Newton-Okounkov bodies of exceptional curve valuations

We prove that the Newton-Okounkov body of the flag $E_{\bullet}:= \left\{ X=X_r \supset E_r \supset \{q\} \right\}$, defined by the surface $X$ and the exceptional divisor $E_r$ given by any divisorial valuation of the complex projective plane $\mathbb{P}^2$, with respect to the pull-back of the line-bundle $\mathcal{O}_{\mathbb{P}^2} (1)$ is either a triangle or a quadrilateral, characterizing when it is a triangle or a quadrilateral. We also describe the vertices of that figure. Finally, we introduce a large family of flags for which we determine explicitly their Newton-Okounkov bodies which turn out to be triangular.


Introduction
Newton polygons are probably the first example of the usage of polyhedral objects to study algebraic varieties, curves in this case. Newton-Okounkov bodies are more complicated objects that pursue the same idea; they associate a convex body to a big divisor on a smooth irreducible normal projective variety X, with respect to a specific flag of subvarieties of X, via the corresponding valuation on the function field of X. They were introduced by Okounkov [24,25,26] and independently developed by Lazarsfeld and Mustaţă [22], on the one hand, and Kaveh and Khovanskii [16], on the other. One of the main advantages of these bodies is that their convex structure helps in the study of the asymptotic behavior of the linear systems given by the divisor and the valuation, the structure of the Mori cone of X, and positivity properties of divisors on X [1,17,18,19,21]. The computation of Newton-Okounkov bodies is a very hard task and, sometimes, their behavior is unexpected [21]. In this regard, little is known about them, even when X is a surface. For these varieties, we know that they are polygons with rational slopes and can be computed from Zariski decompositions of divisors [22].
Let p ∈ P 2 := P 2 C be a point in the complex projective plane and consider a surface X obtained after a sequence of finitely many simple point blow-ups starting with the blow-up of p, where simple means that we only blow points in the exceptional divisor created last. We devote this paper to explicitly describe the Newton-Okounkov body of a flag E • := {X = X r ⊃ E r ⊃ {q}}, with respect to the pull-back H on X of the line bundle O P 2 (1), where E r is the last obtained exceptional divisor and q a point on it. These flags define and are defined by specific representatives ν = ν E• of the so-called exceptional curve valuations. This class of valuations corresponds to one of the five types of valuations of the fraction field of the local ring O P 2 ,p appearing in the classification given by Spivakovsky in [27]; the name comes from [11]. Notice that their rank and rational rank equal 2 and their transcendence degree is 0. These valuations are usually considered up to equivalence [28] and one can attach a Newton-Okounkov body to any representative whose value group is included in R 2 .
Consider the flag E • and let ν be its exceptional curve valuation. The above mentioned Newton-Okounkov body will be denoted by ∆ ν (H) = ∆ ν . The valuation ν has two components: ν = (υ 1 , υ 2 ), where υ 1 := ν r is the divisorial valuation, with Z as value group, defined by the divisor E r . We denote by {β ′ j (ν r )} g+1 j=0 the so-called sequence of Puiseux exponents of the valuation ν r (see Subsection 2.4) which determines, and it is determined by, the dual graph of the valuation ν r ; then we say that ν r has g Puiseux pairs. The pair (ν r , H) comes with an interesting value, denoted byμ(ν r ), which is defined as lim m→∞ m −1 a(mL), where a(mL) is the last value of the vanishing sequence of H 0 (mL) along ν r for a line L of P 2 . This value, introduced in [1], is an analogue of the Seshadri constant for the valuation ν r . We recall that the Seshadri constants were considered by Demailly in [7] in the study of the Fujita conjecture. In our casê where (u, v) are coordinates in an affine chart containing p as the origin.
Returning to Newton-Okounkov bodies, in [4] it is considered a sort of normalized Newton-Okounkov body for valuations ν = ν E• corresponding to valuations ν r with only one Puiseux pair and whose point q is not free. There, the authors prove that these Newton-Okounkov bodies are triangles or quadrilaterals whose vertices depend on the Puiseux exponents of ν r and the valueμ(ν r ). As we will show (see Remark 3.15), the Newton-Okounkov bodies considered in [4] are nothing but those attached to a different representative ν ′ of the valuation ν. The motivation for considering ν ′ is to study the variation of infinitesimal Newton-Okounkov bodies as E r and q vary.
The first main goal in this paper is to give a completely explicit expression of the Newton-Okounkov body ∆ ν attached to any flag as above. Let us explain our result. For a start, recall that the volume of a divisorial valuation ν r is defined as where P α = {f ∈ O P 2 ,p |ν r (f ) ≥ α} ∪ {0}. It is satisfied thatμ(ν r ) ≥ 1/vol(ν r ) and ν r is said to be minimal when the equality holds. An exceptional curve valuation ν is named minimal whenever its first component is minimal.
The following result, which is Theorem 3.8 in the paper, describes ∆ ν for exceptional curve valuations which are minimal.
When ν (and so ν r ) is not minimal, there exists a unique integral curve C given by a polynomial f ∈ C[u, v] such thatμ(ν r ) = ν r (f )/ deg(f ), which is named supra-minimal for ν r (see Lemma 3.10). Setting we are able to prove the following result (Theorem 3.12 in the paper) which describes the Newton-Okounkov bodies of flags E • corresponding to non-minimal valuations ν.
