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Invariants of the graded algebras associated to divisorial
valuations dominating a rational surface singularity
Vincent Cossart, Olivier Piltant, Ana J. Reguera
To cite this version:
DIVISORIAL VALUATIONS DOMINATING A RATIONAL SURFACE SINGULARITY
V. COSSART, O. PILTANT, A. J. REGUERA
1. Introduction
Let (R, M ) be a two dimensional complete Noetherian local domain and K its quotient field. Given a divisorial valuation ν on K∗whose valuation ring dominates
R, we denote by grνR its associated algebra. Then, grνR is finitely generated for all
divisorial valuations ν, if and only if the divisor class group Cl(R, M ) is a torsion group ([Go], [Cu]) and, if the base field is algebraically closed of characteristic zero, this holds if and only if (R, M ) is a rational surface singularity ([Li]).
In this paper we deal with a rational surface singularity (R, M ) and divisorial valuations ν dominating it. We will assume that the residue field R/M is algebrai-cally closed. Our purpose is to obtain invariants of the graded algebra grνR, or
even more, of the Hilbert-Samuel function of grνR. We develop three main ideas
to obtain such invariants. The first one is to embed grνR in a Veronese algebra.
If π : X → Spec R is the minimal desingularization among those on which the center of ν is a curve E and R is the local ring of the singularity obtained by contracting the center of ν in X, then this Veronese algebra is grνR=V(a), where
−a is the self-intersection number of E (theorem 2.2). The second idea is to
un-derstand the period of the Hilbert-Samuel function defined by ν on R (proposition 3.1). The third idea is to determine the self-intersection number of the curve E from the Hilbert-Samuel function of grνR (theorem 3.2). In particular, from the
Hilbert-Samuel function, we determine whether the divisorial valuation ν is essen-tial, i.e. whether is defined by an irreducible component of the exceptional locus of the minimal desingularization of Spec R.
It follows that, from grνR we can recover some local information of the dual
graph of π : X → Spec R around the vertex corresponding to E. For instance, the self-intersection number E2, the number of exceptional curves Ej adjacent to
E and the relative position of the intersection points Ej∩ E in E ∼=P1 (corollary 3.4). A natural question arises : to know whether the dual graph of the minimal desingularization of a rational surface singularity is determined by the set{grνR}ν
of all graded algebras associated to divisorial valuations dominating R. In section 4 we show that the answer is no. More precisely, we give two rational surface singu-larities R1 and R2, which are in fact minimal singularities, whose dual graphs for
the respective minimal desingularizations are not isomorphic, and such that there exists a one to one correspondence θ : V1 → V2 between the sets V1 and V2 of
divisorial valuations dominating each one, such that, for all ν1 ∈ V1, the graded
algebras grν1R1 and grθ(ν1)R2 are isomorphic (proposition 4.3).
2. Embedding of the graded algebra associated to a divisorial valuation in a Veronese algebra
Let (R, M ) be a complete normal ring of dimension two containing an algebrai-cally closed field k isomorphic to its residue field, and having rational singularity. That is, there exists a resolution of singularities X → S of (S, P ) = (Spec R, M) such that R1π∗OX = 0.
Let ν be a divisorial valuation of the quotient field of R centered in R. Let grνR
be its associated algebra, i.e.
grνR =
n∈Φ+
Pn / Pn+
where Φ+= ν(R\{0}) is the semigroup of ν and, for n ∈ N, Pn :={h ∈ R / ν(h) ≥
n} and P+
n :={h ∈ R / ν(h) > n}. In [CPR2] we gave an explicit way to get a finite
generating sequence for ν, i.e. a finite sequence{Qi}iof elements of M whose initial
forms in grνR generate it as k-algebra. Equivalently, every ν-ideal I is generated by
the finite productsjQajj where aj ∈ N are such thatjajν(Qj)≥ ν(I). We will
review that construction. More precisely, we will review the argument in [CPR2] to determine a finite set Σ⊂ Φ+ such that grνR is generated by ⊕n∈ΣPn / Pn+, and
then we will define an embedding of grνR in a Veronese algebra that will allow us
to describe generators of Pn / Pn+, for n∈ Σ.
Among all resolutions of singularities of (S, P ) such that the center of ν in X is a curve, there is a minimal one. Let π : X → S be this minimal resolution and let
E be the center of ν in X. Let{Eγ}γ∈Δbe the set of irreducible components of the
exceptional locus of π. Recall that the dual graph of π is a graph defined from the configuration of the exceptional curves {Eγ}γ as follows : each Eγ is represented
by a vertex eγ and there is a segment joining eγ and eγ if and only if Eγ∩ Eγ = ∅.
The following result, due to Artin, will be applied throughout this work. ([Ar2], p.133) : If D is a divisor on X such that D· Eγ = 0 for all γ ∈ Δ, then
there exists h∈ M such that (h)∗= D, where by (h)∗we mean the total transform in X of the divisor defined by h on S.
LetEXbe the free group generated by the Eγ’s, andE+X the subsemigroup ofEX
of all divisors D such that D·Eγ ≤ 0 for all γ. From the negativity of the intersection
matrix (Eγ· Eγ)γ,γ∈Δit follows that all elements ofE+Xare effective divisors ([Li],
p. 238). Let IX be the semigroup of M -primary complete (i.e. integrally closed)
ideals I such that the sheaf IOX is invertible, with the usual product of ideals
([Li] th. 7.1) For each ideal I ∈ IX, there is a unique divisor DI ∈ E+X such that
IOX =OX(−DI). Then, the map
(1) IX→ E+X, I → DI
defines an isomorphism of semigroups (IX,·) ∼= (E+X, +). Its inverse map isE+X →
IX, D → ID:= π∗OX(−D)P ([Li] prop. 6.2, see also [CPR1] section 1).
