determines the combinatorial type of the minimal embedded
resolution of the curve $C$. The Alexander polynomial is defined for links in
three-dimensional manifolds which are homology spheres. One of them is the
Poincaré sphere which is the intersection of the surface
$S=\\{(z_{1},z_{2},z_{3})\in\mathbb{C}^{3}:z_{1}^{5}+z_{2}^{3}+z_{3}^{2}=0\\}$
(the $\mathbb{E}_{8}$ surface singularity) with the 5-dimensional sphere
$\mathbb{S}_{\varepsilon}^{5}=\\{(z_{1},z_{2},z_{3})\in\mathbb{C}^{3}:|z_{1}|^{2}+|z_{2}|^{2}+|z_{3}|^{2}=\varepsilon^{2}\\}$.
An algebraic link in the Poincaré sphere is the intersection of a germ
$(C,0)\subset(S,0)$ of a complex analytic curve in $(S,0)$ with the sphere
$\mathbb{S}_{\varepsilon}^{5}$ of radius $\varepsilon$ small enough. In [9],
it was analyzed to which extent the Alexander polynomial in several variables
of an algebraic link in the Poincaré sphere determines the topological type of
the link or rather the combinatorial type of the minimal embedded resolution
of the corresponding curve. It was shown that, in general, the Alexander
polynomial does not determine the combinatorial type of the minimal embedded
resolution of the curve defining an algebraic link in the Poincaré sphere.
However, if the strict transform of a curve in $(S,0)$ in the minimal
resolution of $(S,0)$ does not intersect the component corresponding to the
end of the longest tail in the graph $\mathbb{E}_{8}$ of the resolution at a
smooth point of the exceptional divisor, then the Alexander polynomial of the
corresponding link determines the combinatorial type of the minimal resolution
of the curve and therefore the topology of the corresponding link.

A valuation (discrete of rank one) on the ring ${\mathcal{O}}_{V,0}$ of germs
of functions on a complex analytic variety $(V,0)$ is a function
$v:{\mathcal{O}}_{V,0}\to\mathbb{Z}_{\geq 0}\cup\\{+\infty\\}$ such that

  1. 1)

$\nu(\lambda g)=\nu(g)$ for $\lambda\in\mathbb{C}$, $\lambda\neq 0$;

  2. 2)

$\nu(g_{1}+g_{2})\geq\min(\nu(g_{1}),\nu(g_{2}))$;

  3. 3)

$\nu(g_{1}g_{2})=\nu(g_{1})+\nu(g_{2})$.

We permit a valuation to have the value infinity for a non-zero element. (In
this case some authors speak about “semivaluations”).

An irreducible curve germ $(C,0)$ in a germ of a complex analytic variety
$(V,0)$ defines a valuation $v_{C}$ on the ring ${\mathcal{O}}_{V,0}$ of germs
of functions on $(V,0)$ (called a curve valuation). Let
$\varphi:(\mathbb{C},0)\to(V,0)$ be a parametrization (an uniformization) of
the curve $(C,0)$, that is ${\text{Im\,}}\varphi=(C,0)$ and $\varphi$ is an
isomorphism between punctured neighbourhoods of the origin in $\mathbb{C}$ and
in $C$. For a function germ $f\in{\mathcal{O}}_{V,0}$, the value $v_{C}(f)$ is
defined as the degree of the leading term in the Taylor series of the function
$f\circ\varphi:(\mathbb{C},0)\to\mathbb{C}$:

$f\circ\varphi(\tau)=a\tau^{v_{C}(f)}+{\text{\ terms of higher degree,}}$

where $a\neq 0$; if $f\circ\varphi\equiv 0$, one defines $v_{C}(f)$ to be
equal to $+\infty$.

A collection $\\{(C_{i},0)\\}$ of irreducible curves in $(V,0)$,
$i=1,\ldots,r$, defines the collection $\\{v_{C_{i}}\\}$ of valuations. To a
collection $\\{v_{i}\\}$ of discrete rank one valuations on
${\mathcal{O}}_{V,0}$, $i=1,\ldots,r$, one may associate, as in [11], a
Poincaré series
$P_{\\{v_{i}\\}}(t_{1},\ldots,t_{r})\in\mathbb{Z}[[t_{1},\ldots,t_{r}]]$. The
collection $\\{v_{i}\\}$ defines a multi-index filtration on
${\mathcal{O}}_{V,0}$ by

$J({\underline{u}})=\\{g\in{\mathcal{O}}_{V,0}:{\underline{v}}(g)\geq{\underline{u}}\\}\;,$
(1)

where ${\underline{u}}=(u_{1},\ldots,u_{r})\in\mathbb{Z}_{\geq 0}^{r}$,
${\underline{v}}(g)=(v_{1}(g),\ldots,v_{r}(g))$ and
${\underline{u}}^{\prime}=(u^{\prime}_{1},\ldots,u^{\prime}_{r})\geq{\underline{u}}=(u_{1},\ldots,u_{r})$
if and only if $u^{\prime}_{i}\geq u_{i}$ for all $i=1,\ldots,r$. Equation (1)
defines the subspaces $J({\underline{u}})$ for all
${\underline{u}}\in\mathbb{Z}^{r}$. The Poincaré series of the filtration
$\\{J({\underline{u}})\\}$ (or of the collection $\\{v_{i}\\}$ of valuations)
is defined by:

$P_{\\{v_{i}\\}}(t_{1},\ldots,t_{r})=\frac{{\cal
L}(t_{1},\ldots,t_{r})\cdot\prod_{i=1}^{r}(t_{i}-1)}{t_{1}\cdot\ldots\cdot
t_{r}-1}\;,$ (2)

where

${\cal
L}(t_{1},\ldots,t_{r}):=\sum\limits_{{\underline{u}}\in\mathbb{Z}^{r}}\dim(J({\underline{u}})/J({\underline{u}}+\underline{1}))\cdot{\underline{t}}^{\,{\underline{u}}}\,,$

$\underline{1}=(1,1,\ldots,1)\in\mathbb{Z}^{r}$. This definition makes sense
if and only if all the quotients
$J({\underline{u}})/J({\underline{u}}+\underline{1})$ are finite-dimensional.

In [2] it was shown that, for $(V,0)=(\mathbb{C}^{2},0)$, the Poincaré series
$P_{\\{v_{C_{i}}\\}}(t_{1},\ldots,t_{r})$ of a collection of (different) curve
valuations coincides with the Alexander polynomial
$\Delta^{C}(t_{1},\ldots,t_{r})$ in $r$ variables of the algebraic link
defined by the curve $C=\bigcup\limits_{i=1}^{r}C_{i}$ for $r>1$. (For $r=1$
one has $P_{v_{C}}(t)=\frac{\Delta_{C}(t)}{1-t}$.) In [5] it was shown that
the same holds for an algebraic link in the Poincaré sphere. In [7] it was
proved that the Poincaré series of a collection of divisorial valuations on
${\mathcal{O}}_{\mathbb{C}^{2},0}$ (computed in [12]) determines the
combinatorial type of the minimal resolution of the collection. (In general,
this is not the case for a collection consisisting both of curve and
divisorial valuations.)

In [9], it was discussed to which extent the Alexander polynomial in several
variables of an algebraic link in the Poincaré sphere (that is the Poincaré
series of the corresponding curve) determines the topology of the link or
rather the combinatorial type of the minimal (embedded) resolution of the
curve on the $\mathbb{E}_{8}$ surface singularity. It was shown that two
curves (even irreducible ones) with combinatorially different minimal
resolutions may have equal Alexander polynomials. However, if the strict
transform of a curve does not intersect one particular component of the
exceptional divisor of the minimal resolution of the $\mathbb{E}_{8}$ surface
singularity, namely the one corresponding to the end of the longest tail in
the corresponding $\mathbb{E}_{8}$-diagram, then its Poincaré series (that is
the Alexander polynomial of the corresponding link) determines the
combinatorial type of the minimal resolution of the curve and therefore the
topology of the corresponding link. There were given analogues of these
statements for collections of divisorial valuations on the $\mathbb{E}_{8}$
surface singularity.

For other surface singularities (say, for rational ones, for whom one has a
formula for the Poincaré series) the (classical) Poincaré series for a
collection of curve or divisorial valuations does not determine the
combinatorial type of the minimal resolution even in the simplest case of the
$\mathbb{A}_{k}$ singularities. The Poincaré series of a collection of
valuations can be interpreted as an integral with respect to the Euler
characteristic over the space of Cartier divisors (appropriately defined: see
below). We consider an analogue of the Poincaré series which is the same
integral over the space of all (that is Weil) divisors. One has an equation
for this (“Weil–Poincaré”) series similar to the one for the smooth case or
for the $\mathbb{E}_{8}$ surface singularity. (The Weil–Poincaré series of a
collection of curve or divisorial valuations is a fractional power series.) We
show that, except a few cases (somewhat similar to the exception for the
$\mathbb{E}_{8}$ singularity), the Weil–Poincaré series of a collection of
curve valuations or of a collection of divisorial valuations on a rational
double point surface singularity determines the combinatorial type of the
minimal resolution (and thus the topology of the corresponding link in the
curve case) up to the possible symmetry of the minimal resolution graph of the
surface singularity. (It is somewhat curious that exceptions exist for the
$\mathbb{E}_{7}$ and for the $\mathbb{E}_{8}$ surface singularities, i.e.
precisely for those whose minimal resolution graphs have no non-trivial
symmetries.)

##  2 The Weil–Poincaré series

Let $(S,0)$ be a normal surface singularity and let $v_{i}$, $i=1,\ldots,r$,
be either a curve valuation on ${\mathcal{O}}_{S,0}$ defined by an irreducible
curve $(C_{i},0)\subset(S,0)$, or a divisorial valuation on
${\mathcal{O}}_{S,0}$ defined by a component of the exceptional divisor
${\mathcal{D}}$ of a resolution $\pi:({\mathcal{X}},{\mathcal{D}})\to(S,0)$ of
the surface $S$. Assume that $\pi:({\mathcal{X}},{\mathcal{D}})\to(S,0)$ is a
resolution of the surface $S$ which is, at the same time, a resolution of the
collection $\\{v_{i}\\}$ of valuations, that is the total transform of the
union of the curves $C_{i}$ (such that $v_{i}$ is the curve valuation
$v_{C_{i}}$) is a normal crossing divisor on ${\mathcal{X}}$ and each divisor
defining the divisorial valuation from the collection $\\{v_{i}\\}$ is present
in ${\mathcal{D}}$.