Above, ϕ η is a general analytically irreducible element in O P 2 ,p whose strict transform on X is transversal to the exceptional divisor E η (see Definition 2.1).
Recall that the number of vertices of the Newton-Okounkov body defined by a flag and a big divisor on a surface X is bounded by 2̺ + 2, where ̺ is the Picard number of X [21]. This is a consequence of the fact that a ray of the form D − tC, where D is a big divisor and C a curve on X, can only cross ρ + 1 Zariski chambers. It is conjectured in [20] (see also [15]) that the bound could be applied even if the flag is considered on a projective model dominating X (and the Newton-Okounkov body associated to the pull-back of a big divisor on X). Our results can be regarded as new evidence supporting the conjecture.
Our second main goal is presented in Theorem 4.2, where we characterize which Newton-Okounkov bodies are triangular (and, by exclusion, quadrangular) in the case in which ν is not minimal. We prove that the fact of being a triangle or a quadrilateral depends only on the branches of the supraminimal curve C of ν r . Roughly speaking, the divisor E r gives a partition of the dual graph of ν into two connected components, one of them being finite and the other one infinite. The Newton-Okounkov body ∆ ν will be a triangle when all the branches of C go through the same component, and a quadrilateral otherwise. In particular, we prove that the Newton-Okounkov bodies of valuations ν with supraminimal curves having only one branch at p are triangles.
The valuesμ(ν r ) and the supraminimal curves are not easy to compute. An interesting class of divisorial valuations are the so-called non-positive at infinity ones (see the beginning of Section 5). As one can see in [13], they define surfaces with regular cone of curves and, for these surfaces, one can decide (by checking a condition) when their Cox rings are finitely generated. In Corollary 5.2 we prove that the Newton-Okounkov bodies ∆ ν of the valuations ν in this large class are triangles and we give explicitly their vertices. Although we deduce this result from Theorem B, we also provide in Remark 5.3 the Zariski decomposition of those divisors that, according to Theorem 6.4 of [22], would allow us to compute ∆ ν .
The last section of this paper contains three examples, in which Newton-Okounkov bodies of flags of different nature are computed; we hope this will help the readers in a better understanding of our results.

Divisorial and exceptional curve valuations
Thereafter we will see that the valuations defined by those flags we are interested in are exceptional curve valuations and, also, that their first components are divisorial valuations. So we devote this section to study these classes of valuations, providing results which will be useful.
Let P 2 be the complex projective plane. For p ∈ P 2 , let R be the local ring of P 2 at p, and let m be its maximal ideal; moreover, write F for the field of fractions of R. A valuation of F is an onto mapping ν : The valuations we consider are centered at R, that is, such that R ⊆ R ν and R ∩ m ν = m. Finally, the monoid ν(R \ {0}) is called the semigroup of the valuation ν (relative to R). Most of the results in this section are also true when using an algebraically closed field instead of the complex numbers [2,6].
The valuations ν of F centered at R are in one-to-one correspondence with simple sequences of blowing-ups of points whose first center is p: (2.1) Here simple means that we only blow-up points p i , i > 1, belonging to the exceptional divisor created last. The cluster of centers of π will be denoted by C ν = {p = p 1 , p 2 , . . .} and for i > j we say that a point p i is proximate to p j , written p i → p j , whenever p i belongs to the strict transform of the exceptional divisor E j obtained by blowing-up p j . For all i ≥ 1 we denote by E i the exceptional divisor on X i obtained by blowing-up p i , and we say that p i and E i are satellite if p i is proximate to two points of C ν ; otherwise we say that they are free.
The above mentioned valuations (considered up to equivalence) were classified in five types by Spivakovsky [27]. We are interested in one of them, namely the exceptional curve valuations-in the terminology of [11]-, which correspond to Case 3 in [27] and to those of type C of [6]. Divisorial valuations are another remarkable type in Spivakovsky's classification. They correspond to finite sequences π and will be merely instrumental in our development.
More specifically, ν is divisorial if it is defined by the order of vanishing along an exceptional divisor E n obtained from a finite simple sequence of blowing-ups π. In this case C ν = {p i } n i=1 , G ∼ = Z and, when its value group G is included in R, there exists 0 = c ∈ R such that ν(f ) = c · ord En (f ) for all f ∈ F * . Divisorial valuations associated with the same divisor E n but with different constants c are said to be equivalent [28]. Throughout this paper, the representatives we choose for divisorial valuations will correspond to c = 1.
A valuation ν is exceptional curve if C ν is infinite and there exists a point p r ∈ C ν such that p i → p r for all i > r. In this case the group G is non-archimedian, so G ∼ = Z 2 (with lexicographical ordering).
2.1. Dual graphs and Enriques diagrams of divisorial and exceptional curve valuations. For any non-empty finite set W of effective divisors on a nonsingular surface, we define its dual graph Γ W as the graph whose set of vertices is in 1-1 correspondence with W, and two vertices are joined by an edge if and only if the associated divisors intersect. The dual graph Γ ν of a divisorial valuation ν, with associated configuration C ν = {p i } n i=1 , is the dual graph of the set of divisors {E 1 , E 2 , . . . , E n } together with a label i attached to the vertex corresponding to E i ; the symbols E i stand for the strict transforms of the divisors E i on X n ; this dual notation (for divisors and their strict transforms) will be used sometimes throughout this paper. The graph Γ ν is a rooted tree whose root is the vertex that is labeled with 1.