For n∈ N, the ν-ideal Pnis equal to π∗OX(−nE)P. This implies that Pnbelongs
toIX. Let Dn be the element ofE+X corresponding to Pn∈ IX by the isomorphism
in (1), i.e. such that
Recall that Dn∈ E+X is the Laufer divisor associated to nE, obtained by applying
the following algorithm : set D1:= n E and, for i≥ 1, let Dn= Di if Di ∈ E+X, or
else Di+1= Di+ Eγi where γiis such that Di· Eγi > 0 ([CPR2], prop. 2.1).
Let D be the extremal divisor inE+Xcorresponding to the irreducible exceptional component E, i.e. D is the minimal element in E+X such that D· Eγ = 0 for all
Eγ = E. Let P = IDe be the associated complete ideal. Then,
2.1. ([CPR] proposition 2.9 and theorem 2.10)
(i) The ideal P is a ν-ideal, that is, P = Pp where p = ν( P ).
(ii) For n∈ Φ+ and r∈ N, we have
(3) Prp+n= Pr· Pn
Therefore, if Σ := {p} ∪ {n ∈ Φ+ / n− p ∈ Φ+}, then grνR is generated as a
k-algebra by ⊕n∈ΣPn / Pn+.
Now, let R be the local ring of the rational surface singularity obtained by contracting E in X. Then,
Theorem 2.2. (Embedding of grνR in a Veronese algebra)
(i) Let a = −E2. Then grνR is isomorphic to the Veronese algebra V(a) :=
⊕r∈N k[T, T]ra, where T and T are projective coordinates in E ∼=P1, and by k[T, T]α we mean the homogeneous polynomials in T, T of degree α.
(ii) The inclusion R → R induces an inclusion i : grνR → grνR∼=V(a)
such that, if{Ej}j∈ΔE are the exceptional curves intersecting E and, for n∈
Φ+,{cj(n)}j∈ΔE are the coefficients in the Ej’s of the Laufer divisor Dn, and
αn:=−Dn· E, then (4) i(Pn /Pn+) = ⎛ ⎝ j∈ΔE Tjcj(n) ⎞ ⎠ k[T, T]α n
where, if E∩ Ej has coordinates (λj : 1) in E = Spec k[T, T], then Tj =
T + λjT.
Proof : Let n∈ Φ+, and let us consider the exact sequence
0→ OX(−E − Dn)→ OX(−Dn)→ OE⊗ OX(−Dn) ∼=OE(αn)→ 0.
Taking global sections, we have
0→ Pn+→ Pn→ Γ(X, OE(αn)) ∼= k[T, T]αn
and therefore, an injective morphism
(5) ψn: Pn / Pn+→ k[T, T]αn.
Let us prove that ψn is surjective. LetC1 andC2be two nonsingular irreducible
curves in X intersecting transversally E in two different points not belonging to any Eγ = E. We may take projective coordinates T, T in E ∼=P1 so that these
points are (0 : 1) and (1 : 0) respectively. Let us consider the divisors (6) Dn,s= Dn+ sC1+ (αn− s) C2 for 0≤ s ≤ αn.
We have Dn,s· Eγ ≤ 0 for all γ ∈ Δ, and besides
Dn,s· E = Dn· E + s(C1· E) + (αn− s)(C2· E) = −αn+ s + (αn− s) = 0.
For each Eγ = E, let bn,s,γ =−Dn,s·Eγ ∈ N and let Cγbe a nonsingular irreducible
By Artin’s result (see beginning of section 2), there exist Qn,s ∈ R, 0 ≤ s ≤ αn such that (7) (Qn,s)∗= Dn,s+ γ bn,s,γ Cγ.
This implies that Qn,s∈ Pn\ Pn+ and that
ψn(Qn,s) = λ TsTαn−s
for some λ∈ k, λ = 0. Therefore ψn is surjective.
Now, for (i), we consider the new rational surface singularity Spec R and the same valuation ν, whose center is E. We have ν(R\ {0}) = N, the extremal divisor corresponding to E is D1 = E and D1· E = a. Therefore, the isomorphisms (5) in this case are
ψn : Pn / (Pn)+→ k[T, T]na for every n∈ N
where Pn and (Pn)+ are the corresponding ν-ideals of R. From this (i) follows.
For (ii), let n∈ Φ+, and let Qn,s ∈ R, 1 ≤ s ≤ αn, be the elements satisfying
(7). From (6) and (7) it follows that (Qn,s)∗= nE +
j∈ΔE
cj(n)Ej+ sC1+ (αn− s) C2+ D
where D is a divisor whose support does not intersect E. Therefore
ψn(i(Qn,s)) = λs ⎛ ⎝ j∈ΔE Tjcj(n) ⎞ ⎠ TsTαn−s for 1≤ s ≤ α n
where λs∈ k \{0}. From this, (ii) follows and we conclude the proof of theorem 2.2.
