Newton–Okounkov bodies sprouting on the valuative tree

Given a smooth projective algebraic surface X, a point O∈X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O\in X$$\end{document} and a big divisor D on X, we consider the set of all Newton–Okounkov bodies of D with respect to valuations of the field of rational functions of X centred at O, or, equivalently, with respect to a flag (E, p) which is infinitely nearO, in the sense that there is a sequence of blowups X′→X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X' \rightarrow X$$\end{document}, mapping the smooth, irreducible rational curve E⊂X′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E\subset X'$$\end{document} to O. The main objective of this paper is to start a systematic study of the variation of these infinitesimal Newton–Okounkov bodies as (E, p) varies, focusing on the case X=P2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X=\mathbb {P}^2$$\end{document}.


Introduction
The concept of Newton-Okounkov bodies originates in Okounkov's work [25]. Relying on earlier work of Newton and Khovanskii, Okounkov associates convex bodies to ample line bundles on homogeneous spaces from a representation-theoretic point of view. In the generality we know them today, Newton-Okounkov bodies have been introduced by Lazarsfeld-Mustaţȃ [24] and Kaveh-Khovanskii [18].
Given an irreducible normal projective variety X of dimension r defined over an algebraically closed field K of characteristic 0, a big divisor D and a maximal rank valuation v on the function field K (X ) [or, equivalently, an admissible/good flag of subvarieties on some proper birational model of X (see Sect. 2.1)], a convex body v (D) is attached to these data which encodes in its convex geometric structure the asymptotic vanishing behaviour of the linear systems |d D| for d 0 with respect to v. Newton-Okounkov bodies contain a lot of information: from a conceptual point of view, they serve as a set of 'universal numerical invariants' according to a result of Jow [17]. From a more practical angle, they reveal information about the structure of the Mori Cone of X or of its blowups, about positivity properties of divisors (ampleness, nefness, and the like, see for instance Theorem 2.22, Remark 2.23 and [20]), and invariants like the volume or Seshadri constants (see [20,21]).
Not surprisingly, the determination of Newton-Okounkov bodies is extremely complicated in dimensions three and above. They can be non-polyhedral even if D is ample and X is a Mori dream space (see [22]). We point out that the shape of v (D) depends on the choice of v to a large extent: an adequate choice of a valuation can guarantee a more regular Newton-Okounkov body [1]. The case of surfaces, though not easy at all, is reasonably more tractable: the Newton-Okounkov bodies are polygons with rational slopes, and they can be computed in terms of Zariski decompositions (see Sect. 2.3).
In this paper we are mainly interested in infinitesimal Newton-Okounkov bodies, which arise from valuations determined by flags (E, p), with p ∈ E, which are infinitely near a point of the surface X , i.e. there is a birational morphism X → X mapping the smooth, irreducible rational curve E ⊂ X to O. These Newton-Okounkov bodies have already been studied in [20,21], and their consideration is implicit in [11]. Here we intend to connect the discussion in [11] to infinitesimal Newton-Okounkov bodies.
One of the main underlying ideas of [11] is to study the invariant μ (see Sect. 3.4 for the definition), which is roughly speaking an asymptotic multiplicity for quasi-monomial valuations. As such, it can be interpreted as a function on the topological space QM, the valuative tree of quasi-monomial valuations. Spaces of valuations were introduced by Zariski, and the topology we are interested in was originally considered in the celebrated work of Berkovich [4], see also [12]. The tree QM is rooted, and the root corresponds to the multiplicity valuation centred at O, with infinite maximal arcs homeomorphic to [1, ∞) starting from the root, and arcs sprouting from vertices corresponding to integer points (see [12] and Remark 3.6). The function μ is continuous along the arcs of QM. Interestingly enough, infinitesimal Newton-Okounkov bodies can be interpreted as 2-dimensional counterparts of μ.
Here we will focus on the case X = P 2 ; the same questions on other surfaces (general surfaces of degree d in P 3 for instance) are likely to be equally interesting, but we do not treat them in this work in the hope that we will come back to them in the future. A basic property of μ, pointed out in [11], is that μ(s) √ s assuming that s ∈ [1, +∞) is an appropriately chosen parameter on an arc of QM. Furthermore, equality holds unless there is a good geometric reason for the contrary, in the form of a curve C s on X s (for X s the appropriate minimal blow-up of X where the related flag shows up) such that the corresponding valuation takes a value higher than deg(C s ) · √ s. Such a curve is called supraminimal (see Sect. 5.3). Supraminimal curves are geometrically very particular, and give information on the Mori cone of X s ; for instance, it is conjectured in [11] (see also Conjecture 3.13 below) that along sufficiently general arcs of QM, all supraminimal curves are (−1)-curves. If so, μ(s) = √ s for every s 8 + 1/36, which among others implies Nagata's celebrated conjecture claiming that the inverse of the t-point Seshadri constant of P 2 equals √ t for t 9. In this paper we associate a Newton-Okounkov body to each point of the valuative tree and investigate how they change along the arcs of QM. We start by taking a quasimonomial valuation v(C, s) ∈ QM, where C is a curve defining an arc in QM and s ∈ [1, ∞) defines a point on the arc, and associating with it a rank 2 valuation. We take v = (v 1 , v 2 ), where v 1 = v(C, s) and v 2 is the right (or left) derivative of v(C, s) with respect to s. The process is described in Sect. 3.3 (precisely in Proposition 3.10). We obtain two valuations: v + (C, s) and v − (C, s), by taking respectively the right or the left derivative. Once we fix D to be a line in the plane the two valuations lead to the Newton-Okounkov bodies C,s + and C,s − . We focus on describing the properties of C,s + , however the cases are conceptually very similar and many results are obtained simultaneously for both.
The study of the variation of Newton-Okounkov bodies is a natural extension of the study of μ. In particular the projection of C,s + to the first axis is [0, μ(C, s)]. Moreover, the convex geometric behavior of C,s + is essentially simple (it is a certain triangle) whenever μ has the expected value (see Sect. 5.1). Otherwise C,s + exhibits more complicated features. The interesting phenomenon which we study is that, while μ is continuous on QM, the corresponding Newton-Okounkov bodies are not. We explain the concept of continuity and discontinuity (i.e. mutation) of Newton-Okounkov bodies in Sect. 5.3 (in particular Definition 5.16).
Taking into account the relation between Newton-Okounkov bodies and variation of Zariski decompositions (see Theorem 2.17 and [3, Theorem 1]), some discontinuity phenomenon is not unexpected, related to non-differentiability of Zariski decompositions in the big cone (Remark 2.20). We would also like to point out a plausible alternative explanation, provided by higher rank nonarchimedean analytifications, which we however do not use at all in this work. Just as QM parameterizes quasimonomial valuations (of rank 1), topological spaces that parameterize valuations of arbitrary rank have been introduced in the literature, starting with the Zariski Riemann space [30,VI,Sect. 17] (whose topology is however unrelated to the Berkovich topology of QM and so not suitable for our purposes) and most recently and notably the Huber analytification [16] and the Hahn analytification [13] of P 2 (which admit continuous maps to the Berkovich analytification that contains QM). Assigning the rank 2 valuation v + (C, s) to the point v(C, s) ∈ QM determines a map from the tree of quasimonomial valuations to the higher rank analytification, but this map turns out to be nowhere continuous [8]. From this point of view, discontinuities are unsurprising; what is remarkable is the piecewise continuity described in Sects. 5.3 and 5.4.
The main object of our interest is the study of mutations when the valuations move away from the root of QM along a fairly general route, and the results we have been able to obtain are collected in Sect. 5. Our results are partial in the sense that there are intervals in which we have been unable to obtain the appropriate information about mutations occurring there. Our manuscript is far from conclusive, it is simply devoted to lay the ground for future research on the subject.
The paper is organized as follows. In Sect. 2 we collect some basic definitions and results about valuations and Newton-Okounkov bodies, which we recall here to make the paper as self contained as possible. In Sect. 3 we focus on the two dimensional case, and specifically on quasi-monomial valuations, their interpretation in terms of the classical Newton-Puiseux algorithm, and the related clusters of centres. In this section (precisely in Remark 3.6) we briefly recall the structure of the valuation tree QM. In Sect. 5 we provide our computations about the infinitesimal Newton-Okounkov bodies.
In what follows we will mainly work over the field of complex numbers.

