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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:

Vincent Cossart, Olivier Piltant, Ana J. Reguera. Invariants of the graded algebras associated to

divisorial valuations dominating a rational surface singularity. 2012. �hal-00731738�

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 andK its quotient field. Given a divisorial valuationνonK^∗whose valuation ring dominates R, we denote bygrνRits associated algebra. Then,grνRis finitely generated for all divisorial valuations ν, if and only if the divisor class groupCl(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 valuationsν dominating it. We will assume that the residue fieldR/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 ofν inX, then this Veronese algebra isgrνR=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 byν onR(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 gr_νR 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 curvesEj adjacent to E and the relative position of the intersection pointsEj∩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 dominatingR. In section 4 we show that the answer is no. More precisely, we give two rational surface singu- larities R₁ andR₂, 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 andgr_θ(ν1)R₂ are isomorphic (proposition 4.3).

1

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 kisomorphic 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.

Letν be a divisorial valuation of the quotient field ofRcentered inR. Letgr_νR be its associated algebra, i.e.

grνR=

n∈Φ⁺

Pn / P_n⁺

where Φ⁺=ν(R\{0}) is the semigroup ofνand, forn∈N,Pn :={h∈R / ν(h)≥ n}andP_n⁺:={h∈R / ν(h)> n}. In [CPR2] we gave an explicit way to get afinite generating sequence forν, i.e. a finite sequence{Qi}iof elements ofM whose initial forms ingrνRgenerate it ask-algebra. Equivalently, everyν-idealIis generated by the finite products

jQ^a_j^j whereaj ∈Nare such that

jajν(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 / P_n⁺, and then we will define an embedding of grνR in a Veronese algebra that will allow us to describe generators ofPn / P_n⁺, forn∈Σ.

Among all resolutions of singularities of (S, P) such that the center ofν inX is a curve, there is a minimal one. Let π:X →S be this minimal resolution and let E be the center ofνinX. Let{Eγ}_γ∈Δbe the set of irreducible components of the exceptional locus ofπ. Recall that thedual graphofπis a graph defined from the configuration of the exceptional curves {Eγ}γ as follows : eachEγ is represented by a vertexeγ and there is a segment joiningeγ andeγ if and only ifEγ∩Eγ =∅.

The following result, due to Artin, will be applied throughout this work.

([Ar2], p.133) : IfD is a divisor onX such that D·Eγ = 0 for all γ∈Δ, then there existsh∈M such that (h)^∗=D, where by (h)^∗we mean the total transform in X of the divisor defined byhonS.

LetEXbe the free group generated by theEγ’s, andE⁺_X the subsemigroup ofEX

of all divisorsDsuch thatD·Eγ ≤0 for allγ. 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 divisorDI ∈ 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).

Forn∈N, theν-idealPnis equal toπ_∗OX(−nE)P. This implies thatPnbelongs toIX. LetDn be the element ofE⁺_X corresponding toPn∈IX by the isomorphism in (1), i.e. such that

(2) P_n=π_∗OX(−Dn).

Recall that Dn∈E⁺_X is theLaufer divisor associated tonE, obtained by applying the following algorithm : set D₁:=n Eand, fori≥1, letD_n=D_i ifD_i ∈E⁺_X, or else D_i+1=Di+Eγi whereγiis such that Di·Eγi >0 ([CPR2], prop. 2.1).

LetD 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. LetP=I_De be the associated complete ideal. Then,

2.1.([CPR] proposition 2.9 and theorem 2.10)

(i) The idealP is aν-ideal, that is,P=Pp wherep=ν(P).

(ii) For n∈Φ⁺ andr∈N, we have

(3) P_rp+n =P^r·Pn

Therefore, if Σ := {p} ∪ {n ∈ Φ⁺ / n−p ∈ Φ⁺}, then gr_νR is generated as a k-algebra by⊕n∈ΣPn / P_n⁺.

Now, let R be the local ring of the rational surface singularity obtained by contractingE inX. Then,

Theorem 2.2. (Embedding ofgrνR in a Veronese algebra)

(i) Let a = −E². Then grνR is isomorphic to the Veronese algebra V(a) :=

⊕r∈N k[T, T]ra, whereT andT are projective coordinates inE∼=P¹, and by k[T, T]α we mean the homogeneous polynomials inT, T of degreeα.