Let ${\mathcal{D}}=\bigcup\limits_{\sigma\in\Gamma}E_{\sigma}$ be the
representation of the exceptional divisor ${\mathcal{D}}$ as the union of its
irreducible components. For $\sigma\in\Gamma$, let
$\stackrel{{\scriptstyle\circ}}{{E}}_{\sigma}$ be the “smooth part” of the
component $E_{\sigma}$ in the total transform of the curve $\bigcup_{i}C_{i}$,
i. e., the component $E_{\sigma}$ itself minus the intersection points with
all other components of the exceptional divisor ${\mathcal{D}}$ and with the
strict transforms of the curves $C_{i}$. A curvette corresponding to a
component $E_{\sigma}$ of the resolution is the blow-down of a germ of a
smooth curve transversal to $E_{\sigma}$ at a point of
$\stackrel{{\scriptstyle\circ}}{{E}}_{\sigma}$. For $i\in\\{1,2,\ldots,r\\}$,
let $\tau(i)$ be either the index of the component $E_{\tau(i)}$ which
intersects the strict transform of the curve $C_{i}$ (if $v_{i}$ is a curve
valuation), or the index of the component which defines the divisorial
valuation $v_{i}$. Let $(E_{\sigma}\circ E_{\delta})$ be the intersection
matrix of the components $E_{\sigma}$. The diagonal entries of this matrix are
negative integers and a non-diagonal entry is equal to $1$ if the components
$E_{\sigma}$ and $E_{\delta}$ intersect and is equal to $0$ otherwise. Let
$(m_{\sigma\delta}):=-(E_{\sigma}\circ E_{\delta})^{-1}$. The entries
$m_{\sigma\delta}$ are positive rational numbers whose denominators divide the
determinant $d$ of the matrix $-(E_{\sigma}\circ E_{\delta})$. For
$\sigma\in\Gamma$, let
$\underline{m}_{\sigma}:=(m_{\sigma\tau(1)},m_{\sigma\tau(2)},\ldots,m_{\sigma\tau(r)})\in\mathbb{Q}_{>0}^{r}$.

######  Definition 1.

The Weil–Poincaré series (W–Poincaré series for short) of the collection of
valuations $\\{v_{i}\\}$ is

$P^{W}_{\\{v_{i}\\}}({\underline{t}}):=\prod_{\sigma\in\Gamma}\left(1-{\underline{t}}^{\underline{m}_{\sigma}}\right)^{-\chi(\stackrel{{\scriptstyle\circ}}{{E}}_{\sigma})}\in\mathbb{Z}[[t_{1}^{1/d},t_{2}^{1/d},\ldots,t_{r}^{1/d}]]\,,$
(3)

where ${\underline{t}}=(t_{1},t_{2},\ldots,t_{r})$, for
$\underline{m}=(m_{1},\ldots,m_{r})\in\mathbb{Q}_{\geq 0}^{r}$,
${\underline{t}}^{\underline{m}}=t_{1}^{m_{1}}\cdot\ldots\cdot t_{r}^{m_{r}}$.

######  Remark 1.

One can see that the W–Poincaré series $P^{W}_{\\{v_{i}\\}}({\underline{t}})$
is well-defined, i.e., does not depend on the choice of the resolution
$\pi:({\mathcal{X}},{\mathcal{D}})\to(S,0)$. This follows from the fact that a
resolution of the collection $\\{v_{i}\\}$ can be obtained from the minimal
one by additional blow-ups either at smooth points of the total transform of
the curve $\bigcup_{i}C_{i}$ or at intersection points of it.

If $(S,0)$ is smooth or if it is the $\mathbb{E}_{8}$-surface singularity, the
W–Poincaré series of the collection $\\{v_{i}\\}$ coincides with the usual
Poincaré series of $\\{v_{i}\\}$ described above: [2], [5], [9].

######  Remark 2.

For the case when $(S,0)$ was a rational surface singularity and $v_{i}$ were
divisorial valuations corresponding to all the components of the exceptional
divisor of a resolution of $(S,0)$, the series
$P^{W}_{\\{v_{i}\\}}({\underline{t}})$ was defined in [4] and [5] and used in
[6]; see also [14].

##  3 Weil–Poincaré series and integrals with respect to the Euler
characteristic

In [3] it was (essentially) shown that the Poincaré series
$P_{\\{v_{i}\\}}({\underline{t}})$ of a collection $\\{v_{i}\\}$ of valuations
(curve or divisorial ones) on the ring ${\mathcal{O}}_{X,0}$ of germs of
functions on a variety $X$ can be given by the equation

$P_{\\{v_{i}\\}}({\underline{t}})=\int_{\mathbb{P}{\mathcal{O}}_{X,0}}{\underline{t}}^{{\underline{v}}}\,d\chi\,,$

where the right hand side is the inegral with respect to the Euler
characteristic over the projectivization of ${\mathcal{O}}_{X,0}$ (properly
defined), $t_{i}^{\infty}:=0$ (see also [10, Proposition 1.1]). In [3] it was
shown that the Poincaré series of a collection of curve valuations on
${\mathcal{O}}_{\mathbb{C}^{2},0}$ can be written as an integral with respect
to the Euler characteristic over the configuration space of effective divisors
on the smooth part of the exceptional divisors of the embedded resolution of
the union of curves.

Let $(S,0)$ be a normal surface singularity, let $\\{v_{i}\\}$ be a collection
of curve or divisorial valuations on ${\mathcal{O}}_{S,0}$, and let
$\pi:({\mathcal{X}},{\mathcal{D}})\to(S,0)$ be a resolution of the collection
(not the minimal one, in general). Let $E_{\sigma}$, $\sigma\in\Gamma$, be the
irreducible components of the exceptional divisor ${\mathcal{D}}$ and let
$\stackrel{{\scriptstyle\circ}}{{E}}_{\sigma}$ be the “smooth part” of
$E_{\sigma}$ in the total transform of the union $(C,0)=\bigcup_{i}(C_{i},0)$
of the irreducible curves $(C_{i},0)$ defining curve valuations from the
collection (i. e. $E_{\sigma}$ minus the intersection points with other
components of ${\mathcal{D}}$ and with the total transforms of the curves
$C_{i}$). Let

$Y^{\pi}:=\prod_{\sigma\in\Gamma}\left(\coprod_{k=0}^{\infty}S^{k}\stackrel{{\scriptstyle\circ}}{{E}}_{\sigma}\right)=\coprod_{\\{k_{\sigma}\\}\in\mathbb{Z}^{\Gamma}_{\geq
0}}\prod_{\sigma}S^{k_{\sigma}}\stackrel{{\scriptstyle\circ}}{{E}}_{\sigma}$

be the configuration space of effective divisors on
$\stackrel{{\scriptstyle\circ}}{{\cal
D}}=\bigcup_{\sigma}\stackrel{{\scriptstyle\circ}}{{E}}_{\sigma}$.

Let ${\underline{v}}:Y^{\pi}\to\mathbb{Q}_{\geq 0}^{r}$ be the function which
sends the component
$\prod_{\sigma}S^{k_{\sigma}}\stackrel{{\scriptstyle\circ}}{{E}}_{\sigma}$ of
$Y^{\pi}$ to $\sum\limits_{\sigma\in\Gamma}k_{\sigma}\underline{m}_{\sigma}$.
Let ${\mathcal{O}}_{S,0}^{\pi}$ be the set of non-zero function germs on
$(S,0)$ such that the strict transform of the zero-level curve $\\{f=0\\}$
intersects ${\mathcal{D}}$ only at points of
$\stackrel{{\scriptstyle\circ}}{{\cal D}}$. One has a map $I^{\pi}$ from
${\mathcal{O}}_{S,0}^{\pi}$ to $Y^{\pi}$ which sends a function $f$ to the
intersection of the strict transform of the curve $\\{f=0\\}$ with
$\stackrel{{\scriptstyle\circ}}{{\cal D}}$. Let $Y_{\mathcal{C}}^{\pi}$ be the
image of $I^{\pi}$. The set $\\{\pi\\}$ of resolutions of the collection
$\\{v_{i}\\}$ is a partially ordered set: a resolution is bigger than another
one if it can be obtained from the latter by a sequence of blow-ups.

Let ${{\mathfrak{W}}}_{S,0}$ be the set of effective divisors (that is Weil
divisors) on $(S,0)$ and let ${\mathfrak{C}}_{S,0}\subset{\mathfrak{W}}_{S,0}$
be the set of effective Cartier divisors on $(S,0)$. There is a natural map
$J$ from $\mathbb{P}{\mathcal{O}}_{S,0}$ to ${\mathfrak{C}}_{S,0}$. For a
resolution $\pi$ of the collection $\\{v_{i}\\}$, let
${\mathfrak{W}}^{\pi}_{S,0}$ be the set of divisors in ${\mathfrak{W}}_{S,0}$
whose strict transforms intersect the exceptional divisor ${\mathcal{D}}$ only
at points of $\stackrel{{\scriptstyle\circ}}{{\cal D}}$. One has the natural
map ${\check{I}}^{\pi}$ from ${\mathfrak{W}}^{\pi}_{S,0}$ onto $Y^{\pi}$ and
the map $I^{\pi}$ factorizes through it: $I^{\pi}={\check{I}}^{\pi}\circ J$.

Let ${\underline{v}}:{\mathfrak{W}}_{S,0}\to\mathbb{Q}^{r}_{\geq 0}$ be the
composition ${\underline{v}}\circ I$, where
${\underline{v}}:Y^{\pi}\to\mathbb{Q}^{r}_{\geq 0}$ is described above. The
map ${\underline{v}}$ sends ${\mathfrak{C}}_{S,0}$ to $\mathbb{Z}^{r}_{\geq
0}$. For a rational surface singularity $S$ the map
${\underline{v}}:{\mathfrak{W}}_{S,0}\to\mathbb{Q}^{r}_{\geq 0}$ can be
defined in the following way. For any divisor $C\in{\mathfrak{W}}_{S,0}$, a
multiple $kC$ of it, $k>0$, is a Cartier divisor, i.e. the divisor of a
holomorphic function $f:S\to\mathbb{C}$. (It is possible to take
$k=\det(E_{\sigma},E_{\delta})$ for a resolution of the singularity $S$.) Then
${\underline{v}}(C)={\underline{v}}(f)/k$. For two divisors $C$ and
$C^{\prime}$ on $S$ ($C=\bigcup C_{i}$, $C^{\prime}=\bigcup C^{\prime}_{j}$,
where $C_{i}$ and $C^{\prime}_{j}$ are irreducible curves) the (rational)
number $\sum_{i}v_{C_{i}}(C^{\prime})=\sum_{j}v_{C^{\prime}_{j}}(C)$ can be
regarded as the intersection number $(C,C^{\prime})$ of the curves $C$ and
$C^{\prime}$ and will be called (and denoted) in this was. (In these terms the
number $m_{\sigma\delta}$ is the intersection number of curvettes at the
components $E\sigma$ and $E_{\delta}$.)

We shall show that the Poincaré series $P_{\\{v_{i}\\}}({\underline{t}})$ can
be interpreted as an integral

$\int_{{\mathfrak{C}}_{S,0}}{\underline{t}}^{{\underline{v}}}\,d\chi\;$

with respect to the Euler characteristic. Moreover, we shall show that in the
same way the W-Poincaré series $P^{W}_{\\{v_{i}\\}}({\underline{t}})$ is equal
to

$\int_{{\mathfrak{W}}_{S,0}}{\underline{t}}^{{\underline{v}}}\,d\chi\;$

For that we have to define such integrals.