Preliminaries
Newton-Okounkov bodies in the projective geometric setting have been treated in [24], hence this is the source we will primarily follow. Let X be an irreducible normal projective variety of dimension r defined over an algebraically closed field K of characteristic 0 (we will usually have the case K = C in mind), and let D be a big Cartier divisor (or line bundle; we may abuse terminology and identify the two concepts) on X .
Although one first introduces Newton-Okounkov bodies for Cartier divisors, the notion is numerical, even better, it extends to big classes in N 1 (X ) R (see [24,Proposition 4.1]). Newton-Okounkov bodies are defined with respect to a rank r valuation of the field of rational functions K (X ) of X . We refer to [30,Chapter VI and Appendix 5] and [9,Chapter 8] for the general theory of valuations.

Basics on valuations
where G is an ordered abelian group satisfying the following properties: G is called the value group of the valuation. Two valuations v, v with value groups G, G respectively are said to be equivalent if there is an isomorphism ι : The subring

Definition 2.2
The rank of a valuation v is the minimal non-negative integer r such that the value group is isomorphic to an ordered subgroup of R r lex (i.e. R r with the lexicographic order). One can then write with v i : K (X ) * → R for every integer i with 1 i r .
For every integer i with 1 i r , the i-th truncation of v is the rank i valuation The trivial valuation, defined as v( f ) = 0 for all f = 0, has rank zero; it can be considered as the 0-th truncation of all valuations v.