If ΔE≥ 1, i.e. there is at least one exceptional curve Ej adjacent to E, we may
improve the previous argument with the following one : Let us consider the configu-ration of exceptional curves{Eγ}γ∈Δ\{E}. It has as many irreducible components
Γj as exceptional curves Ej intersecting E. For each Γj, let (Sj, Pj) be the rational
surface singularity obtained by contracting Γj, and let Zjbe the fundamental cycle
for the morphism X → Sj. Then, we can replace the divisor Dn,s in (6) by the
following one : if ΔE≥ 2, we choose two exceptional curves E1, E2 adjacent to E,
and we set
(8) Dn,s= Dn+ sZ1+ (αn− s)Z2
which is a divisor with exceptional support (while the divisor in (6) depends on the choice of the curvesCi). If ΔE= 1, then we take the unique exceptional curve E1
adjacent to E and set Dn,s= Dn+ sZ1+ (αn− s)C2. The reason why these new
divisors Dn,s belong toE+X is the following :
Lemma 2.3. The coefficient in Ej of Zj is 1.
Proof : To compute the fundamental cycle Z for the morphism X → S, we
may apply Laufer’s algorithm (see [CPR2] prop. 2.1) to any of the divisors Eγ,
i.e. we consider a sequence Eγ = D1 < . . . < Dt = Z where Di+1 = Di + Eγi
for some Eγi with Di · Eγi > 0. In particular, we may take such a sequence
Ej = D1< . . . < Dt= Z in such a way that the first steps consist of the Laufer’s
algorithm to compute Zj, hence, for some r, 0 ≤ r < t, we have Dr = Zj and
Eγr = E. By [La], theorem 4.2, we have Di· Eγi = 1 for all i, 0≤ i < t, therefore
Remark 2.4. Lemma 2.3 has appeared in [Ok] lemma 3.5, and it is a key point
in Okuma’s work [Ok]. The techniques in theorem 2.2 above have been applied in [Re1],[Re2].
Remark 2.5. The above proof gives an explicit way to determine elements
{Qn,s}αs=0n whose initial forms define a basis of Pn / Pn+, for each n ∈ Φ+. More
precisely, such that i(Qn,s) = TsTαn−s(j∈ΔET
cj(n)
j ). We will proceed as above
for all n ∈ Σ. Besides note that, if n ∈ Σ admits a decomposition n = n1+ n2
where n1, n2∈ Σ, then (9) i(Qn1,s1Qn2,s2) = ⎛ ⎝ j∈ΔE Tcj(n1+n2) j ⎞ ⎠ Ts1+s2 Tαn1+αn2−s1−s2 ⎛ ⎝ j∈ΔE Tbj j ⎞ ⎠
where bj∈ N is the coefficient in Ej of Dn1+ Dn2− Dn1+n2. Therefore, to obtain a
generating sequence for ν, for each n∈ Φ+, we determine elements {Qn,s}swhose
image by i is a basis of ( j∈ΔE Tjcj(n)) k[T, T]αn/{Ts1+s2Tαn1+αn2−s1−s2( j∈ΔE Tjbj)}n=n1+n2,0≤si≤αni,i=1,2
Then the union of such elements is a finite generating sequence for ν.
Remark 2.6. Let R be a regular local ring and ν a divisorial valuation
domi-nating R. Let X be the minimal nonsingular surface domidomi-nating Spec R such that the center of ν in X is a curve E, and let Qg+1 ∈ R define a curve whose strict transform in X is transversal to E in a point not belonging to any other exceptional curve. Let βg+1 := ν(Qg+1), and let β0, . . . βgbe a minimal system of generators of Φ+= ν(R\ {0}). Then, with the notation in 2.1 and 2.2, p = βg+1, αn= 0 for all
n∈ Σ \ {βg+1} and αβ
g+1 = 1. Any n∈ Σ \ {β0, . . . βg} decomposes as n = n1+ n2
for some n1, n2 ∈ Σ. Therefore, to obtain a generating sequence for ν, we have to
define a generator Qi of Pβi/Pβ+
i for 0≤ i ≤ g as in (7) and (8), and, if βg βg+1
then also Qg+1 for Pβg+1/Pβ+g+1. If βg|βg+1, the βg+1 has two different expressions
βg+1 = ngβg = a0β0+ . . . + agβg in terms of β0. . . βg from which, by (9), it
fol-lows that we do not have to add any element in Pβ
g+1/P
+
βg+1(see [Sp], theorem 8.6).
Example 2.7. Let S be the blowing up of the ideal I = (x6, y)· (x2, y + x)·
(x2, y− x) of the regular local ring k[x, y](x,y). The surface S has one singular point P . Let π : X → S be the minimal desingularization of (S, P ). The dual graph of π
is
` ` ` ` `
e1 e2 e3 e4 e5
fig. 1
D1= E1+ E2+ E3+ E4+ E5 `6−3 ` ` ` `6−1 1 1 1 1 1 D2= E1+ 2E2+ 2E3+ 2E4+ E5 `6−2 `6−1 ` `6−1 ` 1 2 2 2 1 D3= E1+ 3E2+ 3E3+ 2E4+ E5 `6−1 `6−2 `6−1 ` ` 1 3 3 2 1 D = D4= E1+ 4E2+ 3E3+ 2E4+ E5 ` `6−4 ` ` ` 1 4 3 2 1 fig. 2
where we have represented with weighted arrows the intersection of each divisor
Dn with the Ei corresponding to the basis of the arrow.