In Figure 1 we have depicted the dual graph of a divisorial valuation ν. Here we have shown only some labels. The vertices different from 1 which are adjacent to a unique vertex are called dead ends (labeled as ℓ 1 , ℓ 2 , . . . , ℓ g in Figure 1), and those adjacent to three vertices are called star vertices. We have labeled the star vertices with st 1 , st 2 , . . . , st g . If the divisor defining ν is free, then there is a finite sequence of vertices corresponding to free divisors which appears after st g (the tail, in Figure 1) and the last vertex is a dead end. Otherwise this tail does not appear and st g is only adjacent to two vertices.       will be called the number of Puiseux pairs of ν in the sequel. Moreover, we will say that an exceptional divisor E i (or the point p i ) belongs to the jth Puiseux pair of ν if its associated vertex is a vertex of Γ j ν . The vertices of each subgraph Γ j ν corresponding to free divisors are some of the first consecutive ones (with respect to the ordering ) and the one labeled by ℓ j ; we call them (respectively, the exceptional divisors E i and points p i associated to them) the free part of Γ j ν (respectively, the free part of the jth Puiseux pair of ν). Notice that, for 1 ≤ j < g, the exceptional divisors E st j belong to two consecutive Puiseux pairs and they are satellite.
The Enriques diagram of the configuration C ν (or of ν) [9, IV.I], denoted by E ν , is a rooted tree (with root O = p 1 ) that has a vertex for each point in C ν and an edge joining each pair of vertices that represent consecutive points p i and p i+1 . These edges are of two different kinds, straight or curved, according to the following rules: ⋄ If p i+1 is free then the edge joining p i and p i+1 is smooth, curved and such that, if p i = O, it has at p i the same tangent as the edge ending at p i . ⋄ Assume that the vertices p i and p i+1 (and the edge joining them) have been represented. Then all the infinitely near to p i+1 points which are proximate to p i , as well as the edges joining them, appear on a straight half-line which starts at p i+1 and is orthogonal to the edge p i p i+1 at p i+1 . To avoid self-intersections of the diagram, such half-lines are alternately oriented to the right and to the left of the preceding edge. Figure 2 shows the Enriques diagram of a divisorial valuation. We have labeled the vertices associated with the points p st j and p ℓ j for all 0 ≤ j ≤ g. It consists of the concatenation of g subgraphs corresponding to the points in the g Puiseux pairs of ν (see Figure 3) and, if the valuation is defined by a free divisor (as in Figure 2), a sequence of vertices corresponding to free points which appear after p stg (the tail).
For an exceptional curve valuation ν, we consider the sequence of dual graphs (respectively, Enriques diagrams) , where ν i is the divisorial valuation defined by the exceptional divisor E i .
Let r be the index such that p i → p r for all i > r. Then, for all i ≥ r + 1, Γ ν i+1 is obtained from Γ ν i by the following procedure (see Sections 5 and 9 of [27]): (1) delete the edge joining the vertices of the divisors E i and E r , (2) add  (3) add two edges joining the new vertex with those corresponding with E i and E r . Also, for all i ≥ r + 1, the graph E ν i+1 is obtained from E ν i adding a new vertex (the one corresponding with p i+1 ) and an edge p i p i+1 orthogonal to the edge p r p r+1 . We define the dual graph (respectively, the Enriques diagram) of ν, denoted also by Γ ν (respectively, E ν ), as the (infinite) graph whose set of vertices is the union of the sets of vertices of Γ ν i (respectively, E ν i ) for all i ≥ 1, with the same labels, and such that two vertices are connected by an edge in Γ ν (respectively, E ν ) if and only if they are connected by an edge in Γ ν i (respectively, E ν i ) for all i sufficiently large.
Notice that E ν is connected. However Γ ν has two connected components and exactly one of them has infinitely many vertices. In Figure 4 we have depicted the dual graph of an exceptional curve valuation ν. If g stands for the maximum of the number of Puiseux pairs of the divisorial valuations ν i , i ≥ 1, then for all j such that 1 ≤ j ≤ g − 1 we denote by Γ j ν the subgraph of Γ ν such that Γ j ν = Γ j ν i for all i large enough. Also, we denote by Γ g ν the subgraph of Γ ν whose set of vertices is the union of the sets of vertices of Γ g ν i for i large enough and whose edges are those of Γ ν joining the mentioned vertices. Notice that all subgraphs Γ j ν are finite except the last one, Γ g ν . As in the case of divisorial valuations, we will say that an exceptional divisor E i (or the point p i ) belongs to the jth Puiseux pair of ν if its associated vertex is a vertex of Γ j ν .

2.2.
Sequence of values. Let ν be a divisorial or exceptional curve valuation of F centered at R, let C ν = {p i } i≥1 be its cluster of centers and, for each i ≥ 1, let m i be the maximal ideal of the local ring Infinitely many horizontal or vertical vertices ℓ g Γ g ν Figure 4. Dual graph of an exceptional curve valuation In the case of exceptional curve valuations, when p i → p r for all i > r, the equality ν(m r ) = p j →pr ν(m j ) (which involves an infinite sum) must be understood as ν(m r ) > n · ν(m r+1 ) for any positive integer n. As a consequence, considering the value group included in Z 2 (with lexicographical ordering), it holds that ν(m r ) = (a, b) and ν(m i ) = (0, c) for all i > r, where a, b, c are integers, a and c being strictly positive. In fact, any choice of values a, b, c with these conditions determines an exceptional curve valuation ω equivalent to ν such that C ω = C ν . Definition 2.1. An i-curvette is an analytically irreducible element of R, which gives rise to a germ of curve whose strict transform on X i is non-singular and meets transversally the divisor E i at a general point.
