Fields Institute Communications Volume 00, 0000

Divisorial valuations dominating rational surface singularities

Vincent Cossart

                                 cossart@math.uvsq.fr

Université de Versailles – St Quentin bât. Fermat, 45 av. des États-Unis F–78035 Versailles cedex (France)

Olivier Piltant

                            piltant@math.polytechnique.fr

École polytechnique F–91128 Palaiseau cedex (France)

Ana J. Reguera-López

                                 areguera@agt.uva.es

Dpto. Álgebra y Geometría Facultad de Ciencias Valladolid 47005 (Spain)

1 Introduction.

Let (R, M, k) be a Noetherian local domain and K be its quotient field. Let ν : K^{∗} → Φ be a valuation centered in R, i.e., if R_{ν} is the valuation ring of ν and M_{ν} its maximal ideal, then R ⊆ R_{ν} and M = M_{ν} ∩ R. One also says that ν dominates R ([ZS] app.2). The following invariants are associated with ν; the rank rk ν := dim R_{ν}, the rational rank rat.rk ν := dim_{Q}(Φ ⊗ Q) and the transcendence degree tr.deg ν := tr.deg(\frac{R_{ν}}{M_{ν}} /k). We have rk ν ≤ rat.rk ν ([ZS] p.50) and Abhyankar’s inequality ([ZS] p.331):

rat.rk ν + tr.deg ν ≤ dim R.

The valuation is said to be divisorial if rk ν = 1 and tr.deg ν = dim R − 1. In this case, the value group Φ is isomorphic to Z and R_{ν} is a noetherian ring. In Zariski’s terminology ([ZS] p.339), a divisorial valuation centered in R is a prime R-divisor of its quotient field K.

Suppose (R, M, k) is a two dimensional normal local ring and ν is a valuation centered in R. Let S := Spec R. Then ν is divisorial if and only if there exists a finite sequence of point blowing-ups followed by normalizations

(1)  S ⟵^{π_{1}} S_{1} ⟵ … ⟵^{π_{n}} S_{n}
\begin{matrix}c \\ ◯ \end{matrix}0000 American Mathematical Society

1991 Mathematics Subject Classification. 14J17, 14C20.

such that the center of ν in S_{n} is a curve, i.e., there is an irreducible component E of the exceptional locus of S_{n} → S such that R_{ν} coincides with O_{S_{n},E} ([Z2] if k is an algebraically closed field. See also [Co] for an arbitrary field k).

In case (R, M, k) is a two dimensional regular local ring, Zariski’s above result establishes a one to one correspondence between the set V of divisorial valuations centered in R and the set C of chains of point blowing-ups of S (i.e., sequences (1) such that, if π_{i} is the blowing-up of the point η_{i} ∈ S_{i−1}, then π_{i}(η_{i+1}) = η_{i}). Moreover, he develops a theory of complete ideals in R ([Z1] and [ZS] app.5). These ideals are local analogues for complete linear systems on projective algebraic varieties. The set I of all simple complete M-primary ideals also is in one to one correspondence with V. His main result is the unique factorization of M-primary complete ideals into simple ones. If ν is a divisorial valuation centered in R, and (1)_{ν} is the chain of point blowing-ups determined by ν via the above correspondence, the following properties hold:

a) the extension to S_{n} of any valuation ideal I ⊂ R of ν (i.e., I = IR_{ν} ∩ R) is locally invertible.

b) those valuation ideals of ν which are indecomposable, that is, which are not a nontrivial product of two valuation ideals of ν, are exactly those simple complete ideals in I corresponding to subchains S ← … ← S_{m}, m ≤ n of (1)_{ν}.

These two facts are of fundamental importance in Spivakovsky’s article ([Sp]), in order to determine a minimal generating sequence of a divisorial valuation centered in a regular local ring of dimension two. Roughly speaking, he shows how the weighted dual graph of the configuration of exceptional curves in (1)_{ν} determines -and is determined by- the isomorphism class of the graded k-algebra associated with the ν-adic filtration. This is achieved by explicitly computing its generators.

In this note, we will consider divisorial valuations centered in the local ring (R, M, k) of a rational surface singularity, in the case where R is complete. In this case, divisorial valuations correspond to the irreducible components of the exceptional locus of some sequence of point blowing-ups π : X → S. Unique factorization does not hold in general in the associated semigroup of M-primary complete ideals, but the weaker property of semifactorization into simple extremal ones ([Gö], [Li], [CPR]). Simple extremal complete ideals are in one to one correspondence with divisorial valuations as in the regular case (definitions are given in section 2).

In section 2, we show that the natural extension of property a) above for R having a rational singularity still holds. This allows us to apply the dictionary given in [Li] between M-primary complete ideals whose extension to X is locally invertible and nef divisors on X with exceptional support for π. In corollary 4, we give a constructive proof of the known fact that the graded algebra gr_{ν}(R) of the ν-adic filtration of R is Noetherian (see e.g. [Cu] p. 60 for a more abstract proof). Our proof also allows us to give a procedure in order to explicitly compute a generating sequence for ν (section 3).

The last section is devoted to the study of the Hilbert function of gr_{ν}(R). The natural extension of property b) above does not hold in general for R having a rational singularity (see example in section 3). However, the set of indecomposable valuation ideals is still determined in some way by the weighted dual graph of the configuration of exceptional curves on X. We also observe that the same fact holds for the Hilbert function of gr_{ν}(R) (theorem 2).