To give the definition we shall consider arcs and divisors on $(S,0)$ as arcs
and divisors on a resolution $\pi:({\mathcal{X}},{\mathcal{D}})\to(S,0)$. In
this case an arc is represented locally (in some local coordinates) by a pair
of power series in a parameter $\tau$. Let ${\cal L}_{S,0}$ be the space of
arcs (that is parametrized curves) on $(S,0)$ and let $B={\cal
L}_{S,0}/{\text{A}ut}(\mathbb{C},0)$ be the space of branches, i.e. non-
parametrized arcs. Let ${\mathcal{B}}=\bigsqcup_{k=0}^{\infty}S^{k}B$. Each
element of ${\mathcal{B}}$ represents a (effective) divisor on $(S,0)$.
However some divisors are represented by different elements of
${\mathcal{B}}$. This can be explained by the following situation. Let
$\check{\gamma}\in B$ be a branch, It is represented by an arc
$\gamma=\gamma(\tau)$. Then the branch $\check{\gamma}^{k}$ represented by the
arc $\gamma(\tau^{k})$ defines the same divisor as the collection of $k$
copies of $\check{\gamma}$. Let us call a branch $\check{\gamma}$ (represented
by an arc $\gamma$) primitive if $\gamma$ is an uniformization of its image
and let $B_{0}\subset B$ be the set of primitive branches. One has
${\mathfrak{W}}_{S,0}=\bigsqcup_{k=0}^{\infty}S^{k}B_{0}$.

Let $J^{m}B$ be the space of $m$-jets of branches on
$({\mathcal{X}},{\mathcal{D}})$. The restriction of the truncation map to
$B_{0}$ is surjective. Therefore the image of
$\bigsqcup_{k=0}^{\infty}S^{k}B_{0}$ in $\bigsqcup_{k=0}^{\infty}J^{m}B$ is
the same as the one of $\bigsqcup_{k=0}^{\infty}B$. This produces a problem to
use the image to define the integral over ${\mathfrak{W}}_{S,0}$ through this
truncation. To avoid this problem, let us consider the subspace
$J^{m}_{prim}B\subset J^{m}B$ consisting of the jets each representative of
whom is primitive (a jet is a class of branches). Let
$\mathbb{J}^{m}=\bigsqcup_{k=0}^{\infty}S^{k}J^{m}_{prim}B$.

Let $w_{i}:{\mathfrak{W}}_{S,0}\to\mathbb{Q}_{\geq 0}\cup\\{\infty\\}$,
$i=1,2,\ldots,r$ be functions on the set of Weil divisors. Let
$w_{i}^{m}:\mathbb{J}^{m}\to\mathbb{Q}_{\geq 0}\cup\\{\infty\\}$ be the
function defined by $w_{i}^{m}([a])=\sup_{a\in[a]}w_{i}(a)$ where
$[a]\subset\bigsqcup_{k=0}^{\infty}S^{k}B_{0}\subset W$ is the equivalence
class of the $m$-jet $a$. Let
$\underline{w}:=(w_{1},\,\ldots,w_{r}):{\mathfrak{W}}_{S,0}\to(\mathbb{Q}_{\geq
0}\cup\\{\infty\\})^{r}$ and
$\underline{w}^{m}:=(w^{m}_{1},\,\ldots,w^{m}_{r}):\mathbb{J}^{m}\to(\mathbb{Q}_{\geq
0}\cup\\{\infty\\})^{r}$. We shall say that the function $\underline{w}$ is
constructible if $\underline{w}^{m}$ is constructible for all $m$ (and
therefore integrable with respect to the Euler characteristic).

######  Definition 2.

The integral with respect to the Euler characteristic of the function
${\underline{t}}^{\underline{w}(-)}$ over the space ${\mathfrak{W}}_{S,0}$ is
defined by

$\int\limits_{{\mathfrak{W}}_{S,0}}{\underline{t}}^{\underline{w}(-)}d\chi=\lim_{m\to\infty}\int\limits_{\mathbb{J}^{m}_{S,0}}{\underline{t}}^{\underline{w}^{m}(-)}d\chi\in\mathbb{Z}[[t_{1}^{1/d},\ldots,t_{r}^{1/d}]]\;;$
(4)

where the limit in the right hand side is in the sense of the $\langle
t_{1},\,\ldots,t_{r}\rangle$-adic topology on
$\mathbb{Z}[[t_{1}^{1/d},\ldots,t_{r}^{1/d}]]$, $\langle
t_{1},\,\ldots,t_{r}\rangle$ is the ideal generated by $t_{1},\,\ldots,t_{r}$.

If the right hand side of (4) makes no sense, i.e. the limit does not exist,
we regard the function ${\underline{t}}^{\underline{w}(-)}$ as a non-
integrable one. For a subset ${\mathcal{A}}\subset{\mathfrak{W}}_{S,0}$ and a
function
$\underline{w}:{\mathcal{A}}\to\mathbb{Z}[[t_{1}^{1/d},\ldots,t_{r}^{1/d}]]$,
the integral $\int_{{\mathcal{A}}}{\underline{t}}^{\underline{w}(-)}d\chi$ is
understood as
$\int_{{\mathfrak{W}}_{S,0}}{\underline{t}}^{\widehat{}\underline{w}(-)}d\chi$,
where $\widehat{\underline{w}}(-)$ is the extension of the function
$\underline{w}$ by $+\infty$ outside of ${\mathcal{A}}$ (recall that
$t^{+\infty}=0$).

Now let $v_{i}$, $i=1,\ldots,r$ be curve and/or divisorial valuations on
$(S,0)$. They define natural maps (also denoted by $v_{i}$) from
${\mathfrak{W}}_{S,0}$ to $\mathbb{Z}[[t_{1}^{1/d},\ldots,t_{r}^{1/d}]]$. (In
this case one has $v_{i}(a+b)=v_{i}(a)+v_{i}(b)$.) Moreover, in this case one
assumes that $d=\det(-(E_{\sigma}\cdot E_{\delta}))$.

Let $\mathbb{J}^{m}_{\pi}$ be the set of jets of divisors which intersect the
exceptional divisor ${\mathcal{D}}$ of the resolution $\pi$ only at points of
$\stackrel{{\scriptstyle\circ}}{{\cal D}}$. There is a natural map from
$\mathbb{J}^{m}_{\pi}$ to $Y^{\pi}$. One can see that the preimage of a point
from $Y^{\pi}$ has the Euler characteristic equal to $1$. This follows from
the following arguments. An arc at a point of
$\stackrel{{\scriptstyle\circ}}{{\cal D}}$ in some local coordinates $(u,v)$
such that $\stackrel{{\scriptstyle\circ}}{{\cal D}}$ is given by the equation
$u=0$ can be written as $u=\tau^{s}$, $v$ is a (truncated) series in $t$. For
a fixed $s$ the set of jets of these arcs has the Euler characteristic equal
to $1$ (being isomorphic to a complex affine space). If $s=1$, the jet belongs
to $J^{\cdot}_{prim}B$. If $s>1$, the set of jets not belonging to
$J^{\cdot}_{prim}B$ has the Euler characteristic equal to $1$ (being also
isomorphic to a complex affine space). Therefore the set of jets belonging to
$J^{\cdot}_{prim}B$ has the Euler characteristic equal to $0$. The preimage of
a point from $Y^{\pi}$ is the union of products of symmetric powers of these
spaces. All of them have the Euler characteristics equal to zero except the
product of the symmetric powers of the spaces of the spaces of arcs with $s=1$
whose Euler characteristic is equal to $1$.

The fact that the preimage of a point from $Y^{\pi}$ has the Euler
characteristic equal to $1$ (alongside with the Fubini formula) implies the
following statements.

######  Proposition 1.

$\int\limits_{{\mathfrak{C}}_{S,0}}{\underline{t}}^{{\underline{v}}(-)}d\chi=\lim_{\\{\pi\\}}\int\limits_{Y^{\pi}_{{\mathcal{C}}}}{\underline{t}}^{{\underline{v}}(-)}d\chi=P_{\\{v_{i}\\}}({\underline{t}})\,.$

######  Proposition 2.

$\int\limits_{{\mathfrak{W}}_{S,0}}{\underline{t}}^{{\underline{v}}(-)}d\chi=\lim_{\\{\pi\\}}\int\limits_{Y^{\pi}}{\underline{t}}^{{\underline{v}}(-)}d\chi=\prod_{\sigma\in\Gamma}(1-{\underline{t}}^{\underline{m}_{\sigma}})^{-\chi(\stackrel{{\scriptstyle\circ}}{{E}}_{\sigma})}\;.$

######  Corollary 1.

$P^{W}_{\\{v_{i}\\}}({\underline{t}})=\int\limits_{{\mathfrak{W}}_{S,0}}{\underline{t}}^{{\underline{v}}(-)}d\chi\,.$

##  4 Curves and divisors on the $\mathbb{E}_{7}$-singularity whose
Weil–Poincaré series do not determine the minimal resolution

######  Example 1.

The dual graph of the minimal resolution of the $\mathbb{E}_{7}$-singularity
is shown on Figure 5. Let $C^{\prime}$ be the curvette at the component
$E_{2}$ and let $C^{\prime\prime}$ be the blow down of a smooth curve
${\widetilde{C}}^{\prime\prime}$ on the surface of the resolution tangent to
the component $E_{7}$ at a smooth point (i. e. not at the intersection point
with $E_{6}$) with the intersection multiplicity equal to 2 (i. e. the
tangency of ${\widetilde{C}}^{\prime\prime}$ and $E_{7}$ is simple). The
minimal resolution of the curve $C^{\prime}$ coincides with the minimal
resolution of the surface. The dual graph of the minimal resolution of the
curve $C^{\prime\prime}$ is shown on Figure 1.

Figure 1: The minimal resolution graph of the curve $C^{\prime\prime}$.

One can show that

$P^{W}_{C^{\prime}}(t)=P^{W}_{C^{\prime\prime}}(t)=\frac{(1-t^{6})(1-t^{8})}{(1-t^{2})(1-t^{3})(1-t^{4})}\,.$

(The data for these computations and for those in the next example can be
taken from the matrix 6 on page 6.) Therefore the W–Poincaré series of a curve
on the $\mathbb{E}_{7}$ surface singularity does not determine, in general,
the combinatorial type of its minimal resolution.

######  Example 2.

Let $D^{\prime}$ be the divisor created by the blow-up of a smooth point of
the component $E_{9}$ of the resolution shown on Figure 1 and let
$D^{\prime\prime}$ be the divisor created after 3 blow-ups starting at a
smooth point of the component $E_{2}$ and produced at each step at a smooth
point of the previously created divisor. One can show that for the divisorial
valuations $v^{\prime}$ and $v^{\prime\prime}$ defined by the divisors
$D^{\prime}$ and $D^{\prime\prime}$ respectively one has

$P^{W}_{v^{\prime}}(t)=P^{W}_{v^{\prime\prime}}(t)=\frac{(1-t^{6})(1-t^{8})}{(1-t^{2})(1-t^{3})(1-t^{4})(1-t^{9})}\,.$

Therefore the W–Poincaré series of a divisorial valuation on the
$\mathbb{E}_{7}$-singularity does not determine, in general, the combinatorial
type of the minimal resolution.