Remark 2.3
The rank of every valuation on K (X ) is bounded by r = dim(X ), and every valuation of maximal rank is discrete, i.e., it has a value group isomorphic to Z r lex ⊂ R r lex [30,VI,Sects. 10 and 14]. Whenever v is a valuation of maximal rank, one may assume that the value group of v equals Z r lex up to equivalence under the action of some order-preserving (i.e., lower-triangular) element of GL(r, R).

Remark 2.4
The rational rank of the valuation v is the dimension of the Q-vector space G ⊗ Z Q, where G is the value group of v; it is well known that the rational rank is bounded below by the rank of v, and above by the dimension of X (see [30, VI, Sect. 10, p. 50]). A valuation v can be of rank 1 and rational rank r > 1. The standard example is in [30, VI, Sect. 14, Example 1, p. 100] (see also Remark 3.3 below).
By [30,VI,Sect. 10,Theorem 15], the rank of a valuation v equals the Krull dimension of its valuation ring R v . More precisely, the ideals in R v are totally ordered by inclusion, and if the rank is r , then the prime ideals of R v are The valuation rings of the truncations satisfy reverse inclusions By the valuative criterion of properness [15,II,4.7], since X is projective, there is a (unique) morphism The image in X of the closed point of Spec(R v ) (or the irreducible subvariety which is its closure) is called the centre of v in X , and we denote it by centre X (v). When the variety X is understood, we shall write centre and centre(v| i ) = σ X,v (p i ). Note that some of the inclusions may be equalities. For a valuation of rank r > 1, the centre of the first truncation v| 1 is called the home of v, following [6].
Example 2.5 (Divisorial valuations) If centre(v) is a divisor V , then v is equivalent to the valuation that assigns to each rational function its order of vanishing along V . Moreover, the residue field K v is the function field of V (see [30,VI,Sect. 14]).

Remark 2.6
Let L be a line bundle on X . On an affine neighborhood U of the centre of v (considered as a schematic point; equivalently, an affine neighborhood of the generic point of centre(v)) L is trivial, and so any section s ∈ H 0 (X, L) − {0} restricted to U can be seen as a non-zero element f ∈ K (X ), and one can set v(s) = v( f ). A different choice of U would give an element in K (X ) differing by a factor of value 0, so v(s) is well defined. By setting v(D) = v(s) whenever D = (s), s ∈ H 0 (X, O X (D)), valuations can be considered to assume values on divisors; effective divisors take nonnegative values.
is called admissible, if codim X (Y i ) = i for all 0 i dim(X ) = r , and Y i is normal and smooth at the point Y r , for all 0 i r − 1. The flag is called good if Y i is smooth for all i = 0, . . . , r .
Let φ ∈ K (X ) be a non-zero rational function, and set One verifies that ν Y • is a valuation of maximal rank, and that the flag (1) given by the centres of the truncations of ν Y • coincides with the flag Y • in (2).