2 Noetherianness of the graded algebra.

Let (R, M) be a complete normal local ring of dimension 2 containing an algebraically closed field k mapping isomorphically to its residue field. Let (S, η) := (Spec R, M). By [Hi], there exists a proper birational map π : X → S with X regular.

Let {E_{γ}}_{γ∈Δ} be the set of irreducible components of the exceptional locus of π. Let E_{X} be the free group generated by the E_{γ}’s and E_{X}^{+} be the subsemigroup of E_{X} of all divisors D such that D.E_{γ} ≤ 0 for all γ ∈ Δ. It follows from the negative definiteness of the intersection matrix (E_{γ}.E_{γ′})_{γ,γ′∈Δ} that all elements of E_{X}^{+} are effective divisors ([Li] p. 238).

Proposition 1 Let D ∈ E_{X}. Among all exceptional divisors D′ ∈ E_{X}^{+} such that D′ ≥ D (i.e., D′ −D is effective), there is a minimal one \overline{D}. One can compute \overline{D} by applying the following algorithm: let D̂_{1} := D, and let for i ≥ 1

\overline{D} := D̂_{i} if D̂_{i}.E_{γ} ≤ 0 for all γ ∈ Δ,

D̂_{i+1} := D̂_{i} + E_{γ_{i}} otherwise, where γ_{i} is such that D̂_{i}.E_{γ_{i}} > 0.

Proof: The fundamental cycle Z of π has positive multiplicity in all of the E_{γ}’s ([Ar]), so rZ ≥ D for r >> 0 and rZ ∈ E_{X}^{+}. Given D′_{1} = ∑ a_{γ}^{1}E_{γ}, D′_{2} = ∑ a_{γ}^{2}E_{γ} belonging to E_{X}^{+}, with D′_{i} ≥ D for i = 1, 2, let D′ = ∑ a_{γ}E_{γ}, where a_{γ} := min{a_{γ}^{1}, a_{γ}^{2}}. For a fixed γ ∈ Δ and, say, a_{γ}^{1} ≤ a_{γ}^{2},

D′.E_{γ} = a_{γ}^{1}E_{γ}^{2} + ∑_{α≠γ} a_{α}(E_{α}.E_{γ}) ≤ D′_{1}.E_{γ} ≤ 0,

since E_{α}.E_{γ} ≥ 0 for α ≠ γ. Therefore D′ ∈ E_{X}^{+} and D′_{i} ≥ D′ ≥ D for i = 1, 2. These facts imply the first assertion in the proposition.

To prove the second statement, it is enough to show that the divisors D̂_{i} satisfy D̂_{i} ≤ \overline{D}. This is trivial for i = 1. Suppose D̂_{i} < \overline{D}. Then D̂_{i+1} = D̂_{i} + E_{γ_{i}} and (\overline{D} − D̂_{i}).E_{γ_{i}} < 0. Since \overline{D} − D̂_{i} is effective, this implies \overline{D} − D̂_{i} ≥ E_{γ_{i}}, whence D̂_{i+1} ≤ \overline{D}.

Remark: Notice that the idea underlying the construction of \overline{D} is the same as that given by Laufer to construct the fundamental cycle ([La] prop. 4.1).

Corollary 1 Let D ∈ E_{X} be effective. The inclusion map O_{X}(−\overline{D}) ⊆ O_{X}(−D) induces the equality

π_{∗}O_{X}(−\overline{D})) = π_{∗}O_{X}(−D).

Proof: It is enough to check the equality on the stalks at the closed point η. Since O_{X}(−D) ⊆ O_{X} and π_{∗}O_{X} = O_{S}, π_{∗}O_{X}(−D)_{η} = Γ(X, O_{X}(−D)) ⊆ Γ(X, O_{X}) = R. Let f ∈ R, f ≠ 0 in π_{∗}O_{X}(−D)_{η}. Write div(f) = (f)′ + E_{f}, where E_{f} ∈ E_{X} is effective, and (f)′ is effective and has support meeting properly all of the E_{γ}’s. We have E_{f} ≥ D and E_{f}.E_{γ} = −(f)′.E_{γ} ≤ 0 for all γ ∈ Δ. This implies that E_{f} ∈ E_{X}^{+}, and consequently E_{f} ≥ \overline{D} by proposition 1. Thus f belongs to π_{∗}O_{X}(−\overline{D})_{η}.

¿From now on and till the end of this article, we will make the additional assumption that R has a rational singularity. This means that there exists a resolution of singularities π : X → S such that R^{1}π_{∗}O_{X} = 0. This property then holds for any resolution of singularities ([Li] prop. 1.2).

Let I_{X} be the semigroup of all M-primary complete (integrally closed) ideals I of R such that IO_{X} is locally invertible. Since R has a rational singularity, the product law in I_{X} is the usual product of ideals ([Li], th. 7.1). We have I_{X} ⊆ I_{X′} whenever X′ dominates X, and hence the I_{X}’s form a direct system of semigroups whose direct limit is the semigroup I of all M-primary complete ideals. For each ideal I ∈ I_{X}, there is a unique divisor D_{I} ∈ E_{X}^{+} such that IO_{X} = O_{X}(−D_{I}). Of fundamental importance is the following fact, due to Lipman ([Li], prop. 6.2, see also [CPR] section 1):

Proposition 2 The above map I_{X} → E_{X}^{+}, I ↦ D_{I} establishes an isomorphism of semigroups (I_{X}, .) ≃ (E_{X}^{+}, +). The inverse isomorphism E_{X}^{+} → I_{X} is the map D ↦ I_{D} := π_{∗}O_{X}(−D)_{η}.