Throughout the paper we will use the symbol ϕ i to denote an i-curvette.
Notice that, in the divisorial case, the proximity equalities (2.2) imply that ν(m i ) = mult p i (ϕ n ) for all i ≥ 1, where E n is the divisor defining ν and, for all f ∈ R \ {0}, mult p i (f ) denotes the multiplicity of the strict transform in R i of the germ of curve with equation f = 0.

2.3.
Noether formula. Let ν be a divisorial or exceptional curve valuation. For every f ∈ R \ {0}, the value ν(f ) can be computed using the so-called Noether formula [3,Theorem 8.1.6]: A straightforward consequence of the above formula is that, when ν is divisorial, ν(f ) coincides with the intersection multiplicity at p of the germs of curve with equations f = 0 and ϕ n = 0, where E n is the divisor defining ν.

Remark 2.2.
Assume that ν is exceptional curve and let C ν = {p i } i≥1 and r be as before. Notice that, due to (2.2) and the Noether formula, the valuation ν is determined by the points p 1 , p 2 , . . . , p r+1 as well as by the values ν(m r ) and ν(m r+1 ). Moreover, for any linear automorphism ψ : R 2 → R 2 , the composition ψ •ν defines a plane valuation that is equivalent to ν (in the sense of [28, page 33]) and whose value group is ψ(Z 2 ) (with the order relation that is induced, through ψ, by the lexicographical ordering in Z 2 ). Furthermore, each plane valuation equivalent to ν with value group contained in R 2 arises in this way [28, page 49].
Consider an index 1 ≤ j ≤ g; then the continued fraction expansion of the Puiseux exponent β ′ j (ν) determines (and is determined by) the Enriques diagram of the jth Puiseux pair of ν. Indeed, writing [a 0 , a 1 , . . . , a s ] := a 0 + 1 are the number of consecutive points p k in the jth Puiseux pair with the same value ν(m k ). Moreover β ′ g+1 (ν) is the number of vertices in the tail of ν plus 1. This allows us to get the Enriques diagram of ν from its Puiseux exponents. Conversely, an analogous reasoning yields that the continued fraction expansion of the values β ′ j (ν) of a divisorial valuation ν can be computed from its Enriques diagram E ν . In fact, for all k > 1, it holds that ν(m k ) = ν(m k−1 ) if and only if the edge p k p k+1 (if any) is orthogonal to the edge p k−1 p k .
To sum up, the Enriques diagram E ν is a union of maximal smooth paths such that there is a one-to-one correspondence between the set of (different) values in {ν(m i )} and this set of smooth paths (see the forthcoming Example 2.4). Moreover, for each j = 1, 2, . . . , g (respectively, g + 1), β ′ j (ν) has a continued fraction expansion [m 0 , m 1 , . . . , m t ], where t+1 is the number of maximal smooth paths in the subgraph of E ν whose vertices correspond to divisors E i belonging to the jth Puiseux pair of ν (respectively, 1). Let us denote the number t by σ j (ν).

2.5.
Intersection numbers between curvettes. In this section we provide formulae to compute the intersection multiplicity at p of two curvettes in terms of the maximal contact values of the divisorial valuations of their defining exceptional divisors E i . This result will be essential in the proof of Theorem 4.2. Fix a divisorial valuation ν, consider its cluster of centers C ν = {p i } n i=1 and the dual graph Γ ν . For each index k ∈ {1, 2, . . . , n} we denote by ρ(k) the maximum natural number m such that the strict transform of ϕ k passes through all points of C ν in the mth Puiseux pair of ν. Now we state the above mentioned result on intersection multiplicity: Proposition 2.5. Let i, j be two indices in {1, 2, . . . , n} such that i < j, and let ν i and ν j be the divisorial valuations defined by the divisors E i and E j . Then: Proof. This result, except for the last part of (b), can be deduced from [5, page 362] adapting the notation to our purposes.
For the remaining part notice that, by ( and this is the case if and only if j i (with equality when ρ(i) < ρ(j)). Indeed, this follows after considering the construction of the dual graph Γ ν from the Enriques diagram of ν (see [3,Proposition 4.2.2] and the paragraphs before Example 2.4), and using the continued fraction expansions of the Puiseux exponents (see Section 2.4) and well-known properties of continued fractions.
3. The Newton-Okounkov body of an exceptional curve valuation 3.1. Newton-Okounkov bodies. General setting. Let X be a smooth irreducible projective variety of dimension n over the complex numbers C and let K(X) be its function field. Consider a flag of subvarieties, i.e., a sequence of subvarieties of X It is a discrete valuation of rank n, and any valuation of maximal rank comes from a flag [4, Theorem 2.9].
Definition 3.1. Given a flag Y • and a big divisor D on X, the Newton-Okounkov body of D with respect to Y • (or ν Y• ) is defined to be the following subset of R n : where { · } stands for the closed convex hull of the set { · }.
Newton-Okounkov bodies are convex and compact sets with nonempty interior. In [21] it is proved that ∆ Y• (D) is a polygon if X is a surface, and that in higher dimensions it can be non-polyhedral. Moreover, where vol R n means Euclidean volume and vol X (D) := lim m→∞ h 0 (X, O X (mD)) m n /n! .