Therefore, to obtain a generating sequence for ν, we need elements Q1∈ P1/P1+,
Q2∈ P2/P2+, Q3∈ P3/P3+ and Q40, Q44∈ P4/P4+whose images by the embedding grνR → V(2) in theorem 2.2 are T T, T T3, T T5, T5T3, T T7 respectively. Since
x, y, x6 y , x2 y + x, x2 y− x
are coordinates inOS,P, we may take
Q1= x, Q2= y, Q3= y 2 y− x = (y + x) + x2 y− x, Q40= x 6 y , Q44= y3 (y− x)(y + x) = y + 1 2 x2 y + x+ 1 2 x2 y− x. Therefore, grνR is isomorphic to k[Q1, Q2, Q3, Q4,0, Q4,4] ⎛ ⎜ ⎜ ⎝ Q22− Q3(Q2− Q1), Q61− Q4,0Q2, Q32− Q4,4(Q22− Q21) ⎞ ⎟ ⎟ ⎠
where Qi has degree i, for 1≤ i ≤ 3, and Q4,0 and Q4,4 have degree 4.
3. Some invariants of the singularity recovered from the graded algebra associated to a divisorial valuation
In this section we will describe some invariants of (S, P ) which can be recovered from the graded algebra grνR of a divisorial valuation ν dominating R. Most of
them will be determined by the Hilbert-Samuel function of grνR.
The Hilbert-Samuel function defined by ν is the function N → N, n → h(n) := l(R/Pn).
Therefore, it is an equivalent data to the Samuel function defined by ν, Φ+→ N, n → l(Pn/Pn+) = αn+ 1
where αn :=−Dn· E (see equality (4)).
The following result improves theorem 4.2 in [CPR2].
Proposition 3.1. For all n ∈ N we have
where Q(n) is a polynomial of degree two in n and ϕ(n) is a periodic function of period p = ν( P ) for n >> 0.
Proof : If, for any Q-Cartier divisor B on X, we set χ(B) := −12B· (B + K),
where K is a canonical divisor on X, then
Q(n) := χ
n pD
is a polynomial of degree two in n. Besides, if we denote by c(n) the coefficient of
Dn in E, then equality (3) implies that, for n >> 0, the function
N → (EX)Q n → Dn:= Dn−c(n)
p D
is periodic and its period divides p, where (EX)Q:=γ∈ΔQEγ. We have
l(R/Pn) = Q(n) + χ( Dn).
(see [CPR2], theorem 4.2). Therefore, l(R/Pn) is expressed as stated in the
pro-position, where ϕ(n) := χ( Dn) is a periodic function, for n >> 0, whose period
divides p. We have to prove that the period of ϕ is p. For n∈ Φ+, let
L(n) := Q(n + 1)− Q(n) ψ(n) := ϕ(n + 1)− ϕ(n).
Then, for n∈ N, l(Pn/Pn+) = L(n) + ψ(n) and
αn+ 1 = L(n) + ψ(n) for n∈ Φ+.
The function ψ(n) is periodic for n >> 0, and its period divides the period of ϕ, hence it divides p.
Let q be the period of ψ, thus q divides p. We have
L(n) = χ n + 1 p D − χ n pD = χ 1 pD − 1 pD · n pD hence, ψ(n) =−Dn· E + 1 − L(n) = −Dn· E + 1 − χ 1 pD +n pD· E. Therefore, ψ(n + q) = ψ(n) is equivalent to (10) (Dn+q− Dn)· E = q p D· E.
This equality for n >> 0 implies that
(11) Drp+kq· E = r + q pk D· E for r >> 0, 1 ≤ k ≤ p q.
In fact, given r >> 0, the previous equality for k = pq is consequence of (3) and, by inverse recurrence, if it holds for k + 1, then (10) implies that it also holds for k.
For r >> 0, let us consider thatQ-divisor
D:= Drp+q− r +q p D.
By (11), D belongs to the cone (E+X)Qin (EX)Qdefined byE+X. Since, for r >> 0,
we have rp + q ∈ Φ+, the coefficient of D in E is rp + q−
r +pq
p = 0. This
implies that D = 0, i.e. Drp+q =
r +qp
D. Then qp D is a divisor and, by the
Theorem 3.2. Let n ∈ N be any multiple of p = ν( P ) such that n− 1 ∈ Φ+. Then
(12) Dn= Dn−1+ E.
Therefore,
(13) −E2= αn− αn−1.
Proof : First, let us show that Supp(Dn− Dn−1) is connected. This will hold for
any n∈ Φ+such that n−1 ∈ Φ+. Let{Eγ}γ∈Δ be the configuration of exceptional
curves defined by the connected component of Supp(Dn−Dn−1) which contains E.
Let D∈ EX be such that D = Dn on∪γ∈ΔEγ and D = Dn−1on∪γ∈Δ\ΔEγ. Let
us show that D∈ E+X. Then, since Dn−1+ E≤ D ≤ Dn, we will have D = Dnand
hence, for any γ∈ Δ \ Δ, the coefficient of (Dn− Dn−1) in Eγ will be 0. Therefore
Supp(Dn− Dn−1) will be connected. By the definition of Δ, for any γ ∈ Δ we
have
D· Eγ = Dn· Eγ ≤ 0.
If γ ∈ Δ is such that Eγ is not adjacent to any Eγ for γ∈ Δ, then
D· Eγ = Dn−1· Eγ ≤ 0.
Finally, if γ ∈ Δ is such that Eγ is adjacent to some Eγ, for γ ∈ Δ, then the
coefficient of D in Eγ is equal to the coefficient of Dn in Eγ. Let{bβ}β∈ΔEγ be the
coefficients of Dn− Dn−1 in the Eβ’s adjacent to Eγ. Then
D· Eγ = Dn· Eγ−
β∈ΔEγ
bβ≤ Dn· Eγ ≤ 0
since bβ ≥ 0 for all β ∈ ΔEγ. Thus, Supp(Dn− Dn−1) is connected.