Corollary 2 Let D ∈ E_{X} be effective, and I := π_{∗}O_{X}(−D)_{η} ⊆ R. Then I ∈ I_{X}. In particular, IO_{X} = O_{X}(−\overline{D}) is locally invertible.

Proof: We have I = π_{∗}O_{X}(−\overline{D})_{η} = I\overline{D} by corollary 1, where \overline{D} ∈ E_{X}^{+}.

In [Li], th. 20.1 and 25.1, Lipman has proved that if char k ≠ 2, 3, 5, the singularities of type E_{8} and the nonsingular germs of surface are the only rational surface singularities for which unique factorization holds in the semigroup I^{∗} of M-primary complete ideals, that is every I ∈ I^{∗} factors in a unique way as a product of simple complete ideals. For any other rational surface singularity over a field k of any characteristic, a weaker notion holds in I^{∗}, or equivalently in I_{X}^{∗} for every resolution of singularities X, that of semifactorization. Semifactorization in E_{X}^{+} means that every element D ∈ E_{X}^{+}, D ≠ 0, can be formally expressed in a unique way as a linear combination with coefficients in Q_{≥0} of pairwise inequivalent extremal elements of E_{X}^{+}. Here D ∈ E_{X}^{+}, D ≠ 0 is extremal if nD = D_{1} + D_{2}, for some n ∈ N, D_{1}, D_{2} ∈ E_{X}^{+} implies that m_{i}D_{i} = n_{i}D, for suitable positive integers m_{i}, n_{i}, i = 1, 2. D and D′ are said to be equivalent if mD = m′D′ for suitable positive integers m, m′.

For each irreducible component E_{γ} of the exceptional locus of π : X → S, let D′_{E_{γ}} ∈ ⨁_{α∈=Δ} QE_{α} be the unique Q-Cartier divisor on X with exceptional support for π such that D′_{E_{γ}} .E_{γ} = −1 and D′_{E_{γ}} .E_{α} = 0 for α ≠ γ. The existence of D′_{E_{γ}} follows from the definiteness of the intersection matrix (E_{γ}.E_{γ′})_{γ,γ′∈Δ}. Let m_{γ} ∈ N be such that D_{E_{γ}} := m_{γ}D′_{E_{γ}} is the least multiple of D′_{E_{γ}} which is a divisor. The negative definiteness of the intersection matrix ensures that the extremal elements of E_{X}^{+} are precisely all integer multiples of all divisors D_{E_{γ}}.

Given D ∈ E_{X}^{+}, one can express in a unique way D = ∑_{γ} q_{γ}D_{E_{γ}}, where q_{γ} ∈ Q_{≥=0}. In fact,

q_{γ} := \frac{D.E_{γ}}{D_{E_{γ}} .E_{γ}} .

This shows the semifactoriality of E_{X}^{+}. Via proposition 2, we obtain that semifactorization holds in I_{X}^{∗} for all X, hence in I^{∗} as well.

Let ν be a divisorial valuation centered in the local ring R of the rational surface singularity (S, η). Let us consider a resolution of singularities π : X → S such that the center of ν in X is a curve E, which is one of the E_{γ}’s.

The valuation ν defines a filtration of R by ν-ideals

(2)  P_{n} := {h ∈ R / ν(h) ≥ n} = π_{∗}O_{X}(−nE)_{η}

for n ∈ N. It is clear that P_{n} is a M-primary complete ideal.

Definition 1 For n ∈ N, let D_{n} be the smallest divisor D′ ∈ E_{X}^{+} such that D′ ≥ nE (that is, D_{n} = \overline{nE} with notations as in proposition 1).

Corollary 3 For n ∈ N, we have P_{n}O_{X} = O_{X}(−D_{n}).

Proof: Obvious from corollary 2.

Remark: Let Φ^{+} := ν(R\{0}) be the semigroup of ν. For any n ∈ N, the coefficient of D_{n} in E is c(n) := ν(P_{n}), which is the smallest element in Φ^{+} greater than or equal to n. Thus n ∈ Φ^{+} if and only if c(n) = n.

Now, let D^{∼} be the extremal divisor D_{E} ∈ E_{X}^{+} corresponding to the irreducible

exceptional component E. Let P^{∼}:= I_{D^{∼}} be the associated complete ideal. We have

Proposition 3 The ideal P^{∼} is a ν-ideal, that is P^{∼}= P_{ν(P^{∼})}, or equivalently

D^{∼}= D_{ν(P^{∼})}.

Proof: Since P^{∼} O_{X} = O_{X}(− D^{∼}), the value ν(P^{∼}) is the coefficient of D^{∼} in

E and, since D^{∼} ∈ E_{X}^{+}, one has D^{∼}≥ D_{ν(P^{∼})}. Moreover E ∉ Supp(D_{ν(P^{∼})}− D^{∼})

because ν(P^{∼}) ∈ Φ^{+} and (D_{ν(P^{∼})}− D^{∼}).E_{γ} = D_{ν(P^{∼})}.E_{γ} ≤ 0 for E_{γ} ≠ E, thus for

E_{γ} ∈ Supp(D_{ν(P^{∼})}− D^{∼}). Therefore D_{ν(P^{∼})}− D^{∼}≥ 0. This concludes the proof.