3.2.
Exceptional curve valuations as flag valuations. From now on in this paper, we will consider any surface X = X r defined by a finite sequence of blowups of points as in (2.1) and its corresponding divisorial valuation ν r . Our goal is to study the Newton-Okounkov body of the flag This flag determines a cluster of centers of blowing-ups i=1 are given directly by the flag and the remaining points satisfy p i → p r for all i > r. It is well known that C ν defines an exceptional curve valuation ν (up to equivalence of valuations). Also, when q is not free, we set η for the index such that η = r and p r+1 ∈ E η .
As already mentioned, the flag E • defines a flag valuation ν E• such that, for , where π r : X r → X 0 is the composition of the first r point blowing-ups with centers in C ν , z r = 0 a local equation for E r and π * (f )/z where, as above, (· , ·) q stands for the intersection multiplicity at q.
Proposition 3.2. Under the above notation it holds that, for all f ∈ R \ {0}, where ν η denotes the divisorial valuation defined by E η .
As a consequence of the above proposition and the Noether formula (2.3), one can deduce the following: Corollary 3.3. Let ν be an exceptional curve valuation and let E • be the flag defined by C ν . Denote also by ν ′ the unique valuation equivalent to ν whose value group is Z 2 lex and such that ν ′ (m r ) = (1, 0) and ν ′ (m r+1 ) = (0, 1). Then ν ′ = ν E• . In spite of their importance, very few explicit examples of Newton-Okounkov bodies can be found in the literature. We are interested in an explicit computation of the Newton-Okounkov bodies ∆ ν E• (H) = ∆ ν E• of flags E • as before, with respect to the divisor H given by the pull-back of the line-bundle O P 2 (1). Therefore the rest of the section is devoted to state and prove results which provide an explicit description of these bodies. Corollary 3.3 allows us to associate them to arbitrary exceptional curve valuations. In fact, we can give a slightly more general definition: Definition 3.4. The Newton-Okounkov body of an exceptional curve valuation ν that takes values in R 2 is defined as If ν(m r ) = (1, 0) and ν(m r+1 ) = (0, 1) we get the equality ∆ ν = ∆ ν E• by Corollary 3.3, i.e., ∆ ν is the Newton-Okounkov body of the flag E • with respect to the line bundle H. Moreover notice that, by Remark 2.2, if ν ′ is a valuation equivalent to ν with value group in R 2 , then there exists a linear automorphism ψ : Therefore, from now on, unless otherwise be stated, we will assume that every considered exceptional curve valuation ν satisfies these conditions: its value group is Z 2 lex , ν(m r ) = (1, 0) and ν(m r+1 ) = (0, 1). Newton-Okounkov bodies of exceptional curve valuations ν with only one Puiseux pair and whose last divisor E r is satellite were studied in [4].
3.3. The Newton-Okounkov body of a minimal exceptional curve valuation. Let (X : Y : Z) be projective coordinates in P 2 and assume that p = (1 : 0 : 0). Consider affine coordinates u = Y /X and v = Z/X around p and let ν be an exceptional curve valuation centered at R = O P 2 ,p . We will keep this framework for all valuations that we will consider from now on.
Let C ν = {p 1 = p, p 2 , . . .} be the cluster of centers of ν and, keeping the notation of the preceding sections, let p r be the point such that p i → p r for all i > r.
An analogue of the Seshadri constant for valuations and line bundles on normal projective varieties was introduced in [1]. For the divisorial valuation ν r and the line bundle H, this analogue can be defined aŝ It is known thatμ(ν r ) ≥ 1/vol(ν r ) (see [1]). The divisorial valuation ν r is called to be minimal ifμ(ν r ) = 1/vol(ν r ) (see [8] and [14] for further information about minimal valuations).
Definition 3.5. An exceptional curve valuation ν is called to be minimal whenever its first component ν r is minimal.
Let us denote by S ν the semigroup of values of ν, that is, endowed with the lexicographical ordering. Let C(ν) be the convex cone of R 2 spanned by S ν and H(ν) be the half-plane {(x, y) ∈ R 2 | x ≤μ(ν r )}. The following result yields a description of C(ν) ∩ H(ν). Let g be the number of Puiseux pairs of the divisorial valuation ν r ; notice that ν has g (respectively, g + 1) Puiseux pairs whenever q = p r+1 is satellite (respectively, free).
We finish this section describing the Newton-Okounkov bodies of minimal exceptional curve valuations ν and proving that this description characterizes this class of valuations. We will need a previous lemma, which is straightforward.
(respectively,μ(ν r ) 2 / 2β g+1 (ν r )) if q = p r+1 is satellite and q ∈ E r ∩ E η (respectively, q is free). The Newton-Okounkov body ∆ ν is contained in C(ν) ∩ H(ν) by Lemma 3.7. The next lemma shows thatμ(ν r ) 2 / 2β g+1 (ν r ) is also the area of C(ν) ∩ H(ν) in the satellite case and this concludes the proof because this area is larger than or equal to 1/2 (the area of ∆ ν by (3.1)), being equal to 1/2 exactly when ν r is minimal. Lemma 3.9. Keeping the above notation, if the point p r+1 is satellite, then Proof. With the notation of Subsection 3.2, consider the divisorial valuations ν r and ν η . We will prove that which allows us to conclude the result. Indeed, when e g−1 (ν η )β g (ν r ) − e g−1 (ν r )β g (ν η ) = 1, it holds that .