Now, let n be as in the statement of the theorem, and let Δ ⊆ Δ be such that Supp(Dn− Dn−1) =∪γ∈ΔEγ. Let S be the surface obtained by contracting
∪γ∈ΔEγ. Then, S has only a singular point which is a rational surface singularity.
Therefore, for any effective divisor D with support in∪γ∈ΔEγ we have pa(D)≤ 0
([Ar1], 1.7).
Let us apply this to the divisor D = Dn− Dn−1. For any γ∈ Δ we have
D· Eγ = (Dn− Dn−1)· Eγ =
−Dn−1· Eγ ≥ 0 if Eγ = E
E2+j∈ΔE∩Δbj if Eγ = E
where, for any γ∈ Δ, bγ is the coefficient of Dn− Dn−1in Eγ, in particular, the
{bj}j∈Δ∩ΔE are the nonzero coefficients of Dn− Dn−1 in the Ej’s adjacent to E
(in the above equality we use the fact that n is a multiple of p, since it implies that
Dn· Eγ = 0 for Eγ = E). From the above equality, it follows that
D2= γ∈Δ,Eγ=E bγ(D· Eγ) + (D· E) ≥ D · E = E2+ j∈ΔE∩Δ bj.
Let K be a canonical divisor on X. Since pa(Eγ) = 0 for all γ∈ Δ, by the adjunction
formula we have K· Eγ =−2 − E2γ, and, since X is the minimal desingularization
of S where the center of ν is a curve E, then −Eγ2≥ 2 for Eγ = E. Thus,
D· K = γ∈Δ bγ(Eγ· K) = γ∈Δ bγ(−2 − Eγ2)≥ −2 − E2.
By the adjunction formula, we have
Then pa(D) ≤ 0 implies that bj = 0 for all Ej adjacent to E. From this, (12)
follows. Equality (13) is obtained by computing the intersection numbers with E of the members of (12).
Corollary 3.3. From the Hilbert-Samuel function of the graded algebra grνR
as-sociated to a divisorial valuation ν dominating R, we can determine whether the valuation ν is essential, i.e. it is defined by an irreducible component of the excep-tional locus of the minimal desingularization.
Proof : It follows directly from theorem 3.2, since E is essential if and only if −E2≥ 2.
Corollary 3.4. Let (S, P ) be a rational surface singularity, ν a divisorial valuation
dominating (S, P ) and π : X → S the minimal desingularization such that the center of ν in X is a curve E. Let {Eγ}γ∈Δ be the irreducible components of the
exceptional locus of π. The following invariants of (S, P ) are determined from the Hilbert-Samuel function of the graded algebra defined by ν :
(i) The semigroup Φ+ = ν(R\ {0}) and the function Φ+ → N, n → αn =
−Dn· E.
(ii) The coefficient of E in the fundamental cycle Z for π : X → S. (iii) The integer p = ν( P ).
(iv) The coefficient of E in the unique canonical divisor of X with exceptional support for π.
(v) The integer a =−E2.
Moreover, from the graded algebra grνR, the following is determined :
(vi) The number vE of exceptional curves Ej which are adjacent to E.
(vii) For each n∈ Φ+, the coefficients{cj(n)}j∈ΔE of Dn in the Ej’s adjacent
to E.
(viii) The relative position of the intersection points{Ej∩ E}j∈ΔE in E ∼=P1.
Therefore, from all the Hilbert-Samuel functions defined by the divisorial valuations dominating (S, P ), we recover :
(ix) The number of irreducible exceptional curves in the minimal desingulariza-tion of (S, P ), and their self-intersecdesingulariza-tions.
(x) The multiplicity of (S, P ).
(xi) The determinant of the intersection matrix of the exceptional curves for the minimal desingularization of (S, P ).
Besides, from all the graded algebras {grνR}ν where ν is any divisorial valuation
dominating (S, P ) we also obtain :
(xii) For each exceptional curve E for the minimal desingularization, the integer vE= ΔE and the relative position of the intersection points{E ∩ Ej}j∈ΔE
Proof : (i) is clear. For (ii) it suffices to note that the coefficient in E of Z is the
smallest element in Φ+. From proposition 3.1 and theorem 3.2, (iii) and (v) follow. For (iv), let K be a canonical divisor on X. By the negativity of the intersection matrix (Eγ·Eγ)γ,γ∈Δ, there exists a uniqueQ-Cartier divisor K0with exceptional
support for π such that
K0· Eγ = K· Eγ for all γ∈ Δ
being this integer equal to−2 − Eγ2 by the adjunction formula. On the other hand,
the integer p = ν( P ) is determined by the Hilbert-Samuel function of grνR and,
by the adjunction formula, l(R/ P ) =−12 D· ( D + K). We have D2=−pαp, hence
we obtain D· K. But coefEK0=−αp D· K0=−αp D · K ∈ Z. Thus K0 is the
Suppose that we know, not only the Hilbert-Samuel function, but also the graded algebra grνR associated to ν. Let us consider the fraction field of grνR, let us fix
n∈ Φ+, and let
Ω ={(s1, s2)∈ (Pn/Pn+)2/ ss1
2 is a αn− power}.
Then, for any embedding j : grνR → V(a) where a = −E2, known by (v), the
grea-test common divisor of j(s1), j(s2), where (s1, s2)∈ Ω, determines j∈ΔET
cj(n)
j
in equality (4) modulo a unit. From this, (vi), (vii) and (viii) follow.