##  5 Main statements

Let $(S,0)$ be a rational double point and let $\\{v_{i}\\}$, $i=1,\ldots,r$,
be a collection of different curve valuations on ${\mathcal{O}}_{S,0}$ defined
by irreducible curves $(C_{i},0)\subset(S,0)$.

Let us make the following aditional assumptions. If the singularity $(S,0)$ is
of type $\mathbb{E}_{7}$ (respectively of type $\mathbb{E}_{8}$), we either
assume that the (minimal) resolution process does not contain a blow-up at a
smooth point of the exceptional divisor of the minimal resolution of $(S,0)$
lying on the component $E_{7}$ (on the component $E_{8}$ respectively) or
assume that does not contain a blow-up at the similar point lying on the
component $E_{2}$ (on the component $E_{6}$ respectively).

######  Theorem 1.

Under the previous assumptions the W–Poincaré series
$P^{W}_{\\{C_{i}\\}}({\underline{t}}):=P^{W}_{\\{v_{i}\\}}({\underline{t}})$,
${\underline{t}}=(t_{1},\ldots,t_{r})$, of the curve
$C=\bigcup_{i=1}^{r}C_{i}$ determines, up to the symmetry of the dual graph of
the minimal resolution of $(S,0)$, the combinatorial type of the minimal
(embedded) resolution of the curve $(C,0)=\bigcup_{i=1}^{r}(C_{i},0)$ and
therefore the topological type of the link $C\cap L$ in
$L=S\cap\mathbb{S}^{5}_{\varepsilon}$, i. e. the topological type of the pair
$(L,C\cap L)$.

Now let $\\{v_{i}\\}$, $i=1,\ldots,r$, be a collection of different divisorial
valuations on ${\mathcal{O}}_{S,0}$. In the cases of $\mathbb{E}_{7}$ and of
$\mathbb{E}_{8}$ singularities we assume the same restrictions on the
resolution process of the collection of divisorial valuations as for curves
above.

######  Theorem 2.

Under the previous assumptions the W–Poincaré series
$P^{W}_{\\{v_{i}\\}}({\underline{t}})$ determines, up to the symmetry of the
dual resolution graph of $(S,0)$, the combinatorial type of the minimal
resolution of the collection $\\{v_{i}\\}$.

######  Remark 3.

To a divisorial valuation $v_{i}$ one can associate a curve $C_{i}$: a
curvette at the component $E_{\tau(i)}$ defining the valuation $v_{i}$.
Theorem 2 implies that the series $P^{W}_{\\{v_{i}\\}}({\underline{t}})$
determines the topological type of the link
$\left(\bigcup_{i=1}^{r}C_{i}\right)\cap L$ in $L$.

######  Remark 4.

In a statement like Theorems 1 and 2 one cannot mix curve and divisorial
valuations in one collection; see an example in [7].

##  6 The case of one valuation

Let $(S,0)$ be a rational double point (of type $\mathbb{A}_{k}$,
$\mathbb{D}_{k}$, $\mathbb{E}_{6}$, $\mathbb{E}_{7}$, or $\mathbb{E}_{8})$)
and let $v$ be either a curve valuation (defined by a curve germ
$(C,0)\subset(S,0)$) or a divisorial valuation on ${\mathcal{O}}_{S,0}$. In
the latter case, let $(C,0)\subset(S,0)$ be a curvette at the divisor defining
the valuation. (A resolution of a divisorial valuation is at the same time a
resolution of the corresponding curvette, but not vise versa.) The minimal
resolution of the valuation $v$ is obtained from the minimal resolution of the
surface $(S,0)$ by a sequence of blow-ups made (at each step) at intersection
points of the strict transform of the curve $C$ and the exceptional divisor.
Let $\pi^{\prime}:({\mathcal{X}}^{\prime},{\mathcal{D}}^{\prime})\to(S,0)$ be
the minimal resolution of the surface $(S,0)$ such that the strict transform
of the curve $C$ intersects the exceptional divisor ${\mathcal{D}}^{\prime}$
at its smooth point. This resolution is obtained from the minimal resolution
of $(S,0)$ by blow-ups made (at each step) at intersection points of the
components of the exceptional divisor.

######  Definition 3.

The resolution
$\pi^{\prime}:({\mathcal{X}}^{\prime},{\mathcal{D}}^{\prime})\to(S,0)$ of
$(S,0)$ will be called the pre-resolution of the valuation $v$.

Let $E_{\sigma_{0}}$ be the component of the exceptional divisor
${\mathcal{D}}^{\prime}$ of the pre-resolution $\pi^{\prime}$ intersecting the
strict transform of curve $C$ and let $\ell$ be the intersection number of
them in the space ${\mathcal{X}}^{\prime}$ of the pre-resolution. We shall use
the numbering of the components of the exceptional divisor of the minimal
resolution of $(S,0)$ shown on Figures 2, 3, 4, 5, and 7 below. (They are at
the same time components of the exceptional divisor of the pre-resolution
$\pi^{\prime}$.) In the case of the $\mathbb{E}_{7}$ ($\mathbb{E}_{8}$)
singularity we either assume that $\sigma_{0}$ is not $7$ ($8$ respectively)
or assume that it is not $2$ ($6$ respectively). Pay atention that in all the
cases excluded from consideration the pre-resolution $\pi^{\prime}$ is the
minimal resolution of the surface $(S,0)$.

######  Lemma 1.

Under the conditions above, the W–Poincaré series $P^{W}_{C}(t)$ determines,
up to the symmetry of the resolution graph of the minimal resolution of
$(S,0)$, the pre-resolution $\pi^{\prime}$, the component $E_{\sigma_{0}}$ of
the exceptional divisor ${\mathcal{D}}^{\prime}$, and the intersection
multiplicity $\ell$.

###### Proof..

The proof is based on the analysis of the matrix $(m_{\sigma\delta})$ which
has to be made separately for different cases. In all the cases, let us write
the Poincaré series $P^{W}_{C}(t)$ in the form

$\prod_{i=1}^{q}(1-t^{m_{i}})^{-1}\prod_{m>0}(1-t^{m})^{s_{m}},$ (5)

where $m_{1}\leq m_{2}\leq\ldots\leq m_{q}$ (thus the first product may have
repeated factors) and, in the second product, the (integer) exponents $s_{m}$
are non-negative and are equal to zero for $m=m_{i}$, $i=1,\ldots,q$. Let us
recall that the representation of the Poincaré series in this form is unique.

Case of $\mathbb{A}_{k}$ singularity. Let $(S,0)$ be the singularity of type
$\mathbb{A}_{k}$. The minimal resolution graph is shown on Fig. 2.

Figure 2: The dual resolution graph of the $\mathbb{A}_{k}$-singularity.

The matrix $(m_{ij})$ is

$\frac{1}{k+1}\cdot\begin{pmatrix}k&k-1&k-2&\cdots&i&\cdots&2&1\\\
k-1&2(k-1)&2(k-2)&\cdots&2i&\cdots&4&2\\\
k-2&2(k-2)&3(k-2)&\cdots&3i&\cdots&6&3\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\vdots\\\
i&2i&3i&\cdots&i(k-i+1)&\cdots&2(k-i+1)&k-i+1\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\
1&2&3&\cdots&k-i+1&\cdots&k-1&k\end{pmatrix}.$

To identify the components of the exceptional divisor
${\mathcal{D}}^{\prime}$, let us mark them by the indices $\sigma$ being
rational numbers inbetween $1$ and $k$, naming the component created by the
blow-up of the intersection point of the components $E_{\sigma_{1}}$ and
$E_{\sigma_{2}}$ by $E_{\frac{\sigma_{1}+\sigma_{2}}{2}}$. (This methods can
be applied to other rational double points under some restrictions.) Since we
have to find $\pi^{\prime}$ and $E_{\sigma_{0}}$ up to the symmetry of the
graph in Fig. 2, we can assume that $\sigma_{0}\geq\frac{k+1}{2}$. We shall
consider the following two cases:

  1. 1)

$\sigma_{0}=k$;

  2. 2)

$\frac{k+1}{2}\leq\sigma_{0}<k$.

Let us consider Case 1. For the curve case either $q=1$ or
$\frac{m_{2}}{m_{1}}>k$ (the first option takes place if $C$ is a curvette at
$E_{k}$). For the divisorial case one has $\frac{m_{2}}{m_{1}}>k$.
In Case 2 one has $q\geq 2$, $m_{1}=\ell m_{\sigma_{0}1}$, $m_{2}=\ell
m_{\sigma_{0}k}$ and therefore $\frac{m_{2}}{m_{1}}<k$ (in contrast with Case
1). The fact that the series $P^{W}_{\bullet}(t)$ determines the component
$E_{\sigma_{0}}$ follows from the fact that the ratio $\frac{m_{2}}{m_{1}}$ is
strictly growing with $\sigma_{0}$.
In both cases the intersection multiplicity $\ell$ is determined by the
equation $m_{1}=\ell m_{\sigma_{0}1}$.

Case of $\mathbb{D}_{k}$ singularity. Let $(S,0)$ be the singularity of type
$\mathbb{D}_{k}$. The minimal resolution graph is shown on Fig. 3.

Figure 3: The dual resolution graph of the $\mathbb{D}_{k}$-singularity.

The matrix $(m_{ij})$ is

$\frac{1}{4}\cdot\begin{pmatrix}4&4&4&\cdots&4&2&2\\\ 4&8&8&\cdots&8&4&4\\\
4&8&12&\cdots&12&6&6\\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\\
4&8&12&\cdots&4(k-2)&2(k-2)&2(k-2)\\\ 2&4&6&\cdots&2(k-2)&k&k-2\\\
2&4&6&\cdots&2(k-2)&k-2&k\end{pmatrix}.$

Assume first that $k>4$. Because of the symmetry of the graph $\mathbb{D}_{k}$
we can assume that $\sigma_{0}$ does not belong to the lower right tail of the
graph, that is $1\leq\sigma_{0}\leq k-1$ (we use the same numbering of newly
created components as above for $\mathbb{A}_{k}$). We shall consider the
following four cases:

  1. 1)

$1<\sigma_{0}\leq k-2$;

  2. 2)

$k-2<\sigma_{0}<k-1$;

  3. 3)

$\sigma_{0}=1$;

  4. 4)

$\sigma_{0}=k-1$.

In Cases 1 and 2 (both in the curve and the divisorial cases) one has $q\geq
3$ and $m_{1}$, $m_{2}$, and $m_{3}$ are $\ell m_{\sigma_{0}1}$, $\ell
m_{\sigma_{0}k-1}$, and $\ell m_{\sigma_{0}k}$ in a certain order.