Proposition 2.8
Let v be a valuation of maximal rank r = dim(X ) whose flag of centres Y • in (1) is admissible. Then v is equivalent to the flag valuation ν Y • .
Proof By induction on r . For r = 0 there is nothing to prove, so assume r 1. Remark 2.5 tells us that v| 1 = ν Y • | 1 (up to equivalence), and that their common residue field is K ( there is an f ∈ K (X ) sitting in R v| 1 whose class modulo m v| 1 isf . Then one setsv(f ) = v( f ) (similarly forν Y • ) and verifies that this is well defined. The value group ofv is the subgroup of the value group of v determined by v 1 = 0 (the "maximal isolated subgroup" in the language of [30, VI, Sect. 10]) and so it has rank r − 1 (maximal for valuations of K (Y 1 )); it is easy to see that its flag of centres isȲ • , withȲ i = Y i+1 for i = 0, . . . , r − 1. Butν Y • has maximal rank r − 1 and flag of centres isȲ • as well, so by inductionv andν Y • are equivalent.
Finally, the valuation ring of v (resp. of Sincev andν Y • are equivalent, then the valuation rings of v and ν Y • are the same, as claimed. Valuations of maximal rank are very well known (see [30,VI,Sect. 14], [29,Examples 5 and 6]) and Theorem 2.9 below is presumably obvious for experts working in the area of resolution of singularities. We include a proof as we lack a precise reference for it. For the case of surfaces, see Sect. 4 below. Theorem 2.9 Let X be a normal projective variety, and v a valuation of the field K (X ) of maximal rank r = dim(X ). There exist a proper birational morphism π :X → X and a good flag Proof Denote by ζ ∈ X the generic point, and set K = K (ζ ) = K (X ). Let 0 = p 0 ⊂ p 1 ⊂ · · · ⊂ p r be the maximal chain of prime ideals in R v , and choose f 1 , . . . , f r ∈ R v ⊂ K so that each f i ∈ p i \p i−1 . Fix projective coordinates [x 0 : . . . : x r ] in P r K ⊂ P r K × X , and let ξ = [1 : f 1 : . . . : f r ] ∈ P r K . Let X 0 be the Zariski closure of ξ in P r K × X . Since its generic point is ξ (which is a closed K-point in P r K ), it has residue field equal to K, and the induced projective morphism X 0 → X is birational.
For i = 1, . . . , r , the restriction of the rational function x i /x 0 to X 0 is f i , which has positive v-value. Therefore the centre of v in X 0 lies in [1 : 0 : . . . : 0] × X , and its local ring O centre X 0 (v) contains f 1 , . . . , f r . Hence as schematic points in X 0 , it follows that the centres of the truncations of v are all distinct. Since there are as many truncations as the dimension of X , the flag (1) in X 1 is a full flag, i.e., dim(centre X 0 (v| i )) = r − i for i = 0, . . . , r . Every birational model of X dominating X 0 will again have this property.
The flag of centres of the truncations in X 0 is usually not good (or even admissible), as X 0 is not necessarily smooth (not even normal) at centre X 0 (v). Using Hironaka's resolution of singularities we know that there is a birational morphism X 1 → X 0 , obtained as a composition of blowups along smooth centres, with X 1 a smooth projective variety. On X 1 we have a full flag like (1) whose codimension 1 term, centre X 1 (v| 1 ), may be singular. But again there is a composition of blowups along smooth centres (contained in centre X 1 (v| 1 )) that desingularizes it; we apply these blowups to X 1 , to obtain X 2 → X 1 . Since the blowup of a smooth variety along a smooth centre is again smooth, X 2 stays smooth, and the divisorial part of the full flag (1) in X 2 is now also smooth. By resolving sequencially the singularities of centre X 2 (v| 2 ), …, centre X r−1 (v| r −1 ) we arrive at a modelX = X r where the flag is good. Now by Proposition 2.8, the valuation v is equivalent to ν Y • as claimed.
Remark 2. 10 We work here in characteristic 0, but a suitable (weaker) version of Theorem 2.9 still holds in any characteristic. The same proof works, by replacing Hironaka's resolution with a sequence of blowups along nonsingular centres given by Urabe's resolution of maximal rank valuations [28]. The members of the resulting flag are not necessarily smooth, but they are non-singular at the centre.
In the situation of Theorem 2.9, we call Y • the good flag associated to v in the modelX . The choice of a flag is not unique, but for two models, the induced rational map between them maps the associated flags into one another.

Newton-Okounkov bodies
Definition 2.11 Let X be an irreducible normal projective variety, D a big divisor on X , and v a valuation of K (X ) of maximal rank r = dim(X ). Define the Newton-Okounkov body of D with respect to v as follows The

Remark 2.12
The properties of valuations yield that if A, B are two distinct valuative points, then any rational point on the segment joining A and B is again a valuative point. This implies that valuative points are dense in v (D) (see [19,Corollary 2.10], for the surface case; the proof is analogous in general). Therefore in (3) it suffices to take the closure in the Euclidean topology of R r .
Alternatively, one defines the Newton-Okounkov body of D with respect to v as where ≡ is the Q-linear equivalence relation. By [24,Proposition 4.1], one can replace Qlinear equivalence by numerical equivalence. Hence, one can define v (ζ ) for any numerical class ζ in the big cone Our definition differs from the one in [24] in that we use valuations of maximal rank instead of those defined by admissible flags on X . But, an admissible flag on X gives rise to a valuation of maximal rank on K (X ) by Example 2.7 (see also [18]). Conversely, by Theorem 2.9, any valuation of maximal rank arises from an admissible flag on a suitable proper birational model of X ; thus maximal rank valuations are the birational version of admissible flags. In conclusion, all known results for Newton-Okounkov bodies defined in terms of flag valuations carry over to Newton-Okounkov bodies in terms of valuations of maximal rank, modulo passing to some different birational model.
In [5,18] one considers Newton-Okounkov bodies defined by valuations of maximal rational rank, an even more general situation which we will not consider here.