The following result will be the key to understanding the graded algebra associated with ν.

Theorem 1 For n ∈ Φ^{+} and r ∈ N, we have P_{rν(P^{∼})+n} = P^{∼}^{r}P_{n}.

Proof: It suffices to show the result for r = 1 and n ≠ 0. Clearly P^{∼}P_{n} ⊆ P_{ν(P^{∼})+n}, so it is enough to show that D_{ν(P^{∼})+n} ≥D^{∼} +D_{n} by corollary 3.

First notice that, since n ∈ Φ^{+} and P_{n}O_{X} = O_{X}(−D_{n}), the coefficient of D_{n} in E is n. Thus, if

D̂_{1} = nE < D̂_{2} < … < D̂_{t} = D_{n}

are obtained as in proposition 1 from the divisor nE, then D̂_{i+1} = D̂_{i} + E_{γ_{i}}, where E_{γ_{i}} ≠ E for 1 ≤ i ≤ t − 1, and D̂_{i}.E_{γ_{i}} > 0. Let us show by induction on i that

D^{∼} +D̂_{i} ≤ D_{ν(P^{∼})+n}. Then the desired inequality is obtained for i = t.

If i = 1, D^{∼}= D_{ν(P^{∼})} < D_{ν(P^{∼})+n}. Moreover, D^{∼} +D̂_{1} =D^{∼} +nE ≤ D_{ν(P^{∼})+n}, since

the coefficient of D_{ν(P^{∼})+n} in E is ν(P^{∼}) + n. Assume D^{∼} +D̂_{i} ≤ D_{ν(P^{∼})+n}. Since

E_{γ_{i}} ≠ E, we have D^{∼} .E_{γ_{i}} = 0. Hence

(D_{ν(P^{∼})+n} − (D̂_{i}+ D^{∼})) .E_{γ_{i}} = D_{ν(P^{∼})+n}.E_{γ_{i}} − D̂_{i}.E_{γ_{i}} < 0.

This implies that E_{γ_{i}} ≤ D_{ν(P^{∼})+n} − (D̂_{i}+ D^{∼}), that is D̂_{i+1}+ D^{∼}≤ D_{ν(P^{∼})+n}. This

concludes the proof.

To the ν-adic filtration are associated its Rees algebra and graded algebra

Gr_{ν}(R) := ⨁_{n∈Φ^{+}} P_{n} gr_{ν}(R) := ⨁_{n∈Φ^{+}} P_{n}/P_{n}^{+},

where P_{n}^{+} := {h ∈ R / ν(h) > n}.

Definition 2 A sequence {Q_{i}} of elements of M is said to be a generating sequence for ν if every ν-ideal I is generated by the set

{finite products ∏_{j} Q_{j}^{a_{j}} / ∑_{j} a_{j}ν(Q_{j}) ≥ ν(I), a_{j} ∈ N}.

Remark: It can be checked that a sequence {Q_{i}} as above is a generating sequence for ν if (and only if) the initial forms of the Q_{i}’s in gr_{ν}(R) generate it as a k-algebra. It is a consequence of the fact that the ν-adic and M-adic topologies are equivalent ([ZS] p.270). Also notice that if ν has a finite generating sequence, the Rees algebra Gr_{ν}(R) is Noetherian.

Corollary 4 There exists a finite generating sequence for ν. In particular, Gr_{ν}(R) is Noetherian.

Proof: Let Σ be the set consisting of ν(P^{∼}) and all n ∈ Φ^{+} such that n − ν(P^{∼}) ∉ Φ^{+}. Clearly, Σ is a finite set. More precisely, since Φ^{+} has a conductor, for 0 ≤ i ≤ ν(P^{∼}) − 1, Σ has exactly one element congruent with i modulo ν(P^{∼}), thus

its cardinal is ν(P^{∼}). For σ ∈ Σ, let {Q_{σ,j_{σ}}} be a finite set of elements whose classes modulo P_{σ}^{+} are a basis of P_{σ}/P_{σ}^{+}. It follows from theorem 1 that the collection of all of the Q_{σ,j_{σ}} are a generating sequence for ν.

Remark: It is shown in [Cu] that for a given complete normal local ring (R, M) of dimension two, R has a rational singularity if and only if every divisorial valuation centered in R has a finite generating sequence.

3 A generating sequence of a valuation.

Corollary 4 provides an explicit way to get a generating sequence for the divisorial valuation ν. We will deal with it in this section.

First, given two M-primary complete ideals I′ ⊂ I, we will show how to find a basis of I/I′. Then we will apply it to the ideals P_{σ}^{+} ⊂ P_{σ} for σ ∈ Σ as in the proof of corollary 4.

Let π′ : X′ → S be any resolution of singularities of (S, η). It factorizes as a sequence of point blowing ups X′ = S_{m}⟶^{π_{m}} … ⟶^{π_{2}}S_{1}⟶^{π_{1}}S where π_{i} is the blowing up of a closed point η_{i} ∈ S_{i−1} ([Ar] and [Li]). For 1 ≤ i ≤ m, let E_{i1}, …, E_{is_{i}} (resp E_{i1}^{∗}, …, E_{is_{i}}^{∗}) be the strict (resp. total) transforms in X′ of the irreducible components of the exceptional locus of π_{i} (see [M] for the definition of total transform of a Weil divisor on a normal surface singularity). Then Δ′ = {(i, j), 1 ≤ i ≤ m, 1 ≤ j ≤ s_{i}} is the set of indices of the irreducible components of the exceptional locus of π′ and ⊕_{γ∈Δ′}QE_{γ} = ⊕_{γ∈Δ′}QE_{γ}^{∗}. The expressions of the E_{γ}^{∗}’s in terms of the E_{γ}’s can be computed from the weighted dual graph of π′ ([Re] th. 1.14).