In Case 1 at least two of the exponents $m_{1}$, $m_{2}$, and $m_{3}$ coincide
(and are equal to $m^{\prime}$). Let us denote the third component by
$m^{\prime\prime}$. We always have
$\frac{m^{\prime}}{m^{\prime\prime}}>\frac{1}{2}$. All three exponents
coincide, that is $m^{\prime\prime}=m^{\prime}$, if and only if
$\sigma_{0}=2$. If $m^{\prime\prime}>m^{\prime}$, then $\sigma_{0}<2$ and
$m_{1}=m_{2}=\ell m_{\sigma_{0}k-1}$, $m_{3}=\ell m_{\sigma_{0}1}$. If
$m^{\prime\prime}<m^{\prime}$, then $2<\sigma_{0}\leq k-2$, $m_{1}=\ell
m_{\sigma_{0}1}$, $m_{2}=m_{3}=\ell m_{\sigma_{0}k-1}$. In this case the ratio
$\frac{m_{\sigma_{0}(k-1)}}{m_{\sigma_{0}1}}$ is strictly growing with
$\sigma_{0}$ and therefore determines $\sigma_{0}$.

In Case 2 the exponents $m_{1}$, $m_{2}$, and $m_{3}$ are different and
moreover $m_{1}=\ell m_{\sigma_{0}1}$, $m_{2}=\ell m_{\sigma_{0}k}$,
$m_{3}=\ell m_{\sigma_{0}(k-1)}$, $\frac{m_{3}}{m_{2}}<\frac{5}{4}$. Again the
ratio $\frac{m_{\sigma_{0}(k-1)}}{m_{\sigma_{0}1}}$ is strictly gowing and
therefore determines $\sigma_{0}$.

The equations above determine $\ell$.

In Case 3 one has: in the curve case either $q=2$ with $m_{1}=m_{2}=2\ell$ or
$m^{\prime}=m_{1}=m_{2}=2\ell$, $m^{\prime\prime}=m_{3}>4\ell$ with
$\frac{m^{\prime}}{m^{\prime\prime}}<\frac{1}{2}$; in the divisorial case
$\frac{m^{\prime}}{m^{\prime\prime}}\leq\frac{1}{2}$, $m_{1}=m_{2}=2\ell$.

In Case 4 one has: in the curve case either $q=2$ with $m_{1}=2\ell$ or
$m_{1}$, $m_{2}$, and $m_{3}$ are different with
$\frac{m_{3}}{m_{2}}>\frac{5}{4}$, $m_{1}=2\ell$; in the divisorial case
$m_{1}$, $m_{2}$, and $m_{3}$ are different with
$\frac{m_{3}}{m_{2}}\geq\frac{5}{4}$, $m_{1}=2\ell$.

Now let $k=4$, i. e. $(S,0)$ is the singularity of type $\mathbb{D}_{4}$.
Because of the symmetry of the graph, we can assume that $1\leq\sigma_{0}\leq
2$. We shall consider the following two case:

  1. 1)

$1<\sigma_{0}\leq 2$.

  2. 2)

$\sigma_{0}=1$;

In Case 1 one has (both for the curve and for the divisorial cases)
$m_{1}=m_{2}=\ell m_{\sigma_{0},3}$, $m_{3}=\ell m_{\sigma_{0},1}$ and
$m_{3}/m_{1}<2$. The ratio $\frac{m_{\sigma_{0}3}}{m_{\sigma_{0}1}}$ is
strictly growing with $\sigma_{0}$ and therefore determines the latter one. In
Case 2 one has: in the curve case either $q=2$ with $m_{1}=m_{2}=2\ell$ or
$m_{1}=m_{2}=2\ell$ and $m_{3}/m_{1}>2$; in the divisorial case
$m_{1}=m_{2}=2\ell$ and $m_{3}/m_{1}\geq 2$.

Case of $\mathbb{E}_{6}$ singularity. Let $(S,0)$ be the singularity of type
$\mathbb{E}_{6}$. The minimal resolution graph is shown on Fig. 4.

Figure 4: The dual resolution graph of the $\mathbb{E}_{6}$-singularity.

The matrix $(m_{ij})$ is

$\begin{pmatrix}4/3&5/3&2&1&4/3&2/3\\\ 5/3&10/3&4&2&8/3&4/3\\\ 2&4&6&3&4&2\\\
1&2&3&2&2&1\\\ 4/3&8/3&4&2&10/3&5/3\\\ 2/3&4/3&2&1&5/3&4/3\end{pmatrix}.$

Because of the symmetry of the graph, we can assume that $\sigma_{0}$ does not
belong to the right tail of the graph, i. e. (using the same rule of numbering
of the components of the exceptional divisor ${\mathcal{D}}^{\prime}$ as
above) $1\leq\sigma_{0}\leq 4$. We shall consider the following four cases:

  1. 1)

$1<\sigma_{0}<3$;

  2. 2)

$3\leq\sigma_{0}<4$;

  3. 3)

$\sigma_{0}=1$;

  4. 4)

$\sigma_{0}=4$.

In Cases 1 and 3 one has $m_{1}<m_{2}$; in Cases 2 and 4 $m_{1}=m_{2}$.

In Case 1 $\frac{m_{3}}{m_{1}}<2$. In Case 3 one has: in the curve case either
$q=2$ or $\frac{m_{3}}{m_{2}}>2$; in the divisorial case
$\frac{m_{3}}{m_{2}}\geq 2$ with $m_{1}=\ell$ in all the cases.

In Case 2 $\frac{m_{3}}{m_{1}}<2$. In Case 4 one has: in the curve case either
$q=2$ or $\frac{m_{3}}{m_{1}}>2$; in the divisorial case
$\frac{m_{3}}{m_{1}}\geq 2$ with $m_{1}=\ell$ in all the cases.

In Case 1 (both for the curve and for the divisorial valuation) one has either
$\frac{m_{2}}{m_{1}}=\frac{3}{2}$ or $\frac{m_{3}}{m_{1}}=\frac{3}{2}$. If
$\frac{m_{2}}{m_{1}}=\frac{3}{2}$, then $\sigma_{0}\leq 1.5$, the ratio
$\frac{m_{3}}{m_{1}}$ is strictly increasing with $\sigma_{0}$ and therefore
determines $\sigma_{0}$; $m_{1}=\ell m_{\sigma_{0}4}$. If
$\frac{m_{3}}{m_{1}}=\frac{3}{2}$, then $\sigma_{0}\geq 1.5$, the ratio
$\frac{m_{2}}{m_{1}}$ is strictly decreasing with $\sigma_{0}$ and therefore
determines $\sigma_{0}$; $m_{1}=\ell m_{\sigma_{0}6}$. In Case 2 (both for the
curve and for the divisorial valuation) the ratio $\frac{m_{3}}{m_{1}}$ is
strictly increasing with $\sigma_{0}$ and therefore determines $\sigma_{0}$;
$m_{1}=\ell m_{\sigma_{0}1}$.

Case of $\mathbb{E}_{7}$ singularity. Let $(S,0)$ be the singularity of type
$\mathbb{E}_{7}$. The minimal resolution graph is shown on Fig. 5.

Figure 5: The dual resolution graph of the $\mathbb{E}_{7}$-singularity.

The matrix $(m_{ij})$ is

$\begin{pmatrix}2&3&4&2&3&2&1\\\ 3&6&8&4&6&4&2\\\ 4&8&12&6&9&6&3\\\
2&4&6&7/2&9/2&3&3/2\\\ 3&6&9&9/2&15/2&5&5/2\\\ 2&4&6&3&5&4&2\\\
1&2&3&3/2&5/2&2&3/2\end{pmatrix}.$ (6)

Let us analyze first the situation when we assume that $\sigma_{0}\neq 7$. We
shall consider the following cases:

  1. 1.

$\sigma_{0}\neq 1,4$;

  2. 2.

$\sigma_{0}=1$;

  3. 3.

$\sigma_{0}=4$.

In Case 1, both for the curve and for the divisorial valuations one has
$m_{3}=\ell m_{\sigma_{0},4}$ and $m_{1}$ and $m_{2}$ are $\ell
m_{\sigma_{0}1}$ and $\ell m_{\sigma_{0},7}$ in a certain order. Moreover
$m_{2}/m_{1}<2$ and $m_{3}/m_{1}<7/3$.

On Figure 6 the ratio
$(m_{\sigma_{0},1}:m_{\sigma_{0},4}:m_{\sigma_{0},7})\in\mathbb{R}\mathbb{P}^{2}$
is shown (by the bold lines) in the affine chart
$(m_{\sigma_{0},1}/m_{\sigma_{0},4},m_{\sigma_{0},7}/m_{\sigma_{0},4})$.

Figure 6: The points
$(m_{\sigma_{0},1}:m_{\sigma_{0},4}:m_{\sigma_{0},7})\in\mathbb{R}\mathbb{P}^{2}$.

(The fact that the edges meeting at a vertex of valency $2$ lie on a straight
line is a general feature of pictures of this sort.) The figure shows that the
ratio $(m_{\sigma_{0},1}:m_{\sigma_{0},4}:m_{\sigma_{0},7})$ determines
$\sigma_{0}$. However the explanation above says that the W-Poincaré series
determines this ratio only up to the exchange of the first and the third
components. The graph obtained by exchanging $m_{\sigma_{0},1}$ and
$m_{\sigma_{0},7}$ is drown by thin lines. One can see that they intersect
only at a point on the diagonal
$\frac{m_{\sigma_{0},1}}{m_{\sigma_{0},4}}=\frac{m_{\sigma_{0},7}}{m_{\sigma_{0},4}}$
and therefore the ratios
$(m_{\sigma_{0},1}:m_{\sigma_{0},4}:m_{\sigma_{0},7})$ determines
$\sigma_{0}$.

In Case 2, one has $m_{2}/m_{1}=2$ (in contrast to Case 1 above and Case 3
below); $m_{1}=\ell$. In Case 3 one has $m_{2}/m_{1}=4/3$ and for the curve
case, either $q=2$ or $m_{3}/m_{1}>7/3$; for the divisorial case
$m_{3}/m_{1}\geq 7/3$; $m_{2}=2\ell$.

In the situation when we assume that $\sigma_{0}\neq 2$, we shall consider the
following four cases:

  1. 1.

$\sigma_{0}\neq 1,4,7$;

  2. 2.

$\sigma_{0}=1$;

  3. 3.

$\sigma_{0}=4$;

  4. 4.

$\sigma_{0}=7$.

The analysis of the Cases 1 to 3 is the same as above. In Case 4 one has
$m_{2}/m_{1}=3/2$. This differs Case 4 from Cases 2 and 3 and in Case 1 the
value $m_{2}/m_{1}=3/2$ holds only if $\sigma_{0}=2$. In this case
$m_{1}=\ell$.

Case of $\mathbb{E}_{8}$ singularity. Let $(S,0)$ be the singularity of type
$\mathbb{E}_{8}$. The minimal resolution graph is shown in Figure 7.