Some properties of Newton-Okounkov bodies
A very important feature of Newton-Okounkov bodies is that they give rise to a 'categorification' of various asymptotic invariants associated to line bundles (see for instance [19,Theorem C] for the corresponding statement for moving Seshadri constants). Recall that the volume of a Cartier divisor D on an irreducible normal projective variety X of dimension r is defined as

Theorem 2.13 (Lazarsfeld-Mustaţȃ [24, Theorem 2.3]) Let X be an irreducible normal projective variety of dimension r , let D be a big divisor on X , and let v be a valuation of the field K
where the volume on the left-hand side denotes the Lebesgue measure in R r .
Remark 2.14 Although the proof of Theorem 2.3 from [24] takes the admissible flags viewpoint, the statement remains valid for Newton-Okounkov bodies defined in terms of valuations of maximal rational rank (with value group equal to Z r ) by the remark above (see also [5,Corollaire 3.9]).
Since the main focus of our work is on the surface case, we will concentrate on surfacespecific properties of Newton-Okounkov bodies.

Theorem 2.15 (Küronya-Lozovanu-MacLean [22]) If dim(X ) = 2, then every Newton-Okounkov body is a polygon.
If dim X = 2, then an admissible flag is given by a pair (C, x), where C is a curve, and x ∈ C a smooth point. If D is a big divisor on X , the corresponding Newton-Okounkov body will be denoted by (C,x) (D).

Remark 2.16
In fact one can say somewhat more about the convex geometry of Newton-Okounkov polygons, see [22,Proposition 2.2]. First, all the slopes of its edges are rational. Second, if one defines For any t ∈ [ν, μ], set D t = D − tC and let D t = P t + N t be the Zariski decomposition of D t . Consider the functions α, β : [ν, μ] → R + defined as follows

Remark 2.18
Note that all the results concerning Newton-Okounkov bodies use Zariski decomposition in Fujita's sense, i.e. for pseudo-effective R-divisors.
As an immediate consequence we have: 19 In the above setting the lengths of the vertical slices of (C,x) Remark 2.20 (See [22, proof of Proposition 2.2]) In the above setting, the function t → N t is nondecreasing on [ν, μ], i.e. N t 2 − N t 1 is effective whenever ν t 1 t 2 μ. This implies that a vertex (t, u) of (C,x) (D) may only occur for those t ∈ [ν, μ] where the ray D − tC crosses into a different Zariski chamber, in particular, where a new curve appears in N t .
Given three real numbers a > 0, b 0, c > 0, we will denote by a,b,c the triangle with vertices (0, 0), (a, 0) and (b, c). We set a,c := a,0,c and a,a := a . Note that the triangle a,b,c degenerates into a segment if c = 0.

Example 2.21
In the above setting suppose that D is an ample divisor. Then, by Theorem 2.17, the Newton-Okounkov body (C,x) (D) contains the triangle μ C (D),D·C , and by Theorem 2.13 one has Equality holds if and only if (C,x)

Remark 2.23
The divisor D is nef (resp. ample) if and only if Neg(D) = ∅ (resp. Null(D) = ∅), so that Theorem 2.22 provides nefness and ampleness criteria for D detected from Newton-Okounkov bodies.
Note that Theorem 2.22 has a version in higher dimension (see [20]). The same papers [19,20] explain how to read the moving Seshardi constant of D at a point x / ∈ Neg(D) from Newton-Okounkov bodies.

Quasimonomial valuations
We will mainly treat the case X = P 2 and D a line, leaving to the reader to make the obvious adaptations for other surfaces.
Let O denote the origin (0, 0) be the field of rational functions in two variables. We will focus on Newton-Okounkov bodies of D with respect to rank 2 valuations v = (v 1 , v 2 ) with centre at O, with the additional condition that either the home of v is a smooth curve through O, or it is equal to O (in which case we call the corresponding body an infinitesimal Newton-Okounkov body) and v 1 is a quasimonomial valuation.
Fix a smooth germ of curve C through O; we can assume without loss of generality that C is tangent to the line y = 0; hence C can be locally parameterized by where θ is transcendental over C. Equivalently, expand f as a Laurent series One has Then f → v 1 (C, s; f ) is a rank 1 valuation which we denote by v 1 (C, s). Such valuations are called monomial if C is the line y = 0 (i.e., ξ = 0), and quasimonomial in general. The point O is the centre of the valuation.
We call (5) the C-expansion of f . Slightly abusing language, s will be called the characteristic exponent of v 1 (ξ, s) (even if it is an integer).
So the rank of v 1 (C, s) is 1, but in the latter case the valuation has rational rank 2. We will be mostly concerned with the rational case. Note that v 1 (C, s) is discrete if and only if s is rational.