This proves that An analogous reasoning shows the case when We finish proving the equality (3.2). By (2.4) we get where i is either r or η. From the definition of the valuesβ j (ν i ), we get that As a consequence,β With the help of the Enriques diagram of ν r and the explanations given in the paragraphs before Example 2.4, it is easily checked that, with the notations in those paragraphs, σ g (ν r ) = σ g (ν η ) + 1. This concludes the proof because of the equality which can be deduced from [23, Theorem 7.5] and the equality above.
3.4. The Newton-Okounkov body of a non-minimal exceptional curve valuation. We have just found the vertices of the Newton-Okounkov body of a minimal exceptional curve valuation. This subsection is devoted to the description of the non-minimal case. Let us start by proving the existence of supraminimal curves for divisorial valuations. This fact was proved in [8] for the case when ν r has only one Puiseux pair and it is defined by a satellite divisor. Then

Moreover, if ν is not a minimal valuation then there exists such an irreducible polynomial f and it is the unique irreducible polynomial (up to product by a nonzero constant) satisfying the above condition.
Proof. To show the first part of the statement, suppose the existence of the mentioned polynomial f . Then it is enough to prove the following assertion: and therefore ν(h) > β g+1 (ν) deg(h). By the assertion, h = f a h 1 , where a > 0 and h 1 ∈ C[u, v] such that f does not divide h 1 . Then which, again by the assertion would imply that f divides h 1 , a contradiction. Now we are going to prove the assertion. Let k ∈ N be any common multiple of ν(h) andβ g+1 (ν). Unloading procedure (see [3]) shows that the ideal P k := {ϕ ∈ R | ν(ϕ) ≥ k} is equal to where m i = ν(m i ) for all i, and E * i stands for the total transform on X of the divisor E i . Moreover, since the divisor − r i=1 m i E * i is nef, there exists a non-empty Zariski subset U of P k such that mult p i (ϕ) = k β g+1 (ν) m i , for all i = 1, 2, . . . , r and for all ϕ ∈ U (see page 60 and Corollary 4.2.4 of [3]). By [3, Lemma 7.2.1] we can also assume that (ϕ, f ) p = w for all ϕ ∈ U, where w := min{(ψ, f ) p | ψ ∈ P k }. Therefore Hence and we conclude that f is a component of h.
Finally, assume that ν is not minimal. Then, there exists a polynomial h ∈ C[u, v] such that ν(h) > β g+1 deg(h) and, therefore, the fact that the valuation of a product is the sum of valuations of the factors shows that, at least an irreducible component f of h, satisfies ν(f ) > β g+1 deg(f ). This proves the existence of the irreducible polynomial satisfying the condition given in the statement. Its uniqueness is a straightforward consequence of the assertion at the beginning of the proof. Definition 3.11. Let ν be a divisorial valuation that is not minimal. Then a curve C defined by an irreducible polynomial f ∈ C [u, v] such that ν(f )/ deg(f ) = µ(ν) is called supraminimal. Notice that we have just proved that this curve is unique. Next, preserving the notation of Subsection 3.2, we will describe the Newton-Okounkov bodies of the flags E • (i.e., of exceptional curve valuations ν). (a) If q = p r+1 ∈ E r ∩ E η is a satellite point, then Therefore ∆ ν is either a triangle or a quadrilateral.
Proof. For a start, we notice that the existence of the supraminimal curve C is a consequence of Lemma 3.10 and therefore Q 3 ∈ ∆ ν . By Theorem 6.4 of [22] and preserving notation, it holds that where µ := sup{s > 0 | H − sE r is big} and, for all t ∈ [0, µ], it is α(t) := ord p r+1 (N t | Er ), β(t) := α(t) + P t · E r ; here P t + N t stands for the Zariski decomposition of the Q-divisor H − tE r . Notice that µ =μ(ν r ) because the big cone and the cone of curves of X have the same closure. Set Indeed, on the one hand, if D is the strict transform on X r of a curve of P 2 defined by a polynomial h ∈ C[u, v] then where the second equality is a consequence of the Noether formula (2.3); on the other hand P t 0 · E i = 0 for all i = 1, 2, . . . , r − 1 by the proximity equalities (2.2). Now, when the point p r+1 is satellite (respectively, free), it holds that and ). Therefore, we have proved that Q 1 , Q 2 ∈ ∆ ν . This concludes the proof since the area of the convex hull of To conclude this section, for every exceptional curve valuation ν we determine an equivalent valuation ν ′ whose Newton-Okounkov body has a side on the Xaxis. First, we will need the following result.
The following result describes the Newton-Okounkov bodies of valuations ν ′ as in the previous lemma, showing that they have a side on the X-axis.
Theorem 3.14. Let E • a flag as above and let ν = ν E• be its attached exceptional curve valuation. Let ∆ ν ′ be the Newton-Okounkov body of the valuation ν ′ equivalent to ν defined in Lemma 3.13.
⋄ If ν is minimal, then ∆ ν ′ is a triangle with vertices (0, 0), where g * stands for the number of Puiseux pairs of ν. ⋄ Otherwise (ν is not minimal), ∆ ν ′ is the convex hull of the following points (0, 0), , 0 , and g * is the number of Puiseux pairs of ν.