Now, suppose that we know all the Hilbert-Samuel functions of the divisorial valuations ν dominating (S, P ). By (v), the data (ix) is determined. Let us consider the set{hγ}γ∈Δ0of all the Hilbert-Samuel functions of the essential valuations and,
for each hγ, let zγ be the coefficient of Z in the corresponding exceptional curve
Eγ, and αγzγ = −Z · Eγ, which are determined from hγ. Then the multiplicity of
(S, P ) is
mult S =−Z2=
γ
zγαγzγ
(see [Ar2] theorem 4). For (xi) it suffices to note that, if pγ is the integer in (iii)
obtained from hγ, then the determinant of the intersection matrix (Eγ·Eγ)γ,γ∈Δ0
is equal to the smallest common multiple of{αγpγ}γ∈Δ0. Finally, if we know all gra-ded algebras {grνR}ν, then, by (v), we may take the ones defined by an essential
valuation, and hence (xii) follows from (vi) and (viii).
4. The dual graph is not recovered from the graded algebras Given a normal surface singularity (S, P ) over an algebraically closed field of characteristic zero, from the set {grνR}ν of all graded algebras associated to
divi-sorial valuations ν dominating (S, P ), we can determine whether (S, P ) is a rational surface singularity. In fact, for any normal surface singularity (S, P ) over any field
k, (S, P ) satisfies the property that, for all ν, the graded algebra grνR is finitely
generated if and only if its group Cl(S, P ) of classes of divisors is a torsion group ([Go], [Cu]) and, if the base field k is algebraically closed of characteristic zero, this is equivalent to (S, P ) being a rational surface singularity ([Li]).
Besides, we have shown in the last section that, from the data {grνR}ν of all
graded algebras associated to divisorial valuations dominating a rational surface singularity (S, P ), we can recover the ones associated to the essential valuations, i.e. the divisorial valuations defined by an irreducible component E of the excep-tional locus of the minimal desingularization of (S, P ). Moreover, we also recover the self-intersection number of these exceptional curves E, the number of other exceptional curves in the minimal desingularization which are adjacent to E, and some other invariants (corollary 3.4). From these observations, a natural question arises :
Question 4.1. Is the dual graph of the minimal desingularization of a rational
surface singularity determined by the set{grνR}νof all graded algebras associated
to divisorial valuations dominating (S, P ) ?
We will show in this section that, in general, the answer to this question is no. More precisely, we will give two minimal singularities (i.e. rational surface singulari-ties whose fundamental cycle is reduced), (S1, P1) and (S2, P2), whose dual graphs
exists a one to one correspondence θ : V1→ V2 between the setsV1 andV2 of di-visorial valuations dominating (S1, P1) and (S2, P2) respectively, such that, for all ν1∈ V1, the graded algebras associated respectively to ν1and θ(ν1) are isomorphic
(proposition 4.3).
Let π : X → S be a desingularization and let {Eγ}γ be the exceptional curves
of π. By weighted dual graph of π : X → S we mean the weighted graph obtained from the dual graph of π by adding, for each γ, the integer −Eγ2 to the vertex eγ
representing Eγ. Note that, if π : X→ S is the minimal desingularization and, for
each Eγ, aγ:=−Eγ2, and vγ := ΔEγ is the number of exceptional curves adjacent
to Eγ, then aγ+ 1≥ vγ for any desingularization of a rational surface singularity
([La], th. 4.2). A given weighted graph is the weighted dual graph of a minimal desingularization of a minimal surface singularity if and only if aγ ≥ vγ.
A cyclic quotient singularity is characterized by the shape of the weighted dual graph for its minimal desingularization, which is
. . . .
` ` ` `
−a1 −a2 −an−1−an
fig. 4
where ai ≥ 2. Therefore, a cyclic quotient singularity is characterized by the
pro-perty that vE = 2 for all essential curves E except for two of them, for which
vE = 1. Hence, it is characterized by the the set {grνR}ν of all graded algebras
associated to divisorial valuations ν dominating (S, P ) which dominate it (corollary 3.4).
Proposition 4.2. Let (S, P ) be a cyclic quotient singularity. Then question 4.1 has
an affirmative answer. More precisely, if (S2, P2) is a rational surface singularity such that there exists a one to one correspondence θ : V1 → V2, where V1 and V2 are the sets of divisorial valuations dominating (S, P ) and (S2, P2) respectively, such that, for all ν ∈ V1, grνOS,P ∼= grθ(ν)OS2,P2, then (S, P ) and (S2, P2) are
isomorphic.
Proof : Let{Ei}ni=1be the irreducible components of the exceptional locus of the
minimal desingularization of (S, P ), where Ei is represented by the i-th vertex in
figure 4, hence−Ei2= ai≥ 2. Let dnbe the determinant of the intersection matrix
(Ei· Ej)1≤i,j,≤n and let dn−1 be the (n− 1) × (n − 1)-minor (Ei· Ej)1≤i,j,≤n−1.
Then, since ai ≥ 2 for 1 ≤ i ≤ n, the sequence (a1, . . . , an) is uniquely
determi-ned by the expression as continued fraction of dn/dn−1. Besides, dn−1=αpp where
p and αp are the data associated to the valuation corresponding to the extreme
En, hence determined from its Hilbert-Samuel function. Thus, from (i), (iii), (vi)
and (xi) of corollary 3.4 it follows that the weighted dual graphs of the minimal desingularizations of (S, P ) and (S2, P2) are the same, and (S, P ), isomorphic to
(S2, P2), is the toric singularity defined by the cone < (1, 0), (dn− dn−1, dn) >⊂ R2
(see [Od], lemma 1.22 and corol. 1.23).