Remark 3.4
The valuation v 1 (C, s) depends only on the s -th jet of C, so for fixed s the series ξ can be assumed to be a polynomial; however, later on we shall let s vary for a fixed C, so we better keep ξ(x) a series.
Remark 3.6 (see [12]) The set QM of all quasi-monomial valuations with centre at O has a natural topology, namely the coarsest topology such that for all There is however a finer topology of interest on the valuative tree QM: the finest topology such that s → v 1 (C, s) is continuous in [1, +∞) for all C. This latter is called the strong topology. With the strong topology, QM is a profinite R-tree, rooted at the O-adic valuation (see [12] for details). To avoid confusion with branches of curves, we will call the branches in QM arcs. Maximal arcs of the valuative tree are homeomorphic to the interval The arcs of QM share the segments given by coincident jets, and separate at integer values of s; these correspond to divisorial valuations on an appropriate birational model. Though we will not use this fact, note that QM is a sub-tree of a larger R-tree V with the same root, called the valuation tree, which consists of all real valuations of K with centre O. Ramification on V occurs at all rational points of the arcs, rather than only at integer points, because of valuations corresponding to singular branches. The tree QM is obtained from V by removing the arcs corresponding to singular branches and all ends (see [12,Chapter 4] for details).

Quasimonomial valuations and the Newton-Puiseux algorithm
We recall briefly the Newton-Puiseux algorithm (see [9, Chapter 1] for a full discussion).
, and a curve C as in Sect. 3.1, we want to investigate the behavior of the function Returning to (5), consider the convex hull NP(C, f ) in R 2 (with (t, u) coordinates) of all points (i, j) + v ∈ R 2 such that a i j = 0, and v ∈ R 2 + . The boundary of NP(C, f ) consists of two half-lines parallel to the t and u axes, respectively, along with a polygon NP(C, f ), named the Newton polygon of f with respect to C.
We will denote by V(C, f ) (resp. by E(C, f )) the set of vertices (resp. of edges) of NP(C, f ), ordered from left to right, i.e., We will denote by V the germ of the curve f = 0. Then, such that whenever γ is a branch of V whose Puiseux expansion with respect to C starts as y − ξ(x) = ax τ + . . . , with τ ∈ Q and τ 1 (i.e., γ is not tangent to the x = 0 axis nor contained in C) then the edge l = ϕ V (γ ) has slope sl(l) = − 1 Consider now the line s with equation t + su = 0 and slope − 1 s . By (6), the valuation v 1 (C, s; f ) is computed by those vertices in V(C, f ) with the smallest distance to s , i.e. for any such vertex v = (i, j), one has v 1 (C, s; f ) = i + s j. Note that there will be only one such point, unless s is parallel to one of the edges l ∈ E(C, f ) (hence s is rational), in which case there will be two: the vertices of l, whose slope sl(l) = − 1 s . From the above discussion its not hard to deduce the following statement: Example 3.8 Let C be the conic x 2 − 2y = 0, so that ξ(x) = x 2 /2, and let The C-expansion of f is then The Newton polygon of f with respect to C is depicted in Fig. 1. It has three sides and four vertices, corresponding to the "monomials" (y − ξ(x)) 6 , x 2 (y − ξ(x)) 2 , x 4 (y − ξ(x)) and x 8 . The curve V : f (x, y) = 0 has four branches through O, all smooth; two of them are transverse to C and map to the first side of the Newton polygon; one of them is tangent to C with intersection multiplicity 2, and maps to the second side; the last one is tangent to C and has intersection multiplicity 4 with it, and maps to the third side.

Quasimonomial valuations and the associated rank 2 valuations
We keep the above notation. As we saw in Sect. 3.2, we have a finite sequence s 0 := 1 < s 1 < · · · < s h < s h+1 := +∞ such that v 1 (C, s; f ) is linear (hence differentiable) in each of the intervals (s k , s k+1 ), for k = 0, . . . , h. The derivative in these intervals is constant and integral. At s k , with k = 0, . . . , h + 1, there are the right and left derivatives of v 1 (C, s; f ) (at s 0 = 1 (resp. at s h+1 = +∞) there is only the right (resp. left) derivative). So we have:  [1, +∞)) left (resp. right) derivative. We will denote them by

Proposition 3.10
For any curve C smooth at O, every s ∈ Q, s > 1 and every f ∈ This defines two rank 2 valuations v − (C, s) and v + (C, s) with home at O. For s = 1, the valuation v + (C, s) defined as above is also a rank 2 valuation with home at O.
Proof Let f ∈ K [x, y] and let (x, ξ(x)) be a local parametrization of C. With notation as in (5), then (6) holds, thus The Obviously v − (C, s) and v + (C, s) have rank at most 2. We will show that they have rank greater than 1. Let f 0 ∈ K [x, y] be such that f 0 = 0 is an equation of C (this, for fixed s, is no restriction by Remark 3.4). We have v ± (C, s; f 0 ) = (s, ±1) (see Example 3.5). Moreover if s = p q for coprime positive integers p, q and f 1 = f q 0 x p then v ± (C, s; f 1 ) = (0, ±q). Thus for every positive integer k we have which is impossible for a rank 1 valuation.