Remark 3.15. The Newton-Okounkov body ∆ ν ′ for ν ′ as above when ν has only one Puiseux pair and p r+1 is satellite was described in [4, Corollary 5.8] by a very different procedure.

Characterization of triangular Newton-Okounkov bodies of exceptional curve valuations
Keep the above notation and recall that the Newton-Okounkov body ∆ ν of an exceptional curve valuation ν is either a quadrilateral or a triangle. We have seen that ∆ ν is a triangle when ν is minimal. In this section, we assume that ν is not minimal and we characterize the cases when ∆ ν is also a triangle.
For a start, let Γ ν be the (infinite) dual graph of ν (see Subsection 2.1) and let C be the supraminimal curve of ν r . Define Γ * ν as the graph obtained from Γ ν by adding a new vertex (associated to C) which is joined (by an edge) with those vertices of Γ ν given by divisors E i such that, for all k large enough, the strict transforms of E i and C in the surface X k have non-empty intersection. Proof.
be the cluster of centers of h, that is, the set of infinitely near points such that q 1 = p and, for each i ≥ 2, q i is the intersection point between the exceptional divisor E i−1 and the strict transform of the germ of curve defined by h. Then C ν ∩ C h = {p 1 , p 2 , . . . , p i 0 }. The proximity equalities for germs of curves [3, Theorem 3.5.3] yield that mult q j (h) = q k mult q k (h) = 0, where the summation runs over all points q k which are proximate to q j .
For each j ∈ {1, 2, . . . , i 0 } we define e j := mult p j (h) − p i mult p i (h), where p i runs over the set of points p i ∈ C ν ∩ C h which are proximate to p j . Notice that e j ≥ 0 for all j = 1, 2, . . . , i 0 .
Let P be the complete ideal of R given by the stalk at p of the sheaf where π : X i 0 → X 0 is the composition of the blowing-ups at C ν ∩ C h . By Zariski's theory of complete ideals of a 2-dimensional regular local ring, the ideal P decomposes as a product of simple complete ideals in the form . . .
where P j is the simple complete ideal of R defined by the divisor E j . Notice that ν(h) coincides with the valuation of a general element of P. Since ϕ j is a general element of P j for all j, it holds that where Ω := {j ∈ {1, 2, . . . , i 0 } | e j > 0}. Notice that i 0 ∈ Ω. We will see that the cardinality |Ω| of Ω is at most 2 and that, when |Ω| = 2, the associated exceptional divisors correspond to vertices of Γ ν lying on the same connected component. Write q := q i 0 +1 . If q is free it is clear that Ω = {i 0 }; therefore we assume that q is satellite and consider the divisorial valuation ν i 0 +1 . It is sufficient to prove that either Ω = {i 0 }, or Ω = {i 0 , j 0 } with i 0 , j 0 i 0 + 1 or i 0 , j 0 i 0 + 1 in the dual graph Γ ν i 0 +1 .
LetẼ ν i 0 +1 be the Enriques diagram obtained from E ν i 0 +1 by adding a new vertex (associated with the point q) which is joined (by an edge) with p i 0 (straight or curved according with the definition of Enriques diagram).
Suppose that p i 0 is free, then the unique possibility forẼ ν i 0 +1 is the one depicted in Figure 6 (notice that we are assuming that q is satellite).
Otherwise, p i 0 is satellite and then the unique possibilities forẼ ν i 0 +1 are those depicted in Figure 7, and those obtained replacing the edge with origin at p s by a curved edge and/or the edge p i 0 p i 0 +1 by a curved edge that is tangent to The proximity equalities for h in each case prove that |Ω| ≤ 2. Moreover, when Ω = {i 0 , j 0 } (j 0 = i 0 ), the position of the vertices associated with the divisors E i 0 and E j 0 in the dual graph Γ ν i 0 +1 shows that either i 0 , j 0 i 0 + 1 or i 0 , j 0 i 0 + 1. This proves the lemma.
First suppose that p r+1 is satellite. By Lemma 3.7, the point Q 1 (respectively, Q 2 ) belongs to the line passing through the origin and with slopeβ 0 (ν η )/β 0 (ν r ) (respectively,β g * (ν η )/β g * (ν r )), or viceversa; hence ∆ ν is a triangle if and only if i.e., if and only if, there is an index j ∈ {0, g * } such that, for each analytic branch of C at p (say, with equation h = 0), υ(h)/ν r (h) =β j (ν η )/β j (ν r ), where ν(h) = (ν r (h), υ(h)). Let us prove that this last condition holds if and only if all the edges of Γ * ν that are incident with C meet Γ ν at the same connected component (that is, Γ * ν is not connected). Let us denote by Γ ν (r) the connected component of Γ ν that contains the vertex r (which is finite), and by Γ ′ ν (r) the other one. Fix a sufficiently large natural number k > r and consider the dual graph Γ ν k of the divisorial valuation ν k . Let h = 0 be the equation of an analytic branch of C at p and assume that its strict transform in X k meets the strict transform of the exceptional divisor E i . Notice that, by Lemma 4.1, we can assume without lost of generality that h = ϕ i .