Figure 7: The dual resolution graph of the $\mathbb{E}_{8}$-singularity.

The matrix $(m_{ij})$ is:

$\begin{pmatrix}4&7&10&5&8&6&4&2\\\ 7&14&20&10&16&12&8&4\\\
10&20&30&15&24&18&12&6\\\ 5&10&15&8&12&9&6&3\\\ 8&16&24&12&20&15&10&5\\\
6&12&18&9&15&12&8&4\\\ 4&8&12&6&10&8&6&3\\\ 2&4&6&3&5&4&3&2\end{pmatrix}$

The situation when we assume that $\sigma_{0}\neq 8$ was analyzed in [9]. One
can see that, in this case, the possibility to restore $\ell$ easily follows
from the discussion therein. In the situation when we assume that
$\sigma_{0}\neq 6$, we shall consider the following four cases:

  1. 1.

$\sigma_{0}\neq 1,4,8$;

  2. 2.

$\sigma_{0}=1$;

  3. 3.

$\sigma_{0}=4$;

  4. 4.

$\sigma_{0}=8$.

In Case 1 one has $m_{2}/m_{1}\leq 2$. The way to determine $\sigma_{0}$ in
this case is described in [9]; $\mathfrak{m}_{1}=\ell m_{\sigma_{0}8}$. In
Case 2 one has $m_{2}/m_{1}=5/2$. In Case 3 one has $m_{2}/m_{1}=5/3$ and
either $q=2$ (in the curve case) or $m_{3}/m_{1}\geq 8/3$. This can be met in
Case 1 when $3\leq\sigma_{0}<4$ (i.e. if $\sigma_{0}$ is on the lower tail of
the diagram). In this case one has $5/2\leq m_{3}/m_{1}<8/3$. In case 3
$m_{1}=3\ell$.

In Case 4 one has $m_{2}/m_{1}=3/2$. This differs Case 4 from Cases 2 and 3.
In Case 1 the value $m_{2}/m_{1}=3/2$ holds only if $\sigma_{0}=6$. In Case 4
$m_{1}=2\ell$.

∎

Let $(C,0)$ be an irreducible curve on a rational double point $(S,0)$.

######  Proposition 3.

Under the described assumptions, the W-Poincaré series $P^{W}_{C}(t)$
determines the combinatorial type of the minimal embedded resolution of the
curve $C$ and therefore the topological type ot the knot $C\cap L$ in
$L=S\cap\mathbb{S}^{5}_{\varepsilon}$.

Let $v$ be a divisorial valuation on $(S,0)$.

######  Proposition 4.

Under the described assumptions, the W-Poincaré series $P^{W}_{\\{v\\}}(t)$
determines the combinatorial type of the minimal resolution of the valuation
$v$.

Proofs of Propositions 3 and 4 are literally the same as of Theorems 1 and 2
in [9].

Assume that $C_{1}$ and $C_{2}$ are two curves with the (known) W–Poincaré
series $P^{W}_{C_{i}}(t)$ such that the components $E_{\sigma_{0}^{i}}$ of the
exceptional divisors ${\mathcal{D}}^{\prime}_{i}$ of the pre-resolutions
emerge from the parts (one can say “tails” in all the cases but
$\mathbb{A}_{k}$) of the resolution graph of the minimal resolution of $(S,0)$
exchangable by a symmetry of the graph acting non-trivially on them. (Pay
attention that in this case $S$ is one of the singularities $\mathbb{A}_{k}$,
$\mathbb{D}_{k}$, and $\mathbb{E}_{6}$. It cannot be a singularity of type
$\mathbb{E}_{7}$ or $E_{8}$ when the series W–Poincaré series determines the
component $E_{\sigma_{0}}$ of the pre-resolution only under additional
conditions.)

######  Lemma 2.

In the discribed situation (for $(S,0)$ of the type $\mathbb{A}_{k}$,
$\mathbb{D}_{k}$, or $\mathbb{E}_{6}$), The W–Poincaré series
$P^{W}_{C_{i}}(t)$, $i=1.2$, alongside with the intersection number
$(C_{1}\circ C_{2})$ determines whether the strict transforms of the curves
$C_{1}$ and $C_{2}$ intersect the same part of the graph of the minimal
resolution of the surface singularity $(S,0)$ or different ones.

###### Proof..

If the strict transforms intersect the same part of the graph, then
$(C_{1}\circ C_{2})\geq\ell_{1}\cdot\ell_{2}\cdot
m_{\sigma_{0}^{1}\sigma_{0}^{2}}$. (The sign $>$ may hold only if
$\sigma_{0}^{1}=\sigma_{0}^{2}$.) If the strict transforms intersect different
parts of the graph, one has $(C_{1}\circ C_{2})=\ell_{1}\cdot\ell_{2}\cdot
m_{\sigma_{0}^{1}\sigma_{0}^{2}}$. Moreover, from the matrices
$(m_{\sigma\delta})$ above one can see that, for fixed up to symmetry
$\sigma_{0}^{1}$ and $\sigma_{0}^{2}$, the intersection number
$m_{\sigma_{0}^{1}\sigma_{0}^{2}}$ for the components $\sigma_{0}^{1}$ and
$\sigma_{0}^{2}$ from the same part is strictly larger than the one for the
components from different parts. ∎

##  7 The case of several valuations

The idea of the proofs of Theorems 1 and 2 is the same as of Theorems 3 and 4
in [9]. Moreover, the proof of Theorem 2 is literally the same modulo one
remark related with the symmetry of the minimal resolution graph of the
surface singularity: see at the end of the section. This is explained by the
fact that the “projection formula” (the equation connecting the Weil–Poincaré
series of a collection of valuations with the one for the collection with one
valuation excluded) is much simpler for a collection of divisorial valuations.
In this case the Weil–Poincaré series of the smaller collection is obtained
from the other one simply by putting the value 1 for the corresponding
variable $t_{i}$. Thus the Weil–Poincaré series of the smaller collection is
determined by the same series for the larger one. This is not the case, in
general, for a collection of curve valuations. In this case the projection
formula includes a certain factor for the excluded valuation which, in
general, cannot be directly obtained from the initial Weil–Poincaré series.
(This also explains the fact that, in [9], the proof for divisorial valuations
is much shorter than for curve ones.) For the curve case (Theorem 1) the proof
contains some differences. Therefore we include some parts of it.

Let $C$ be an irreducible curve germ in the surface $(S,0)$ and let
$\pi:({\mathcal{X}},{\mathcal{D}})\to(S,0)$ be an embedded resolution of the
curve $(C,0)\subset(S,0)$. Let $\Gamma$ be the minimal resolution graph of
$C$, i. e., the dual graph of $\pi$. Let $\overline{C}$ be the total transform
of the curve $C$ in ${\mathcal{X}}$. One has

$\overline{C}=\widetilde{C}+\sum_{\sigma\in\Gamma}m_{\sigma}E_{\sigma}\;,$

where $\widetilde{C}$ is the strict transform of the curve $C$. The rational
numbers $\\{m_{\sigma}:\sigma\in\Gamma\\}$ are uniquely determined by the
system of linear equations

$\\{\widetilde{C}\cdot
E_{\alpha}+\sum_{\sigma\in\Gamma}m_{\sigma}\;E_{\sigma}\cdot E_{\alpha}=0\ :\
E_{\alpha}\in\cal{D}\\}\;.$ (7)

(Each equation is a consequence of the fact $\overline{C}\cdot E_{\alpha}=0$,
see [13, Equation (1)]).

Let $\rho\in\Gamma$ be an end and let
$\Delta=\\{\rho=\alpha_{0},\alpha_{1},\ldots.\alpha_{s}=\sigma\\}$ be the
corresponding dead arc in $\Gamma$ (i.e. the minimal connected subgraph of
$\Gamma$ such that $\sigma$ is a star vertex). Then one has:

######  Lemma 3.

If $-E_{\alpha_{i}}^{2}\neq 1$ for $i=0,\ldots,s-1$, then there exists an
integer $N>1$, independent of the branch $C$, such that
$m_{\sigma}=Nm_{\rho}$.

###### Proof..

Equation (7) for $E_{\rho}$ gives
$m_{\alpha_{1}}=(-E_{\rho}^{2})m_{\rho}=N_{1}m_{\rho}$ with an integer
$N_{1}>1$. Again, the same equation for $E_{\alpha_{i}}$ gives

$\displaystyle m_{\alpha_{i+1}}$
$\displaystyle=(-E_{\alpha_{i}}^{2})m_{\alpha_{i}}-m_{\alpha_{i-1}}=-E_{\alpha_{i}}^{2}N_{i}m_{\rho}-N_{i-1}m_{\rho}=$
$\displaystyle=(-E_{\alpha_{i}}^{2}N_{i}-N_{i-1})m_{\rho}=N_{i+1}m_{\rho}\;.$

Since $E_{\alpha_{i}}^{2}\neq-1$, one has $N_{i+1}\geq 2N_{i}-N_{i-1}>N_{i}$
provided we assume $N_{i}>N_{i-1}$ by induction. ∎

Let $C=\bigcup_{i=1}^{r}C_{i}$ be a reducible (that is, $r>1$) curve germ in
the surface $(S,0)$ and let $\pi:({\mathcal{X}},{\mathcal{D}})\to(S,0)$ be the
minimal embedded resolution of the curve $(C,0)\subset(S,0)$. Let $\Gamma$ be
the minimal resolution graph of $C$, i. e., the dual graph of $\pi$. Let
$\tau(i)$ be the vertex of $\Gamma$ such that the component $E_{\tau(i)}$ of
the exceptional divisor ${\mathcal{D}}$ intersects the strict transform
$\widetilde{C}_{i}$ of the curve $C_{i}$ and let
$m_{\sigma}^{i}:=m_{\sigma\tau(i)}$. One has
$\underline{m}_{\sigma}=(m_{\sigma}^{1},\ldots,m_{\sigma}^{r})$. The reason
(somewhat psychological) for that is the fact that, for a multi-exponent of a
term of the Poincaré series $P_{C}(t_{1},\,\ldots,t_{r})$ or of a factor of
its decomposition, one knows its components $m_{\sigma}^{1}$, …,
$m_{\sigma}^{r}$, but does not know the vertex $\sigma$. One can say that our
aim is to find vertices $\tau(i)$ corresponding to the curve.

Let $\overline{C}_{k}$ ($k=1,\ldots,r$) be the total transform of the curve
$C_{k}$ in ${\mathcal{X}}$. One has

$\overline{C}_{k}=\widetilde{C}_{k}+\sum_{\sigma\in\Gamma}m_{\sigma}^{k}E_{\sigma}\;,$

where $\widetilde{C}_{k}$ is the strict transform of the curve $C_{k}$.

Let us fix a pair of branches $C_{i}$ and $C_{j}$ and let
$q:\Gamma\to\mathbb{Q}$ be the function defined by
$q(\alpha)=m_{\alpha}^{j}/m_{\alpha}^{i}$ for $\alpha\in\Gamma$. One has the
following statement.