Remark 3.11
For irrational s, the expressions v − and v + (as defined in Proposition 3.10), are valuations with home at O, but they are both equivalent to v 1 (and so have real rank 1 and rational rank 2). We will not need this fact, and we leave the proof to the interested reader.
In this case, we will denote by the Newton-Okounkov bodies associated to the line bundle O P 2 (1) with respect to the valuation v − (C, s) and v + (C, s) respectively.
Since v ± (C, s) have maximal rank but their value groups do not equal Z 2 lex , the volumes of Newton-Okounkov bodies associated to these valuations need not satisfy Theorem 2.13. However, there are order preserving elements of GL(2, Q) relating the v ± valuations to valuations with values in Z 2 lex . In Sect. 4.3 below we compute these lower triangular matrices, which turn out to have determinant 1, and so preserve the volume. Therefore Theorem 2.13 also applies to v ± (C, s), and vol C,

The µ invariant
Let v 1 be a rank 1 valuation centred at a smooth point x of a normal irreducible projective surface X , and let D be a big Cartier divisor on X . Following [11], we set is a valuation of rank 2 centred at x, then v (D) lies in the strip and its projection to the t-axis lies the interval [0, μ D (v 1 )], coinciding with it if and only if x / ∈ Neg(D) (see Theorem 2.22). In order to simplify notation, we will set μ D (C, s) = μ D (v 1 (C, s)), and μ D (C, s) = μ D (v 1 (C, s)).
If X = P 2 , x = O and D is a line, we drop the subscript D for μ D (C, s) and we write μ d (C, s) instead of μ d D (C, s) for any non-negative integer d.
From [11] we know that the function μ : QM → R is lower semicontinuous for the weak topology and continuous for the strong topology, i.e., μ(C, s) is continuous for s ∈ [1, +∞) (see [11,Proposition3.9]). Moreover μ(C, s) √ s [11]. If μ(C, s) = √ s, then v 1 (C, s) is said to be minimal (the concept of minimal valuation is more general, see [11], but we will not need it here). We recall from [11] the following: Remark 3.14 According to [11,Proposition 5.4], this Conjecture (actually a weaker form of it, considering only s 9 and C any curve), implies Nagata's Conjecture.
• If C is a line, then The values of μ above are computed using the series of Orevkov rational cuspidal curves (see [26] and Proposition 5.25 below). There are a few more sporadic values of s in the range [7 + 1/9, 9] where the value of μ is known, see [11] for details.
• If s is an integer square and C is a general curve of degree at least √ s, then one has μ(C, s) = √ s.

Cluster of centres and associated flags
In this section the main goal is to introduce the geometric structures related to valuations v 1 (C, s) and v ± (C, s). We give a full description of how to find the birational model of X (the cluster of centres together with their weights) on which these two valuations are equivalent to a flag valuation on this model.

Weighted cluster of centres
As usual, we will refer to the case Each valuation v with centre O ∈ P 2 determines a cluster of centres as follows. Let P 1 = centre X 0 (v) = O. Consider the blowup π 1 : X 1 → X 0 of P 1 and let E 1 ⊂ X 1 be the corresponding exceptional divisor. Then centre X 1 (v) may either be E 1 or a point P 2 ∈ E 1 . Iteratively blowing up the centres P 1 , P 2 , . . . of v we may end up, after k 1 steps, with a surface X k dominating P 2 , where the centre of v is the exceptional divisor E k . In this case v is discrete of rank 1, given by the order of vanishing along E k , by Remark 2.5. Otherwise, this process goes on indefinitely. In particular, for quasimonomial valuations v 1 (C, s), the process terminates if and only if the characteristic exponent s is rational.
Let v = (v 1 , v 2 ) be now a rank 2 valuation whose truncation v 1 is quasimonomial. From Abhyankar's inequalities, [12, p. 12], one concludes that v 1 has rational rank 1. Hence, by Remark 3.3, we have v 1 = v 1 (C, s) for some s ∈ Q. By the above then, the sequence of centres of v is infinite, whereas the sequence of 1-dimensional homes (centres of v 1 ) terminates at a blowup X k where centre X k (v 1 ) = E k is an exceptional divisor. In particular, v is equivalent to the valuation ν Y • , defined by the flag The punchline of all this is that the process of blowing up all 0-dimensional centres of the truncation provides an effective method to find a model where a given rank 2 valuation becomes a flag valuation. By Theorem 2.9, such a model exists for every valuation of maximal rank on a projective variety. The above method works for any valuation of rank 2 on any projective surface (i.e., not necessarily P 2 ).
For each centre P i of a valuation v, general curves on X i−1 through P i and smooth at P i have the same value e i = v(E i ), which we call the weight of P i for v. Following [9, Chapter 4], we call the (possibly infinite) sequence K v = (P e 1 1 , P e 2 2 , . . .) the weighted cluster of centres of v. In general a sequence like K = (P e 1 1 , P e 2 2 , . . .) is called a weighted cluster of points and supp(K) = (P 1 , P 2 , . . .) is called its support.
If v is a valuation with centre at O, then its weighted cluster of centres completely determines v. Indeed, for every effective divisor Z on P 2 , one has where Z i is the proper transform of Z on X i , whenever the sum on the right has finitely many non-zero terms. This is always the case unless v is a rank 2 valuation with home at a curve through O and Z contains this curve; in particular, for valuations of rank 1, such as v 1 (C, s), formula (9) always computes v(Z ) [9, Sect. 8.2]. As usual, with the above notation, we say that a curve Z passes through an infinitely near point P i ∈ X i if its proper transform Z i on X i contains P i .