(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, forn∈ Φ⁺,{cj(n)}_j∈ΔE are the coefficients in theEj’s of the Laufer divisorDn, and αn:=−Dn·E, then

(4) i(Pn /P_n⁺) =

⎛

⎝

j∈ΔE

T_j^cj(n)

⎞

⎠ k[T, T]αn

where, if E ∩E_j has coordinates (λ_j : 1) in E = Spec k[T, T], then T_j = T+λ_jT.

Proof :Letn∈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→P_n⁺→Pn→Γ(X,OE(αn))∼=k[T, T]αn

and therefore, an injective morphism

(5) ψn:Pn / P_n⁺→k[T, T]αn.

Let us prove thatψn 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) D_n,s=D_n+sC₁+ (α_n−s)C₂ for 0≤s≤α_n. We have Dn,s·Eγ ≤0 for allγ∈Δ, and besides

Dn,s·E=Dn·E+s(C1·E) + (αn−s)(C2·E) =−αn+s+ (αn−s) = 0.

For eachEγ =E, letbn,s,γ =−Dn,s·Eγ ∈Nand letCγbe a nonsingular irreducible curve inX intersecting transversallyE_γ in a point not belonging toE_γ forγ =γ.

By Artin’s result (see beginning of section 2), there exist Qn,s ∈ R, 0≤s ≤ αn

such that

(7) (Q_n,s)^∗=D_n,s+

γ

b_n,s,γ Cγ. This implies that Qn,s∈Pn\P_n⁺ and that

ψn(Qn,s) =λ T^sT^αn−s for some λ∈k, λ = 0. Thereforeψ_n is surjective.

Now, for (i), we consider the new rational surface singularity Spec R and the same valuationν, whose center isE. We haveν(R\ {0}) =N, the extremal divisor corresponding to E is D₁=E and D₁·E =a. Therefore, the isomorphisms (5) in this case are

ψ_n :P_n / (P_n)⁺→k[T, T]na for everyn∈N

where P_n and (P_n)⁺ are the correspondingν-ideals ofR. From this (i) follows.

For (ii), let n∈Φ⁺, and let Q_n,s ∈R, 1 ≤s≤α_n, 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 intersectE. Therefore

ψ_n(i(Qn,s)) =λs

⎛

⎝

j∈ΔE

T_j^cj(n)

⎞

⎠ T^sT^αn−s for 1≤s≤αn

whereλ_s∈k\{0}. From this, (ii) follows and we conclude the proof of theorem 2.2.

IfΔE≥1, i.e. there is at least one exceptional curveEj adjacent toE, 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 curvesEj intersectingE. For each Γj, let (Sj, Pj) be the rational surface singularity obtained by contracting Γj, and letZjbe the fundamental cycle for the morphism X → Sj. Then, we can replace the divisor Dn,s in (6) by the following one : ifΔE≥2, we choose two exceptional curvesE₁, E₂ adjacent toE, 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). IfΔE= 1, then we take the unique exceptional curveE₁ adjacent toE and setDn,s=Dn+sZ₁+ (αn−s)C2. The reason why these new divisorsDn,s belong toE⁺_X is the following :

Lemma 2.3. The coefficient inE_j of Z_j is1.

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 D_i+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 alli, 0≤i < t, therefore the coefficient of Zj in Ej isZj·E= 1.

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}^α_s=0ⁿ whose initial forms define a basis ofPn / P_n⁺, for each n∈ Φ⁺. More precisely, such that i(Q_n,s) =T^sT^αn−s(

j∈ΔET_j^cj(n)). 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(Q_n1,s1Q_n2,s2) =

⎛

⎝

j∈ΔE

T_j^cj(n1+n2)

⎞

⎠ T^s1+s2 T^{αn1+αn2−s1−s2}

⎛

⎝

j∈ΔE

T_j^bj

⎞

⎠

wherebj∈Nis the coefficient inEj ofDn1+Dn2−Dn1+n2. Therefore, to obtain a generating sequence for ν, for eachn∈Φ⁺, we determine elements {Qn,s}swhose image byi is a basis of

(

j∈ΔE

T_j^cj(n)) k[T, T]αn/{T^s1+s2T^{αn1+αn2−s1−s2}(

j∈ΔE

T_j^bj)}_{n=n1+n2,0≤s}i≤αni,i=1,2

Then the union of such elements is a finite generating sequence forν.