Corollary 4.3. Let (S, P ) be a normal surface singularity over an algebraically
closed field of characteristic zero. From the set{grνR}ν of all graded algebras
asso-ciated to the divisorial valuations ν dominating (S, P ), we can determine whether
(S, P ) is a cyclic quotient singularity. Moreover, a cyclic quotient singularity is
de-termined up to isomorphism by the collection of graded algebras{grνR}ν associated
Let us consider the following two weighted dual graphs, that will be called res-pectively G1 and G2, ` ` F −3 −3 F F F ` ` F −3 −3 F F F fig. 6 where F and F are respectively
F ` ` ` ` ` −2 −2 −2 −4 −3 F’ ` ` ` ` ` −2 −3 −3 −2 −3 fig. 7
and they are joined in figure 6 by the small segments in the corners. Both G1
and G2 satisfy aγ ≥ vγ for all γ. Therefore, there exist two minimal singularities
(S1, P1) and (S2, P2) whose weighted dual graphs for their minimal
desingulariza-tions are respectively G1 and G2. Moreover, we may also ask (S1, P1) and (S2, P2)
to satisfy the following property : Let {Eα}α∈Λ (resp.{Eα}α∈Λ) be the
exceptio-nal curves of the minimal desingularization of (S, P ) (resp. (S2, P2)), let Eβ1, Eβ2
(resp. Eβ1, Eβ2) be the curves corresponding to the −3-vertices in fig. 6, and, for
j = 1, 2, let αj,1, αj,2 ∈ Λ \ {β1, β2} be such that Eαj,1 and Eαj,2 (resp. Eαj,1 and
Eαj,2) intersect Eβj (resp. Eβj). Then, for j = 1, 2, the relative position of the three
points Eαj,1∩ Eβj, Eαj,2∩ Eβj, Eβ1∩ Eβ2 in Eβj ∼=P1k is the same as the relative
position of the three points Eαj,1∩ Eβj, Eαj,2∩ Eβj, Eβ1 ∩ Eβ2 in Eβj ∼=P1k, and
moreover, this relative position for j = 1 is the same as the one for j = 2.
Let R1:=OS1,P1 and R2=OS2,P2, and, for i = 1, 2, letVibe the set of divisorial
valuations dominating Ri. Then,
Proposition 4.4. There exists a one to one correspondence θ : V1 → V2 such that, for all ν1∈ V1, the graded algebras grν1R1 and grθ(ν1)R2 are isomorphic. But
the dual graphs of the minimal desingularizations of the singularities (S1, P1) and
(S2, P2) are not isomorphic.
In order to prove proposition 4.3, let us first study a way to obtain the Hilbert-Samuel function in terms of some local data, which holds under certain conditions, in particular for minimal surface singularities. Let π : X → S be a desingularization of a rational surface singularity (S, P ), and let{Eγ}γ∈Δbe the irreducible
compo-nents of the exceptional locus of π. Let E be one of them,{Ej}j∈ΔEthe exceptional
curves which are adjacent to E, and aj :=−E2j. Let {Γj}j∈ΔE be the connected
components of∪γ∈ΔEγ\ E where, for each j ∈ ΔE, Ej is contained in Γj. For each
j ∈ ΔE, let Σj the configuration of curves obtained from Γj by substituting Ej by
another curve Ej, also isomorphic to P1 and with the same intersection numbers with the other curves in Γj, but with self-intersection−(aj− 1). Suppose that there
exists a rational surface singularity (Yj, Qj) and a desingularization πj : Xj → Yj
such that the configuration of exceptional curves for πj is Σj, in particular this
implies that aj ≥ 1. It happens, for instance, if (S, P ) is a minimal surface
exceptional divisor. Let hj be the Hilbert-Samuel function of the valuation νj on
Sj defined by Ej. Then
Lemma 4.5. The Hilbert-Samuel function h of the valuation on (S, P ) defined by
E and, moreover, the coefficients {cj(n)}j∈ΔE in the Ej’s adjacent to E of the
divisors {Dn}n∈Φ+ associated to this valuation, can be determined from {hj}j∈ΔE
and E2.
Proof : Let p = ν( P ) and let r >> 0 be such that rp is greater or equal to the
conductor of the semigroup of Φ+. For n∈ N ∪ {0} \ Φ+, let
Dn := Drp+n− r D
(see (3)) and, for n∈ Φ+, let Dn:= Dn. For any n∈ N, let αn:=−Dn· E. Then,
the functionN → N, n → αn− αn−1is a periodic function (for all n∈ N) of period
p (proof of proposition 3.1).
For any j ∈ ΔE, let us do the same construction as before, obtaining thus divisors
{Dj
n}n∈N∪{0} on Σj and a functionN → N, n → αnj, such that n → αjn− αjn−1is
periodic, let pjbe its period. We will show by induction on n that for all n∈ N, the
coefficients{cj(n)}j∈ΔE, and hence the integer αn, are determined from{hj}j∈ΔE
and a :=−E2.