Assume that r η in Γ ν k (the reasoning is similar otherwise). This means that the part of the graph Γ ν corresponding to the two last Puiseux pairs has the shape that is shown in Figure 8. Figure 8. Dual graph of Γ ν if r η in Γ ν k (proof of Theorem 4.2)

Infinitely many vertices
We will distinguish two cases: Here it must be i ≤ r. The Enriques diagram E ν (labeled with the sequence of values of ν) and the Noether formula (2.3) prove that ν(ϕ i ) = (ν r (ϕ i ), ν η (ϕ i )). Moreover ρ(i) < g * . Then, we consider two subcases: Case 1.1 : p i is free and i < ℓ ρ(i)+1 which implies that i ≤ η. Applying Part (a) of Proposition 2.5, it holds that where d is the length of the path [st ρ(i) + 1, i] in Γ ν k and the last equality holds because, as ρ(i) < g * , one gets the following chain of equalities Case 1.2 : Either p i is satellite or i = ℓ ρ(i)+1 (notice that the last condition cannot happen if ρ(i) = g * − 1 because r ℓ g * in Γ ν k ). By Part (b) of Proposition 2.5 we have that As a consequence, we have deduced, for this Case 1, that the slope of the line of R 2 joining the origin and ν(ϕ i ) isβ 0 (ν η )/β 0 (ν r ). So, it only remains to study the following case.
To conclude the proof, assume that p r+1 is free. Then, the result can be proved using a similar reasoning as above after taking into account the part (b) in Theorem 3.12 and the fact that ν(ϕ i ) = (mult pr (ϕ i )β g * (ν r ), m), where m = max{mult pr (ϕ i ), 0}.
An immediate consequence of the above result is the following.

Newton-Okounkov bodies of non-positive at infinite valuations
Keeping the notations of the preceding sections, set (X : Y : Z) projective coordinates in P 2 , assume that the coordinates of the point p are (1 : 0 : 0) and take affine coordinates u = Y /X and v = Z/X around p. Also, consider coordinates x = X/Z and y = Y /Z in the affine chart defined by Z = 0. Then a divisorial valuation as above (say ν r ) is called to be non-positive at infinity whenever r ≥ 2, the line L with equation Z = 0 passes through p 1 = p and p 2 , and ν r (h) ≤ 0 for all h ∈ C[x, y] \ {0}.
Definition 5.1. A exceptional curve valuation ν is named non-positive at infinity when its attached divisorial valuation ν r is non-positive at infinity.
When ν r is non-positive at infinity, it holds thatμ(ν r ) = ν r (v) and the line Z = 0 is a supraminimal curve [13,Theorem 1]. This result in [13] also provides an easy way to decide whether ν r is non-positive at infinity. In fact, ν r is nonpositive at infinity if and only if ν r (v) 2 ≥β g+1 (ν r ). In view of Theorem 4.2 and applying Theorem 3.12, we deduce the following result, which gives explicitly the vertices of the Newton-Okounkov body of this large family of exceptional curve valuations.
Corollary 5.2. Let E • := {X = X r ⊃ E r ⊃ {q}} be a flag whose divisor E r defines a non-positive at infinity valuation. Then the Newton-Okounkov body of the exceptional curve valuation ν = ν E• is a triangle whose vertices are (0, 0), Q 2 and Q 3 , where the latter points have the following coordinates: (a) If ν is not minimal: and Q 3 = (ν r (v), 0) , whenever q = p r+1 is a free point in E r .
whenever q is a satellite point in E η ∩ E r and η r.
whenever q is a free point in E r . otherwise.
Remark 5.3. Corollary 5.2 was announced in [15]. We have deduced it from Theorem 3.12. However, in [15] we deduced the same result computing directly the Zariski decompositions of the R-divisors D t := H − tE r for all t ∈ [0, ν r (v)] and applying then Theorem 6.4 of [22]. For the sake of completeness we show now the decompositions D t = P t + N t , where P t is the positive part and N t the negative part. Let us begin with non-minimal exceptional curve valuations which are nonpositive at infinity. This means that the valuation ν r satisfies the inequality ν r (v) 2 ≥β g+1 (ν r ) (see [13]).
(b) If t ≥β g+1 (ν r )/ν r (v), then P t = b r (t)D r + c(t)(H − L * ) and N t = a 0 (t)L + where, as above, L * means total transform on X and L stands for the strict transform on X of the line L given by Z = 0, and Finally assume that ν is a minimal exceptional curve valuation which is nonpositive at infinity, i.e., ν r (v) 2 =β g+1 (ν r ). Then, for real numbers t in [0, ν r (v)], it holds

Examples
We finish this paper using our results to compute several examples of Newton-Okounkov bodies of exceptional curve valuations. Example 1. Let ν r be a divisorial valuation centered at the local ring of a point p of P 2 whose sequence of maximal contact values is {β j (ν r )} 3 j=0 = {9, 30, 101, 303}. Let C νr = {p i } 12 i=1 (with p = p 1 ) be its cluster of centers and let L be the projective line passing through p and whose strict transform passes through p 2 . By means of an affine change of coordinates, if necessary, we can assume that L is defined, in projective coordinates (X : Y : Z), by the equation Z = 0. Moreover, assume that L does not pass through p 3 . Then, in the notation of Section 5 we have that ν r (v) = 18 and ν r (v) 2 = 324 > 303 =β 3 (ν r ). Thus ν r is non-positive at infinity by [13,Theorem 1]. Let ν = ν E• be the valuation defined by the flag E • = {X = X 12 ⊃ E 12 ⊃ {q = p 13 }}, where q ∈ E 10 ∩ E 12 . The Enriques diagram E ν of ν is displayed in Figure 9.