Remark 2.6. LetRbe a regular local ring andν a divisorial valuation domi- natingR. LetX be the minimal nonsingular surface dominatingSpec R such that the center of ν in X is a curve E, and let Q_g+1 ∈R define a curve whose strict transform inXis transversal toEin a point not belonging to any other exceptional curve. Letβ_g+1 :=ν(Q_g+1), and letβ₀, . . . β_gbe a minimal system of generators of Φ⁺=ν(R\ {0}). Then, with the notation in 2.1 and 2.2,p=β_g+1,αn= 0 for all n∈Σ\ {β_g+1}andα_β

g+1= 1. Anyn∈Σ\ {β₀, . . . β_g}decomposes asn=n₁+n₂ for some n₁, n₂ ∈Σ. Therefore, to obtain a generating sequence for ν, we have to define a generator Qi ofP_βi/P_β⁺

i for 0≤i≤g as in (7) and (8), and, ifβ_gβ_g+1 then alsoQ_g+1 forP_β

g+1/P_β⁺

g+1. Ifβ_g|β_g+1, theβ_g+1has two different expressions β_g+1 =ngβ_g =a₀β₀+. . .+agβ_g in terms of β₀. . . β_g from which, by (9), it fol- lows that we do not have to add any element inP_β

g+1/P_β⁺

g+1(see [Sp], theorem 8.6).

Example 2.7. LetS be the blowing up of the idealI = (x⁶, y)·(x², y+x)· (x², y−x) of the regular local ringk[x, y]_(x,y). The surfaceS has one singular point P. Letπ:X→S be the minimal desingularization of (S, P). The dual graph ofπ is

` ` ` ` ` e₁ e₂ e₃ e₄ e₅

fig. 1

where ei represents an exceptional curve Ei, and we have E₁² = −4, E₂² = E²₃ = E₄²=E₅²=−2. Let us consider the divisorial valuationν defined byE₂. Then

D₁=E₁+E₂+E₃+E₄+E₅ `6<sup>−3</sup> ` ` ` `6⁻¹

1 1 1 1 1

D₂=E₁+ 2E₂+ 2E₃+ 2E₄+E₅ `6<sup>−2</sup> `6⁻¹ ` `6⁻¹ `

1 2 2 2 1

D₃=E₁+ 3E₂+ 3E₃+ 2E₄+E₅ `6<sup>−1</sup> `6⁻² `6<sup>−1</sup> ` `

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 D_n with theE_i corresponding to the basis of the arrow.

Therefore, to obtain a generating sequence forν, we need elementsQ₁∈P₁/P₁⁺, Q₂∈P₂/P₂⁺,Q₃∈P₃/P₃⁺ andQ₄₀, Q₄₄∈P₄/P₄⁺whose images by the embedding grνR → V(2) in theorem 2.2 areT 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= y²

y−x= (y+x) + x² y−x, Q₄₀= x⁶

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, Q_4,0, Q_4,4]⎛

⎜⎜

⎝

Q²₂−Q₃(Q₂−Q₁), Q⁶₁−Q_4,0Q₂, Q³₂−Q_4,4(Q²₂−Q²₁)

⎞

⎟⎟

⎠

where Q_i has degree i, for 1≤i≤3, andQ_4,0 andQ_4,4 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 byν 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(P_n/P_n⁺) =α_n+ 1

where αn :=−Dn·E (see equality (4)).

The following result improves theorem 4.2 in [CPR2].

Proposition 3.1. For alln∈Nwe have

l(R/P_n) =Q(n) +ϕ(n)

where Q(n) is a polynomial of degree two in n andϕ(n) is a periodic function of period p=ν(P) forn >>0.

Proof : If, for any Q-Cartier divisor B on X, we set χ(B) := −¹₂B·(B+K), where Kis a canonical divisor onX, then

Q(n) :=χ n

pD

is a polynomial of degree two inn. Besides, if we denote by c(n) the coefficient of Dn in E, then equality (3) implies that, forn >>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 ofϕisp.

Forn∈Φ⁺, let

L(n) :=Q(n+ 1)−Q(n) ψ(n) :=ϕ(n+ 1)−ϕ(n).