Lemma 1 Given an effective divisor D ∈ E_{X′}, D ≠ 0, if D = ∑_{γ∈Δ′} ρ_{γ}E_{γ}^{∗}, then there exists γ = (i, j) such that ρ_{γ} > 0 and ρ_{γ′} = 0 for all γ′ = (i′, j′) with i′ < i. Moreover, for any such γ, ρ_{γ} is an integer, thus ρ_{γ} ≥ 1.

Proof: Let i, 1 ≤ i ≤ m, such that ρ_{i′j′} = 0 for all i′ < i, 1 ≤ j′ ≤ s_{i′} and maximal with this property. Since E_{ij}^{∗} = E_{ij} + ∑_{i′>i,j′} p_{i′j′}E_{i′j′} for some rational numbers p_{i′j′}, for all j, 1 ≤ j ≤ s_{i}, the coefficient of E_{ij} in D is ρ_{ij}, thus it is a nonnegative integer. Besides, D ≠ 0 implies ρ_{ij} > 0 for some j.

Proposition 4 ([Re] th. 3.15) Let I and I′ be two M-primary complete ideals in R such that I ⊃ I′. There is a chain of M-primary complete ideals I = I_{0} ⊃ I_{1} ⊃ … ⊃ I_{r} = I′ such that dim_{k}(I_{t}/I_{t+1}) = 1. Thus, if for each t we take h_{t} ∈ I_{t} − I_{t+1}, then the classes of {h_{t}}_{t=0}^{r−1} in I/I′ determine a basis of I/I′.

Proof: Let us construct this chain. We define inductively I_{t+1} from I_{t} as follows : let π′ : X′ → S be a resolution of singularities of (S, η) such that both I_{t}O_{X′} and I′O_{X′} are invertible sheaves. Let I_{t}O_{X′} = O_{X′}(−D′_{t}) and I′O_{X′} = O_{X′}(−D′), thus D′_{t}, D′ ∈ E_{X′}^{+}. Let D′_{t} = ∑ ρ_{γ}^{t}E_{γ}^{∗} and D′ = ∑ ρ_{γ}E_{γ}^{∗} be the expressions of D′_{t} and D′ in terms of {E_{γ}^{∗}}_{γ∈Δ′}. Since I_{t} ⊃ I′, we have D′_{t} < D′ and by lemma 1, there exists γ = (i, j) ∈ Δ′ such that ρ_{γ}^{t} < ρ_{γ} and ρ_{γ′}^{t} = ρ_{γ′} for all γ′ = (i′, j′) with i′ < i. We choose a closed point ξ_{t} ∈ E_{γ} such that ξ_{t} ∉ E_{γ′} for all γ′ ∈ Δ′ − {γ} and consider the surface X′′ obtained by blowing up ξ_{t} in X′, the induced resolution of singularities π′′ : X′′ → S and the Q-Cartier divisor D′′_{t+1} = D′′_{t} + E_{ξ_{t}} where D′′_{t} is the total transform in X′′ of D′_{t} and E_{ξ_{t}} is the exceptional locus of the blowing up X′′ → X′. We define I_{t+1} := π′′_{∗}O_{X′′}(−D′′_{t+1})_{η}.

Since D′′_{t} < D′′_{t+1} we have I_{t} ⊃ I_{t+1}. There is an exact sequence

0 ⟶ π′′_{∗}O_{X′′}(−D′′_{t+1})_{η} ⟶ π′′_{∗}O_{X′′}(−D′′_{t})_{η} ⟶ H^{0}(O_{E_{ξ_{t}}}) ≅ k.

Therefore dim(I_{t}/I_{t+1}) = 1. Besides, by lemma 1, ρ_{γ} ≥ ρ_{γ}^{t} +1, hence D′ ≥ D′_{t} +E_{γ} and, since ξ_{t} ∈ E_{γ}, D′′ ≥ D′′_{t} + E_{ξ_{t}} = D′′_{t+1} where D′′ is the total transform in X′′ of D′, thus I′ ⊆ I_{t+1}. Therefore we conclude the result, since it is clear that the process finishes after a finite number of steps.

Now, let us describe a procedure to find a generating sequence for the divisorial valuation ν. Recall that π : X → S is a resolution of singularities of (S, η) such that the center of ν in X is a curve E, and keep the notations as in section 2. We proceed as follows:

1. We compute the extremal divisor D^{∼} corresponding to E and hence ν(P^{∼}).

2. We compute the elements n_{1}, n_{2}, …, n_{ν(P^{∼})} of Σ in the following way:

First, we apply the algorithm in proposition 1 to E to obtain the divisor D_{1} = \overline{E}. Let n_{1} be the coefficient of D_{1} in E, so that D_{n_{1}} = D_{1}. Note that n_{1} is the smallest positive element in Φ^{+}. Now, suppose we have obtained n_{1}, …, n_{i} and the divisors D_{n_{1}}, …, D_{n_{i}}. If i = ν(P^{∼}) we stop. If not, we define n_{i+1} as follows: We compute the divisor \overline{D_{n_{i}} + E} applying proposition 1 to D_{n_{i}} + E. Let m_{i} > n_{i}

be its coefficient in E, so that D_{m_{i}} = \overline{D_{n_{i}} + E}. If the classes modulo ν(P^{∼}) of n_{1}, …, n_{i}, m_{i} are different, we set n_{i+1} = m_{i}. If not, we compute \overline{D_{m_{i}} + E} and

consider its coefficient m′ in E. Again, if the classes modulo ν(P^{∼}) of n_{1}, …, n_{i}, m′

are different, we set n_{i+1} = m′ and if not, we continue this way. Since i < ν(P^{∼}),

n_{i+1} is defined after a finite number of steps. Finally, also for i = ν(P^{∼}), we compute the divisor D_{m_{ν(^{∼} P)}} obtained applying proposition 1 to D_{n_{ν(^{∼} P)}} + E.

3. For i = 1, 2, …, ν(P^{∼}) we consider the complete ideals P_{n_{i}}^{+} = P_{m_{i}} ⊂ P_{n_{i}}, and we apply the algorithm in proposition 4 to define a chain of M-primary complete ideals

P_{n_{i}} = I_{i,0} ⊃ I_{i,1} ⊃ … ⊃ I_{i,r_{i}} = P_{m_{i}}

such that dim(I_{i,t}/I_{i,t+1}) = 1. Moreover, once we have defined I_{α} for α = (j, r) ≤ (i, t) with the lexicographic order, in order to define I_{i,t+1}, if it is possible we choose as point ξ_{t} in the algorithm any of the points already chosed to define the previous ideals.

4. We take {Q_{s}}_{s=0}^{N} such that, for i = 0, …, ν(P^{∼}), the classes of {∏_{s} Q_{s}^{a_{s}}/ ∑ a_{s}ν(Q_{s}) ≥ n_{i}} in P_{n_{i}}/P_{n_{i}}^{+} determine a basis, as follows:

Let Q_{0} be any element in P_{n_{1}} − I_{1,1}. Let (i, t), 0 ≤ t < r_{i} and suppose by induction that we have defined {Q_{s}}_{s=0}^{M} such that, for all α = (j, r), r < r_{j} and α < (i, t), there exists an element of the form ∏_{s=0}^{M} Q_{s}^{a_{s}} in I_{j,r} − I_{j,r+1} Then, one of the following possibilities hold:

4.1. There exists an element of the form ∏_{s=0}^{M} Q_{s}^{a_{s}} in I_{i,t} − I_{i,t+1}. We do not add any Q_{s}.

4.2. There is no ∏_{s=0}^{M} Q_{s}^{a_{s}} in I_{i,t} − I_{i,t+1}. Then we define Q_{M+1} to be any element in I_{i,t} − I_{i,t+1}.

Corollary 5 The set {Q_{s}}_{s=0}^{N} obtained in the previous procedure is a generating sequence for ν.

Remark: Note that the procedure described does not give a unique way of finding a generating sequence. In fact, in the algorithm in proposition 4 there are several possibilities of defining a chain of complete ideals and of finding elements in I_{t} − I_{t+1}. In the previous procedure we have tried to find a generating sequence with few elements.

Example: Let (S, η) be the rational double point of type E_{6} defined by x^{4} + y^{3} + z^{2} = 0 in a neighbourhood of the origin in k^{3}, and let π : X → S be the minimal resolution of singularities of (S, η). The dual graph of π is

[figure: E_{21}; E_{31}; E_{4}; E_{32}; E_{22}; E_{1}]

where we have labelled the irreducible components of the exceptional locus of π in the following way: π is the composition of four point blowing ups X = S_{4}⟶^{π_{4}}S_{3}⟶^{π_{3}}S_{2}⟶^{π_{2}}S_{1}⟶^{π_{1}}S. The exceptional locus of π_{1} (resp. π_{4}) has one irreducible component whose strict transform in X is E_{1} (resp. E_{4}) and the exceptional locus of π_{2} (resp. π_{3}) has two irreducible components whose strict transforms in X are E_{21}, E_{22} (resp. E_{31}, E_{32}). All self intersections E_{γ}^{2} are equal to −2.

Let ν be the divisorial valuation defined by E_{1}. Then, D^{∼} = D_{E_{1}} = 2E_{1} +

E_{21} + E_{22} + 2E_{31} + 2E_{32} + 3E_{4}, so ν(P^{∼}) = 2. We represent D^{∼} in the dual graph as follows:

[figure: D^{∼} = D_{E_{1}}; 1; 2; 3; 2; 1; 2; −1]

where the arrow and the -1 mean that D^{∼}.E_{1} = −1. For step 2 in the procedure, we apply the algorithm in proposition 1 first to E, and obtain D^{∼}, so that n_{1} = 2, D_{2} =D^{∼}, and then to D_{2} + E, and obtain the divisor D_{3}, which is the extremal divisor D_{E_{4}}, represented by