Let us first show that{cj(1)}j∈ΔEare determined from{hj}j∈ΔE and a. Fix j∈
ΔE. If Dj=Eγ∈ΣjcγEγis a divisor on Σj, let us set D
j|Γ
j =
Eγ∈Σj,Eγ=EjcγEγ+
cjEj, which is a divisor on Γj hence also on X. Since Drp= r D, the divisor on Γj
obtained by restriction of Drp has intersection 0 with all Eγ ∈ Γj\ {Ej}, hence it
is equal to Djspj|Γj for some s∈ N, in particular cj(rp) = spj. Then,
0 = Drp· Ej = Djspj|Γj + rpE · Ej =−αjspj+ rp− spj. Therefore, for k≥ 1, Djspj+k|Γj + (rp + 1)E ·Ej=−αjspj+k+rp−spj−(k−1) = −(α j spj+k−α j spj)−(k−1)
and this implies that the smallest k such that−(αjspj+k− αjspj)− (k − 1) ≤ 0 is the
coefficient in Ej of Drp+1− Drp, i.e. it is cj(1).
Now, let n≥ 1 and let us suppose that we know {cj(n)}j∈ΔE. For any j∈ ΔE,
we have 0≥ Dn· Ej = Djcj(n)|Γj+ nE · Ej=−αjcj(n)+ n− cj(n) and, for k≥ 0, Dcjj(n)+k|Γj + (n + 1)E · Ej=−αjcj(n)+k+ n− cj(n)− (k − 1).
Hence, cj(n + 1) = cj(n) + min{k / − αjcj(n)+k+ n− cj(n)− (k − 1) ≤ 0}. From
this, the lemma follows.
Now, let us consider the weighted dual graphs
obtained from the weighted graphs in figures 6 and 7 by the process described before lemma 4.4. There exist two minimal surface singularities (Y, Q) and (Y, Q) whose weighted dual graphs for their minimal desingularizations are GΣand GΣ
respec-tively. Let {Ei}5i=1 and {Ei}5i=1 be the irreducible components of the exceptional
locus of the minimal desingularization of (Y, Q) and (Y, Q) respectively, which are represented by {ei}5i=1 and{ei}5i=1 in figure 8. Let ν and ν be the valuations on
(Y, Q) and (Y, Q) respectively determined by E4 and E4, and let h and h be the
respective Hilbert-Samuel functions.
Lemma 4.6. The Hilbert-Samuel functions h : N → N and h : N → N are the
same.
Proof : The exceptional divisors{Dn}n and{Dn}n for the valuations ν and ν
respectively are D1= E1+ E2+ E3+ E4+ E5 D1= E1 + E2 + E3 + E4 + E5 D2= E1+ 2E2+ 2E3+ 2E4+ E5 D2= E1+ 2E2+ E3+ 2E4+ E5 D3= E1+ 2E2+ 3E3+ 3E4+ E5 D3= E1+ 2E2+ E3+ 3E4+ 2E5 D = D4= E1+ 2E2+ 3E3+ 4E4+ E5 D = D 4= E1+ 2E2+ E3+ 4E4+ 2E5.
Hence, the period of both h and h is 4 and we have
−D1· E4 =−D1· E4 = 0 −D2· E4 =−D2· E4 = 1
−D3· E4 =−D3· E4 = 2 −D4· E4 =−D4· E4 = 4.
Since h(n+1)−h(n) = −Dn·E4+1 and the same holds for h, we conclude the result. End of proof of proposition 4.3 : Let ν1 be a divisorial valuation dominating
(S1, P1). Let π1: X1 → S1 be the minimal desingularization of (S1, P1) where the
center of ν1is a curve E, and let{Ej}j∈ΔE be the exceptional curves for π adjacent
to E. From lemmas 4.4 and 4.5 it follows that there exists a valuation ν2dominating (S2, P2) such that, if π2 : X2 → S2 is the minimal desingularization of (S2, P2) where the center of ν2is a curve E, and{Ej}j∈ΔE are the exceptional curves for
π adjacent to E, then we may identify ΔE and ΔE, the coefficients{cj(n)}j∈ΔE
and {cj(n)}j∈ΔE of the divisors Dn and Dnfor ν1and ν2respectively coincide,
and hence the Hilbert-Samuel functions h1 and h2of ν1and ν2are the same.
Now, note that vE≤ 3 by the definition of (S1, P1) and of X1, and that vE= 3
if and only if E is either Eβ1 or Eβ2, and hence E is either Eβ1 or Eβ2. Hence, we
may order {Ej}j∈ΔE and {Ej}j∈ΔE and take coordinates T1, T2 in E (resp E)
such that Ej ∩ E and Ej ∩ E have the same coordinates for all j ∈ ΔE = ΔE.
Then, from (4) in theorem 2.2 it follows that
grν1R1∼=⊕n∈N Tc1(n) 1 T2c2(n)T3c3(n) k[T1, T2]h1(n+1)−h1(n)−1∼= grν2R2
and we conclude the result.
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Vincent Cossart,
Laboratoire de Math´ematiques LMV UMR8100, Univ. de Versailles, Saint-Quentin 45 avenue des ´Etas-Unis, 78035 Versailles Cedex (France)
E-mail : cossart@math. uvsq.fr Olivier Piltant,
CNRS, LMV, UMR8100, CNRS-UVSQ, Universit´e de Versailles, Saint-Quentin 45 avenue des ´Etas-Unis, F-78035 Versailles Cedex (France)
E-mail : piltant@math. uvsq.fr Ana J. Reguera,
Dep. de ´Algebra, Geometr´ıa y Topolog´ıa, Fac. Ciencias, Universidad de Valladolid, Paseo Bel´en 7, 47011 Valladolid, Spain.