Then, forn∈N,l(Pn/P_n⁺) =L(n) +ψ(n) and

αn+ 1 =L(n) +ψ(n) forn∈Φ⁺.

The function ψ(n) is periodic forn >> 0, and its period divides the period of ϕ, hence it dividesp.

Letqbe the period ofψ, thusqdivides 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) (D_n+q−Dn)·E =q

p D·E.

This equality forn >>0 implies that (11) D_rp+kq·E =

r+q

pk

D·E forr >>0, 1≤k≤ p q.

In fact, givenr >>0, the previous equality fork= ^p_q is consequence of (3) and, by inverse recurrence, if it holds fork+ 1, then (10) implies that it also holds fork.

Forr >>0, let us consider thatQ-divisor D:=D_rp+q−

r+q

p D.

By (11),D belongs to the cone (E⁺_X)_Qin (EX)_Qdefined byE⁺_X. Since, forr >>0, we have rp+q ∈ Φ⁺, the coefficient of D in E is rp+q−

r+_p^q

p= 0. This implies that D = 0, i.e.D_rp+q =

r+^q_p

D. Then ^q_p D is a divisor and, by the definition of D, it follows that pdividesq, henceq=p.

Theorem 3.2. Letn∈Nbe any multiple ofp=ν(P)such thatn−1∈Φ⁺. Then

(12) D_n=D_n−1+E.

Therefore,

(13) −E²=α_n−α_n−1.

Proof :First, let us show that Supp(Dn−D_n−1) is connected. This will hold for anyn∈Φ⁺such thatn−1∈Φ⁺. Let{Eγ}_γ∈Δ be the configuration of exceptional curves defined by the connected component of Supp(Dn−D_n−1) which containsE.

LetD∈EX be such thatD=Dn on∪γ∈ΔEγ andD=D_n−1on∪_γ∈Δ\ΔEγ. Let us show thatD∈E⁺_X. Then, sinceD_n−1+E≤D≤Dn, we will haveD=Dnand hence, for anyγ∈Δ\Δ, the coefficient of (Dn−D_n−1) inEγ will be 0. Therefore Supp(Dn−D_n−1) will be connected. By the definition of Δ, for any γ ∈ Δ we have

D·Eγ =Dn·Eγ ≤0.

Ifγ ∈Δ is such thatEγ is not adjacent to anyEγ forγ∈Δ, then D·Eγ =D_n−1·Eγ ≤0.

Finally, if γ ∈ Δ is such that Eγ is adjacent to some Eγ, for γ ∈ Δ, then the coefficient ofD inEγ is equal to the coefficient ofDn inEγ. Let{bβ}_β∈ΔEγ be the coefficients ofDn−D_n−1 in theEβ’s adjacent toEγ. Then

D·Eγ =Dn·Eγ−

β∈ΔEγ

bβ≤Dn·Eγ ≤0 sincebβ ≥0 for all β∈ΔEγ. Thus, Supp(Dn−D_n−1) is connected.

Now, let n be as in the statement of the theorem, and let Δ ⊆ Δ be such that Supp(Dn−D_n−1) =∪γ∈ΔEγ. LetS 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 divisorD with support in∪γ∈ΔEγ we havepa(D)≤0 ([Ar1], 1.7).

Let us apply this to the divisorD=Dn−D_n−1. For any γ∈Δ we have D·Eγ = (Dn−D_n−1)·Eγ =

−D_n−1·E_γ ≥0 ifE_γ =E E²+

j∈ΔE∩Δb_j ifE_γ =E

where, for anyγ∈Δ,bγ is the coefficient ofDn−D_n−1 in Eγ, in particular, the {bj}_j∈Δ∩ΔE are the nonzero coefficients ofDn−D_n−1 in theEj’s adjacent to E (in the above equality we use the fact thatnis a multiple ofp, since it implies that Dn·Eγ = 0 forEγ =E). From the above equality, it follows that

D²=

γ∈Δ,Eγ=E

b_γ(D·E_γ) + (D·E)≥D·E=E²+

j∈ΔE∩Δ

b_j.