######  Lemma 4.

Let $E_{\alpha}$ be a component of the exceptional divisor ${\mathcal{D}}$
such that $\widetilde{C}_{i}\cdot E_{\alpha}=0$ and let
$\\{\rho_{1},\,\ldots,\rho_{s}\\}\subset\Gamma$ be the set of all vertices
connected by an edge with $\alpha$. Let us assume that either
$\widetilde{C}_{j}$ intersects $E_{\alpha}$ or there exists $\rho_{i_{0}}$
such that $q(\rho_{i_{0}})>q(\alpha)$. Then there exists $\rho_{k}$ such that
$q(\alpha)>q(\rho_{k})$.

###### Proof..

Assume that $q(\rho_{k})\geq q(\alpha)$ for any $k=1,\ldots,s$. Applying (7)
to $C_{j}$ and $C_{i}$ one gets:

$\displaystyle 0$ $\displaystyle=\widetilde{C}_{j}\cdot
E_{\alpha}+m_{\alpha}^{j}E_{\alpha}^{2}+\sum_{k=1}^{s}m_{\rho_{k}}^{j}\geq$
$\displaystyle\geq\widetilde{C}_{j}\cdot
E_{\alpha}+m_{\alpha}^{j}E_{\alpha}^{2}+\sum_{k=1}^{s}q(\alpha)m_{\rho_{k}}^{i}=$
$\displaystyle=\widetilde{C}_{j}\cdot
E_{\alpha}+q(\alpha)(m_{\alpha}^{i}E_{\alpha}^{2}+\sum_{k=1}^{s}m_{\rho_{k}}^{i})={\widetilde{C}}_{j}\cdot
E_{\alpha}\geq 0$

The inequality is strict if $\widetilde{C}_{j}\cdot E_{\alpha}>0$ or if there
exists $i_{0}$ such that $q(\rho_{i_{0}})>q(\alpha)$. This implies the
statement. ∎

Let $[\tau(j),\tau(i)]\subset\Gamma$ be the (oriented) geodesic from $\tau(j)$
to $\tau(i)$ and let $\\{\Delta_{p}\\}$, $p\in\Pi$, be the connected
components of $\Gamma\setminus[\tau(j),\tau(i)]$. For each $p\in\Pi$ there
exists a unique $\rho_{p}\in[\tau(j),\tau(i)]$ connecting $\Delta_{p}$ with
$[\tau(j),\tau(i)]$, i. e., such that
$\Delta^{*}_{p}=\Delta_{p}\cup\\{\rho_{p}\\}$ is connected.

######  Proposition 5.

With the previous notations, one has:

  1. 1.

The function $q$ is strictly decreasing along the geodesic
$[\tau(j),\tau(i)]$.

  2. 2.

For each $p\in\Pi$, the function $q$ is constant on $\Delta^{*}_{p}$.

###### Proof..

Let $\alpha$ and $\beta$ be two vertices of $\Gamma$ connected by an edge and
let $q(\alpha)>q(\beta)$. Lemma 4 permits to construct a maximal sequence
$\alpha_{0},\alpha_{1},\ldots,\alpha_{k}$ of consecutive vertices starting
with $\alpha$ and $\beta$ (i. e., $\alpha_{0}=\alpha$, $\alpha_{1}=\beta$)
such that $q(\alpha_{i})>q(\alpha_{i+1})$. (We will call a sequence of this
sort a decreasing path. If the inequality is in the other direction, the path
will be called increasing.) The maximality means that either $\alpha_{k}$ is a
deadend of $\Gamma$ or $\widetilde{C}_{i}\cdot E_{\alpha_{k}}\neq 0$. If
$\alpha_{k}$ is a deadend, $\alpha_{k-1}$ is the only vertex connected with
$\alpha_{k}$ and Lemma 4 implies that $q(\alpha_{k})=q(\alpha_{k-1})$.
Therefore the constructed path finishes by the vertex $\alpha_{k}=\tau(i)$.
Note that, if $\alpha\in[\tau(j),\tau(i)]$ and $\beta\notin[\tau(j),\tau(i)]$,
the end of a maximal decreasing (or increasing) path has to finish at a
deadend and therefore $q(\alpha)=q(\beta)$. In particular, this implies that
the function $q$ is constant on each connected set $\Delta^{*}_{p}$.

Assume that $\tau(i)\neq\tau(j)$. Lemma 4 implies that there exists a vertex
$\alpha_{1}$ connected with $\tau(j)$ such that $q(\tau(j))>q(\alpha_{1})$.
Therefore the maximal decreasing path starting with $\tau(j)$ and $\alpha_{1}$
coincides with the geodesic $[\tau(j),\tau(i)]$. ∎

######  Remark 5.

Let $\rho\in\Gamma$ be an end and let
$\Delta=\\{\rho=\alpha_{0},\alpha_{1},\ldots.\alpha_{s}=\sigma\\}$ be the
corresponding dead arc in $\Gamma$. In this case Proposition 5 implies that
the ratio $m^{i}_{\alpha}/m^{j}_{\alpha}$ is constant for $\alpha\in\Delta$
and for any pair $i,j\in\\{1,\ldots,r\\}$. In fact, from Lemma 3, one can
easily deduce that $\underline{m}_{\sigma}=N\underline{m}_{\rho}$ for an
integer $N>1$, in particular, $\underline{m}_{\sigma}>\underline{m}_{\rho}$.

Proof of Theorem 1. We have to show that the Weil-Poincaré series
$P^{W}_{\\{C_{i}\\}}({\underline{t}})$ determines the minimal resolution graph
$\Gamma$ of $C$. In the case under consideration one has a projection formula
different of the one for divisorial valuations.

Let $i_{0}\in\\{1,\ldots,r\\}$. The A’Campo type formula (3) for
$P^{W}_{\\{C_{i}\\}}({\underline{t}})$ implies that

$P^{W}_{\\{C_{i}\\}}({\underline{t}})_{|_{t_{i_{0}}=1}}=P^{W}_{C\setminus\\{C_{i_{0}}\\}}(t_{1},\ldots,t_{i_{0}-1},t_{i_{0}+1},\ldots,t_{r})\cdot(1-{\underline{t}}^{\underline{m}_{\tau(i_{0})}})_{|_{t_{i_{0}}=1}}\;.$
(8)

Applying (8) several times one gets

$P^{W}_{C}({\underline{t}})_{|_{t_{j}=1\text{ for }j\neq
i_{0}}}=P^{W}_{C_{i_{0}}}(t_{i_{0}})\cdot\prod_{i\neq
i_{0}}(1-t_{i_{0}}^{m_{\tau(i)}^{i_{0}}})\;.$ (9)

Pay attention to the fact that $m_{\tau(i)}^{i_{0}}=m_{\tau(i_{0})}^{i}$ and
therefore the series $P^{W}_{C_{i_{0}}}(t_{i_{0}})$ can be determined from the
Weil-Poincaré series $P_{C}({\underline{t}})$ if one knows the multiplicity
$\underline{m}_{\tau(i_{0})}$. The strategy of the proof follows the steps
from [7] (see also [8]):

  1. 1)

To detect an index $i_{0}$ for which one can find the corresponding
multiplicity $\underline{m}_{\tau(i_{0})}$ from the A’Campo type formula for
$P^{W}_{C}({\underline{t}})$. Then Theorem 2 and equation (9) permit to
recover the minimal resolution graph $\Gamma_{i_{0}}$ of the curve
$C_{i_{0}}$. Equation (8) gives the possibility to compute the Poincaré series
$P^{W}_{C\setminus\\{C_{i_{0}}\\}}(t_{1},\ldots,t_{i_{0}-1},t_{i_{0}+1},\ldots,t_{r})$
of the curve $C\setminus\\{C_{i_{0}}\\}$. By induction one can assume that the
resolution graph $\Gamma^{i_{0}}$ of the curve $C\setminus\\{C_{i_{0}}\\}$ is
known. Moreover, Lemma 2 implies that, for each $j$, the multiplicity
$\underline{m}_{\tau(i_{0})}$ determines whether the vertices of the minimal
resolution graph coresponding to the curves $C_{i_{0}}$ and $C_{j}$ are on the
same part from those exchanged by symmetries or on different ones.

  2. 2)

To determine the separation vertex of the curves $C_{i_{0}}$ and $C_{j}$ for
$j\neq i_{0}$ in order to join the graphs $\Gamma_{i_{0}}$ and
$\Gamma^{i_{0}}$ to obtain the resolution graph $\Gamma$.

The second step literally repeats the same one in the proof of Theorem 3 in
[9]. Therefore we omit an analysis of it here.

Proposition 5 implies that, for any fixed $i_{0}$ and for any $j\neq i_{0}$
and $\sigma\in\Gamma$, one has $m_{\sigma}^{j}/m_{\sigma}^{i_{0}}\geq
m_{\tau(i_{0})}^{j}/m_{\tau(i_{0})}^{i_{0}}$. Therefore one has

$\frac{1}{m_{\sigma}^{i_{0}}}\underline{m}_{\sigma}\geq\frac{1}{m_{\tau(i_{0})}^{i_{0}}}\underline{m}_{\tau(i_{0})}\;.$

Let
$P^{W}_{C}({\underline{t}})=\prod_{k=1}^{p}(1-{\underline{t}}^{{\underline{n}}_{k}})^{s_{k}}$
be the Weil-Poincaré series of the curve $C$, where $s_{k}\neq 0$ for all $k$.
Note that the only case in which $P^{W}_{C}({\underline{t}})=1$ corresponds to
the singularity $\mathbb{A}_{k}$ and two branches $C_{1}$, $C_{2}$ in such a
way that they are curvettes at the end points named $1$ and $k$ of the dual
graph of $\mathbb{A}_{k}$. In the sequel we omit this trivial situation. For
$i\in I=\\{1,\ldots,r\\}$ let $\kappa:I\to\\{1,\ldots,p\\}$ be the map defined
by $k=\kappa(i)$ be such that $s_{k}>0$ and

$\frac{1}{n_{j}^{i}}{\underline{n}}_{j}\geq\frac{1}{n_{k}^{i}}{\underline{n}}_{k}$
(10)

for all $j$. Note that if the inequalities (10) hold for
${\underline{n}}_{k}=\underline{m}_{\rho}$, $\rho$ a deadend of $\Gamma$, then
(see Proposition 5) one has the same condition for $\sigma$, the star vertex
of $\Gamma$ more close to $\rho$. Let $E=\kappa(I)\subset\\{1,\ldots,p\\}$ be
the set of indices $k$ such that $k=\kappa(i)$ for some $i\in\\{1,\ldots,r\\}$
and for $k\in E$ let $A(k)\subset\\{1,\ldots,r\\}$ denote the set of indices
$i$ such that $k=\kappa(i)$. Note that $A(k)$ contains all the indices
$i\in\\{1,\ldots,r\\}$ such that
${\underline{n}}_{k}=\underline{m}_{\tau(i)}$. Let $B(k)$ be the subset of
such indices. Our aim is to show that one can find $k\in E$ such that
$B(k)\neq\emptyset$.