[figure: D_{3} = D_{E_{4}}; 2; 4; 6; 4; 2; 3; −1]

thus n_{2} = 3. We compute the divisor D_{m_{2}} := \overline{D_{3} + E} which is 2 D^{∼}. For step 3, we obtain the chains P_{2} ⊃ I_{1,1} ⊃ P_{3} and P_{3} ⊃ P_{4}, where the ideal I_{11} is defined as follows: We have D_{2} = 2E_{1}^{∗} and D_{3} = 3E_{1}^{∗} + \frac{1}{2}(E_{21}^{∗} + E_{22}^{∗} + E_{31}^{∗} + E_{32}^{∗}), so we take any point ξ ∈ E_{1}, ξ ∉ E_{γ} for γ ≠ 1, consider the blow up X′ of X with center ξ and the divisor D′ = 2E′_{1} + E′_{21} + E′_{22} + 2E′_{31} + 2E′_{32} + 3E′_{4} + 2E_{ξ} where E_{ξ} is the exceptional locus of X′ → X and by E′_{γ} we denote the strict transform of E_{γ} in X′. The ideal I_{1,1} is π′_{∗}O_{X′}(−D′)_{η} where π′ is the resolution of singularities X′ → S. Finally, for step 4 we may consider x ∈ P_{2} − I_{11}, y ∈ I_{11} − P_{3} and z ∈ P_{3} − P_{4}. Therefore {x, y, z} is a generating sequence for ν and gr_{ν}R ≃ k[X, Y, Z]/(Y^{3} + Z^{2}) where X, Y have degree 2 and Z has degree 3.

Now, let us consider the divisorial valuation ν′ defined by E_{21}. Then D′^{∼}= D_{E_{21}} is represented by

[figure: D′^{∼} = D_{E_{21}}; 4; 5; 6; 4; 2; 3; −3]

so that ν′(P′^{∼}) = 4. Applying proposition 1 we obtain the divisors D′_{1} = D_{E_{1}} and

[figure: D′_{2}; 2; 3; 4; 3; 2; 2; −1; −1; D′_{3}; 3; 5; 6; 4; 2; 3; −1; −1]

so, by corollary 3 and theorem 1, all ν′-ideals P′_{n} are defined by the divisors

{r D′^{∼}, D′_{i} + r D′^{∼}}_{1≤i≤3,r∈N}. Therefore, the unique ν′-ideals which are extremal are

the ones associated to D′^{∼}= D_{E_{21}} and D′_{1} = D_{E_{1}} and the valuation ideals P′_{2} and P′_{3} corresponding to D′_{2} and D′_{3} are indecomposable but not extremal. A generating sequence for ν′ is {x, y, w = x^{2} + iz} and gr_{ν′}R ≃ k[X, Y, W]/(2X^{2}W + Y^{3}) where deg X = 1, deg Y = 2, deg W = 4.

4 The Hilbert function.

In this section, we give an expression for the Hilbert-Samuel function

N ⟶ N, n ↦ l (R/P_{n})

defined by a divisorial valuation ν centered in the local ring R of a rational surface singularity. We keep notations as in section 2.

First note that, since P_{n}O_{X} = O_{X}(−D_{n}) (corollary 3), we have l (R/P_{n}) = χ(O_{D_{n}}) = h^{0}(O_{D_{n}}) (see [Li] lemma 23.1). By the adjunction formula

(3)  l (R/P_{n}) = −\frac{1}{2}D_{n}.(D_{n} + K),

where K is a canonical divisor on X.

For n ∈ N, let us consider the Q-Cartier divisor

D^{∼}_{n} := D_{n} − \frac{c(n)}{ν(P^{∼})} D^{∼},

where c(n) is the coefficient of D_{n} in E (see remark after corollary 3).

Remark: Let us consider the blow-up Y of Spec R along P^{∼}. Then Y is a surface having only rational singularities, and there is a morphism X ⟶ Y contracting all of the E_{γ}’s which are exceptional for π : X ⟶ Spec R except E (see [CPR]

lemma 2.4). Now, D^{∼}_{n} is a Q-Cartier divisor with support in ⋃_{E_{γ}≠E} E_{γ} such that,

for E_{γ} ≠ E, d_{γ} := −D^{∼}_{n}.E_{γ} = −D_{n}.E_{γ} ∈ N. Since R is complete, one can choose curves C_{γ,1}, …, C_{γ,d_{γ}} on X, each of which is smooth, transversal to E_{γ}, and does not intersect any other exceptional curve of X ([Li], p.226). Let C_{n} be the projection

on Y of C := ∑_{E_{γ}≠E} (C_{γ,1} + ⋯ + C_{γ,d_{γ}}). Then C_{n} is a Weil divisor on Y and D^{∼}_{n} is the exceptional part of the total transform of C_{n} in X.

Given a Q-Cartier divisor B on X, we set

χ(B) := −\frac{1}{2}B.(B + K).

Note that for B, B′ two Q-Cartier divisors on X,

χ(B + B′) = χ(B) + χ(B′) − B.B′.
Applying this to the decomposition D_{n} = \frac{c(n)}{ν(P^{∼})} D^{∼} +D^{∼}_{n}, we obtain

χ(D_{n}) = χ (\frac{c(n)}{ν(P^{∼})} D^{∼}) + χ(D^{∼}_{n}) − \frac{c(n)}{ν(P^{∼})} D^{∼} .D^{∼}_{n}.

Now, by definition of D^{∼}, D^{∼} .D_{n} = c(n) D^{∼} .E, since c(n) is the coefficient of D_{n}

in E. Similarly, D^{∼} . D^{∼}= ν(P^{∼}) D^{∼} .E because the coefficient of D^{∼} in E is ν(P^{∼}).

Therefore D^{∼} .D^{∼}_{n} = 0 and we conclude that

(4)  l (R/P_{n}) = χ (\frac{c(n)}{ν(P^{∼})} D^{∼}) + χ(D^{∼}_{n}).