LetKbe a canonical divisor onX. Sincepa(Eγ) = 0 for allγ∈Δ, by the adjunction formula we have K·Eγ =−2−E²_γ, and, sinceX is the minimal desingularization ofS where the center of ν is a curve E, then −E_γ²≥2 forEγ =E. Thus,

D·K=

γ∈Δ

bγ(Eγ·K) =

γ∈Δ

bγ(−2−E_γ²)≥ −2−E². By the adjunction formula, we have

pa(D) = 1 +1

2D·(D+K)≥1 +1 2

⎛

⎝E²+

j∈ΔE∩Δ

bj−2−E²

⎞

⎠=1 2

j∈ΔE∩Δ

bj.

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 valuationν 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 byν :

(i) The semigroup Φ⁺ = ν(R\ {0}) and the function Φ⁺ → N, n → α_n =

−Dn·E.

(ii) The coefficient ofE in the fundamental cycleZ forπ:X →S.

(iii) The integer p=ν(P).

(iv) The coefficient of E in the unique canonical divisor of X with exceptional support forπ.

(v) The integer a=−E².

Moreover, from the graded algebragrνR, the following is determined : (vi) The numbervE of exceptional curves Ej which are adjacent toE.

(vii) For each n∈Φ⁺, the coefficients{cj(n)}j∈ΔE of Dn in the Ej’s adjacent toE.

(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-intersections.

(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 curveEfor 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 inEofZ 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 onX. By the negativity of the intersection matrix (E_γ·E_γ)_γ,γ∈Δ, there exists a uniqueQ-Cartier divisorK₀with exceptional support for πsuch that

K₀·Eγ =K·Eγ for allγ∈Δ

being this integer equal to−2−E_γ² 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 haveD²=−pαp, hence we obtain D·K. ButcoefEK₀=−αp D·K₀=−αp D ·K ∈Z. Thus K₀ is the unique canonical divisor with exceptional support forπ, and (iv) follows.

Suppose that we know, not only the Hilbert-Samuel function, but also the graded algebra grνR associated toν. Let us consider the fraction field ofgrνR, let us fix n∈Φ⁺, and let

Ω ={(s1, s₂)∈(Pn/P_n⁺)²/ s₁

s₂ is a αn−power}.

Then, for any embeddingj:grνR → V(a) wherea=−E², known by (v), the grea- test common divisor of j(s₁), j(s₂), where (s₁, s₂)∈Ω, determines

j∈ΔET_j^cj(n) 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 valuationsνdominating (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 α^γ_zγ = −Z·Eγ, which are determined from hγ. Then the multiplicity of (S, P) is

mult S=−Z²=

γ

zγα^γ_zγ

(see [Ar2] theorem 4). For (xi) it suffices to note that, if pγ is the integer in (iii) obtained fromh_γ, then the determinant of the intersection matrix (E_γ·E_γ)_γ,γ∈Δ0 is equal to the smallest common multiple of{α^γ_pγ}_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 valuationsν 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 allν, the graded algebragrνR 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 fieldkis 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 of the respective minimal desingularizations are not isomorphic, and such that there

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 toν₁andθ(ν₁) are isomorphic (proposition 4.3).

Let π:X →S be a desingularization and let {Eγ}γ be the exceptional curves of π. Byweighted dual graph of π:X →S we mean the weighted graph obtained from the dual graph ofπ by adding, for eachγ, the integer−E_γ² to the vertex e_γ representing E_γ. Note that, ifπ:X →S is the minimal desingularization and, for eachE_γ,a_γ:=−E_γ², andv_γ :=Δ_Eγ is the number of exceptional curves adjacent to Eγ, thenaγ+ 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 ifaγ ≥vγ.

A cyclic quotient singularity is characterized by the shape of the weighted dual graph for its minimal desingularization, which is

. . . .

` ` ` `

−a₁ −a₂ −a_n−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 valuationsνdominating (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 ∼= gr_θ(ν)OS2,P2, then (S, P) and (S₂, P₂) are isomorphic.

Proof :Let{Ei}ⁿ_i=1be the irreducible components of the exceptional locus of the minimal desingularization of (S, P), whereEi is represented by thei-th vertex in figure 4, hence−E_i²=ai≥2. Letdnbe the determinant of the intersection matrix (Ei·Ej)_1≤i,j,≤n and let d_n−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/d_n−1. Besides, d_n−1=_α^p_p 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−d_n−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 to the divisorial valuationsν dominating it.