######  Proposition 6.

Let us assume that $\\#E\geq 2$. Then there exists $k\in E$ and $i\in A(k)$
such that

  1. 1.

$n^{i}_{k}\geq n^{j}_{k}$ for any $j\in A(k)$

  2. 2.

$n^{j}_{k}\leq n^{i}_{k^{\prime}}$ for any $k^{\prime}\in E$, $k^{\prime}\neq
k$, and $j\in A(k^{\prime})$.

Moreover, for any pair $(k,i)$ satistying conditions 1) and 2) above, one has
that $i\in B(k)$ and therefore $B(k)\neq\emptyset$.

###### Proof..

Let $j\in A(k)$, $j\notin B(k)$. One has

$\frac{1}{n_{k}^{j}}{\underline{n}}_{k}>\frac{1}{m_{\tau(j)}^{j}}\underline{m}_{\tau(j)}$

and therefore $\chi(\stackrel{{\scriptstyle\circ}}{{E}}_{\tau(j)})=0$. This
implies that $\tau(j)$ is connected with only one vertex in $\Gamma$ (plus the
arrow corresponding to $\widetilde{C}_{j}$), i. e., $\tau(j)$ is a deadend of
the resolution graph of the curve $C\setminus\\{C_{j}\\}$. Let
$\sigma\in\Gamma$ be such that ${\underline{n}}_{k}=\underline{m}_{\sigma}$
and assume that $i\in B(k)\neq\emptyset$. In this case
${\underline{n}}_{k}=\underline{m}_{\tau(i)}$ and one has
$n^{i}_{k}=m^{i}_{\tau(i)}=Nm^{i}_{\tau(j)}$, with an integer $N>1$, being
$\tau(j)$ a deadend of the dual graph of $C\setminus\mathbb{C}_{j}\\}$ (see
3). In particular $n^{i}_{k}>m^{i}_{\tau(j)}=m^{j}_{\tau(i)}=n^{j}_{k}$.
Notice that if $i,s\in B(k)$ one has also that
${\underline{n}}_{k}=\underline{m}_{\tau(s)}$ and therefore
$n^{i}_{k}=n^{s}_{k}$. As a consequence, if $B(k)\neq\emptyset$ one can fix an
index $i(k)\in B(k)$ taking a maximal one of the set $\\{n^{i}_{k}:i\in
A(k)\\}$ as in the condition 1) of the statement.

Let $k^{\prime}\in E$, $k^{\prime}\neq k$, and $j\in A(k^{\prime})$. If $j\in
B(k^{\prime})$, then ${\underline{n}}_{k^{\prime}}=\underline{m}_{\tau(j)}$
and therefore
$n^{i(k)}_{k^{\prime}}=m^{i(k)}_{\tau(j)}=m^{j}_{\tau(i(k))}=n^{j}_{k}$.
Otherwise $j\in A(k^{\prime})\setminus B(k^{\prime})$ and one has that
$n^{j}_{k}=m^{j}_{\tau(i(k))}=m^{i(k)}_{\tau(j)}$ and also
$n^{i(k)}_{k^{\prime}}=Nm^{i(k)}_{\tau(j)}$ for a positive integer $N$. As a
consequence $n^{i(k)}_{k^{\prime}}>n^{j}_{k}$ and therefore $i(k)$ satisfies
the second requirement of the proposition.

In order to finish the proof one has to prove that $B(k)\neq\emptyset$ for
some $k\in E$. Let us assume that $B(k)=\emptyset$ for some $k\in E$ and
${\underline{n}}_{k}=\underline{m}_{\sigma}$, for some $\sigma\in\Gamma$ with
$v(\sigma)\geq 3$ (here $v(\sigma)$ is the valency of the vertex $\sigma$).
Let $\pi^{\prime}:({\mathcal{X}}^{\prime},{\mathcal{D}}^{\prime})\to(S,0)$ be
the pre-resolution of $C$ and let $\Gamma^{\prime}$ be the resolution graph of
$\pi^{\prime}$. For any $j\in A(k)$ one has that $\tau(j)$ is an end of
$\Gamma\setminus\\{\widetilde{C}_{j}\\}$. We will distinguish two cases:

Let us assume that $\sigma\notin\Gamma^{\prime}$. In this case also
$\tau(j)\notin\Gamma^{\prime}$ is a deadend of the dual resolution graph of
the curve $C\setminus\\{C_{j}\\}$ and $A(k)=\\{j\\}$. In particular, the
vertex $\sigma$ appears after $\tau(j)$ in the resolution process of a certain
branch $C_{i}$, $i\neq j$, which is not a curvette at $E_{\sigma}$. It is
clear that in this case
$(1/n^{i}_{k}){\underline{n}}_{k}>(1/m^{i}_{\tau(i)})\underline{m}_{\tau(i)}$
and $k^{\prime}=\kappa(i)$ provides a new element $k^{\prime}\in E$. Using
this new element $k^{\prime}$ we can repeat the argument up to the moment when
we reach $e\in E$ such that $B(e)\neq\emptyset$ (note that
${\underline{n}}_{k}<{\underline{n}}_{k^{\prime}}$).

Assume that $\sigma\in\Gamma^{\prime}$. If there exists an irreducible
component $C_{i}$ such that its strict transform by $\pi^{\prime}$ intersects
${\mathcal{D}}^{\prime}$ at $E_{\sigma}$ one can repeat the same argument of
the above case for $k^{\prime}=\kappa(i)$ and so there exists an element $e\in
E$ such that $B(e)\neq\emptyset$. Otherwise $\sigma\in\Gamma^{\prime}$ must be
a star vertex of $\Gamma^{\prime}$ and all the elements of $A(k)$ correspond
to curvettes at some of the end points of $\Gamma^{\prime}$. However, there is
(at most) only one vertex in $\Gamma^{\prime}$ with such conditions: the
vertex named $k-2$ if $S=\mathbb{D}_{k}$, the vertex named $3$ for
$\mathbb{E}_{6},\mathbb{E}_{7}$ and $\mathbb{E}_{8}$ and no one for the case
$\mathbb{A}_{k}$. Being $\\#E\geq 2$ in order to finish it suffices to take a
new $k^{\prime}\in E$, $k^{\prime}\neq k$. ∎

In order to finish the proof of the Theorem it remains only to treat the case
when $\\#E=1$, i.e. $k\in E$ is such that $A(k)=\\{1,\ldots,r\\}$. First of
all we will identify the cases when $B(k)=\emptyset$. Taking into account the
discussion in the proof of the Proposition above about the indices $j\in
A(k)\setminus B(k)$, the only possibility to have this situation is when
${\underline{n}}_{k}=\underline{m}_{\sigma}$ for $\sigma\in\Gamma^{\prime}$
being a star vertex and all the branches are curvettes at some end points of
$\Gamma^{\prime}$ and no two of them are in the same vertex. This situations
can be described (and so detected) one by one for each of the singularities
$\mathbb{D}_{k}$, $\mathbb{E}_{6},\mathbb{E}_{7},\mathbb{E}_{8}$ (note that
$\mathbb{A}_{k}$ does not appear in this situation). The table below describe
all the possible choices of the ends and the corresponding Weil-Poincaré
series:

$\begin{matrix}\mathbb{D}_{k}&\\{1,k\\}&\mapsto&(1-t_{1}t_{2}^{(k-2)/2})(1-t_{1}^{1/2}t_{2}^{(k-2)/4})^{-1}\\\
&\\{k-1,k\\}&\mapsto&(1-t_{1}^{(k-2)/2}t_{2}^{(k-2)/2})(1-t_{1}^{1/2}t_{2}^{1/2})^{-1}\\\
&\\{1,k-1,k\\}&\mapsto&(1-t_{1}t_{2}^{(k-2)/2}t_{3}^{(k-2)/2})\\\
\mathbb{E}_{6}&\\{1,4\\}&\mapsto&(1-t_{1}^{2}t_{2}^{3})(1-t_{1}^{2/3}t_{2}^{1})^{-1}\\\
&\\{1,6\\}&\mapsto&(1-t_{1}^{2}t_{2}^{2})(1-t_{1}t_{2})^{-1}\\\
&\\{1,4,6\\}&\mapsto&(1-t_{1}^{2}t_{2}^{3}t_{3}^{2})\\\
\mathbb{E}_{7}&\\{1,4\\}&\mapsto&(1-t_{1}^{4}t_{2}^{6})(1-t_{1}t_{2}^{3/2})^{-1}\\\
&\\{1,7\\}&\mapsto&(1-t_{1}^{4}t_{2}^{3})(1-t_{1}^{2}t_{2}{3/2})^{-1}\\\
&\\{4,7\\}&\mapsto&(1-t_{1}^{6}t_{2}^{3})(1-t_{1}^{2}t_{2})^{-1}\\\
&\\{1,4,7\\}&\mapsto&(1-t_{1}^{4}t_{2}^{6}t_{3}^{3})\\\
\mathbb{E}_{8}&\\{1,4\\}&\mapsto&(1-t_{1}^{10}t_{2}^{15})(1-t_{1}^{2}t_{2}^{3})^{-1}\\\
&\\{1,8\\}&\mapsto&(1-t_{1}^{10}t_{2}^{6})(1-t_{1}^{5}t_{2}{{}^{3}})^{-1}\\\
&\\{4,8\\}&\mapsto&(1-t_{1}^{15}t_{2}^{6})(1-t_{1}^{5}t_{2}^{2})^{-1}\\\
&\\{1,4,8\\}&\mapsto&(1-t_{1}^{10}t_{2}^{15}t_{3}^{6})\\\ \end{matrix}$

Now, if the Weil-Poincaré series is not one of the above, then
$B(k)\neq\emptyset$ and the we can choose an index $i\in B(k)$ as in the
Proposition above. This finishes the proof of Theorem 1. $\Box$

For the proof of Theorem 2 (an analogue of Theorem 1 for divisorial
valuations) the projection formula permits to reduce (to split) the case of
$r$ valuations to the cases of one valuation $v_{1}$ and $(r-1)$ remaining
valuations. For the $\mathbb{E}_{8}$-singularity in [9] this finishes the
proof (due to the absence of symmetries of the $\mathbb{E}_{8}$ graph). In the
case under consideration the multiplicity $\underline{m}_{1i}$ determines
whether the vertices of the minimal resolution graph coresponding to the
divisorial valuations $v_{1}$ and $v_{i}$ are on the same part from those
exchanged by symmetries or on different ones.

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Addresses:

A. Campillo and F. Delgado: IMUVA (Instituto de Investigación en Matemáticas),
Universidad de Valladolid, Paseo de Belén, 7, 47011 Valladolid, Spain.
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
S.M. Gusein-Zade: Moscow State University, Faculty of Mathematics and
Mechanics, Moscow, GSP-1, 119991, Russia.
E-mail<EMAIL_ADDRESS>