Theorem 2 With notations as above, we have for all n ∈ N

(5)  l (R/P_{n}) = Q(n) + φ(n),

where Q(n) is a polynomial of degree two in n and φ(n) satisfies φ(n+ν(P^{∼})) = φ(n) for n ∈ Φ^{+}. Thus φ(n) is a periodic function for n >> 0.

Proof: Let

(6)  Q(n) := χ (\frac{n}{ν(P^{∼})} D^{∼}) = \frac{− D^{∼} . D^{∼}}{2 ν(P^{∼})^{2}} n^{2} − \frac{D^{∼} .K}{2 ν(P^{∼})}n,

and φ(n) := l(R/P_{n}) − Q(n). We have − D^{∼} . D^{∼}> 0 so that Q(n) has degree two as stated.

Let n ∈ Φ^{+}, so that c(n) = n. Then

(7)  φ(n) = χ(D^{∼}_{n})

by (4) and D_{n+ν(P^{∼})} =D^{∼} +D_{n} by corollary 3 and theorem 1. Besides, n + ν(P^{∼})

∈ Φ^{+} and consequently c(n + ν(P^{∼})) = n + ν(P^{∼}). Therefore

D^{∼}_{n+ν(P^{∼})} =D^{∼} +D_{n} − \frac{n + ν(P^{∼})}{ν(P^{∼})} D^{∼}= D^{∼}_{n}

and we conclude that φ(n + ν(P^{∼})) = φ(n).

Remark: The decomposition l (R/P_{n}) = Q(n) + φ(n), with Q(n) a polynomial of degree two in n and φ(n) a periodic function for n >> 0 is a consequence of

corollary 4. As for the period of φ, theorem 2 shows that it divides ν(P^{∼}). In a

forthcoming paper we will prove that this period is exactly ν(P^{∼}). The aim of this future paper is to determine invariants of S from the data of the graded k-algebras gr_{ν}R for all divisorial valuations ν dominating R.

Among all resolutions of singularities X of S such that the center of ν in X is a curve, there is a minimal one, denoted by X_{ν}. This can be seen as follows; let X_{min} be the minimal resolution of S. If the center of ν in X_{min} is a curve, then let X_{ν} := X_{min}. If the center of ν in X_{min} is a point, then blow it up and repeat the process. This terminates after a finite number of steps.

Let Γ_{ν} be the weighted dual graph of the configuration of exceptional curves on X_{ν}. Vertices of Γ_{ν} correspond to irreducible curves with exceptional support for X_{ν} → S, an edge joins two vertices whose corresponding curves intersect on X_{ν}. Each vertex is weighted by the self intersection of the corresponding curve.

Let E_{ν} be the vertex of Γ_{ν} corresponding to the center of ν in X_{ν}. We define the valence of ν to be the number of edges of Γ_{ν} containing E_{ν}. Note that, if X_{ν} is not the minimal desingularization of S (i.e., ν is not essential) then the valence of ν is either 1 or 2.

The pointed weighted dual graph (Γ_{ν}, E_{ν}) is the data consisting of Γ_{ν} together with the vertex E_{ν}. Two pointed weighted dual graphs (Γ_{ν}, E_{ν}) and (Γ_{ν′}, E_{ν′}) are isomorphic if there exists an isomorphism of graphs Γ_{ν} ≃ Γ_{ν′} mapping E_{ν} to E_{ν′} and preserving weights.

It has been shown in section 2 that the collection of divisors {D_{n}}_{n∈N} only depends on the pointed weighted dual graph of ν. By corollary 3, we have P_{n} := π_{∗}O_{X_{ν}} (−D_{n})_{η} for all n ∈ N. The above formulas (6) and (7) moreover show that the Hilbert function of gr_{ν}(R) is determined by the pointed weighted dual graph of ν. This makes it plausible that the only if part of the following problem is true.

Problem 1 Let ν and ν′ be two divisorial valuations of valence 1 or 2 centered in the local ring of a rational surface singularity S. Is it true that ν and ν′ have isomorphic pointed weighted dual graphs if and/or only if their graded k-algebras gr_{ν}(R) and gr_{ν′}(R) are isomorphic?

Remark: In case R is regular, the if and only if version follows from [Sp]. The dual graph Γ_{ν} is an equivalent data to the Zariski exponents \overline{β}_{0}, …, \overline{β}_{g+1} associated with ν ([Sp], (6.1)). With notations as in section 6 of [Sp], there is an isomorphism

gr_{ν}(R) ≃ \frac{k[Q_{0}, …, Q_{g+1}]}{< \{Q_{j}^{n_{j}} − Q_{0}^{a_{0,j}} … Q_{j_{1}}^{a_{j−1,j}}\}_{1≤j≤g} >}.

We have ν(Q_{j}) = \overline{β}_{j}, the n_{j}’s are determined by

n_{j} := \frac{g.c.d.(\overline{β}_{0}, …, \overline{β}_{j−1})}{g.c.d.(\overline{β}_{0}, …, \overline{β}_{j})},

and the nonnegative integers a_{i,j}’s are uniquely determined by the relations

n_{j}\overline{β}_{j} = a_{0,j}\overline{β}_{0} + ⋯ + a_{j−1,j}\overline{β}_{j−1}

subject to the restrictions 0 ≤ a_{i,j} < n_{i} for 1 ≤ i < j.

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