The cluster associated to v 1 (C, s)
The description of the cluster K (C,s) := K v 1 (C,s) is classical and we refer for complete proofs to [9]. Here, we merely focus on the construction of the cluster of centres for v 1 (C, s) and its main properties that will be used in the next section. The cluster K (C,s) is a very specific one, and we will need the following definition to make things more clear. Definition 4.1 With notation as above, the centre P i ∈ X i−1 is called proximate to P j ∈ X j , for 1 j < i k, (and one writes P i P j ) if P i belongs to the proper transform E i−1, j on X i−1 of the exceptional divisor E j+1 := E j+1, j over P j ∈ X j−1 . For the cluster K (C,s) , each P i , with i 2, is proximate to P i−1 and to at most one other centre P j , with 1 j < i − 1; in this case P i = E i−1, j ∩ E i−1 and P i is called a satellite point. A point which is not satellite is called free.
We know that the support of the cluster K (C,s) = K v 1 (C,s) is determined by the continued fraction expansion s = p q = [n 1 ; n 2 , . . . , n r ] = n 1 + 1 where p, q are coprime and r ∈ Z >0 . Before moving forward, let's fix some notation. Let k i = n 1 + · · · + n i and k = k r . We denote by s i = p i q i = [n 1 ; n 2 , n 3 , . . . , n i ], for i = 1, . . . , r the partial fractions of s, where p i , q i are coprime positive integers. First, the cluster K (C,s) consists of k = n i centres (if s is irrational there are infinitely many centres). Set K = K (C,s) and for each i = 0, . . . , k − 1 let π i : X i+1 → X i be the blow-up of X i at the centre P i+1 with exceptional divisor E i+1 . As usual we start with X 0 := P 2 . Denote X K := X k and let π : X R → X be the composition of the k blowups.
With this in hand, we explain the algorithm for the construction of K. If s = n 1 (so that r = 1), then the centre P i+1 is the point of intersection of the proper transform of C through the map X i → X 0 and the exceptional divisor E i of π i−1 , for each i = 1, . . . , n 1 − 1. When r > 1, then the first n 1 + 1 (including P 1 ) centres of K are obtained as in the case when s was integral, i.e. these points are chosen to be free. The rest are satellites: starting from P n 1 +1 there are n 2 + 1 points proximate to P n 1 , i.e. each P j is the point of intersection of the proper transform of E n 1 and the exceptional divisor E j−1 . Thus, E n 1 plays the same role for these centres as C did in the first step. Then, one chooses n 3 + 1 points proximate to P n 1 +n 2 and so on. Since r < ∞, then the last n r points (not n r + 1) are proximate to P n 1 +···+n r−1 . The final space X K is where v 1 (C, s) becomes a divisorial valuation, defined by the order of vanishing along the exceptional divisor E k ⊆ X K . Finally note that C plays a role only in the choice of the first n 1 centres. This is due to Remark 3.4, saying that the valuation v 1 (C, s) depends only on the s -th jet of C.
The weights in K (C,s) are proportional to the multiplicities of the curve with Puiseux series y = ξ(x) + θ x s at the points of supp(K (C,s) ). These and the continued fraction expansion are computed as follows. Consider the euclidean divisions where m 0 := p, m 1 := q. Then the first n 1 points of K (C,s) have weight e 1 = e 2 = · · · = e n 1 = m 1 q = 1, the subsequent n 2 points have weight m 2 /q, . . ., the final n r points have weight m r /q = 1/q. Therefore the proximity equality holds for all j = 0, . . . , k − 1. Conversely, for every weighted cluster K with finite support, in which every point is infinitely near the previous one, no satellite point precedes a free point, and the proximity equality holds, there exist a smooth curve through O and a rational number s such that K = K (C,s) .

v ± (C, s) and the associated flag valuation
In order to describe the flag valuation associated to v ± (C, s), it is necessary to understand first the intersection theory of all the proper and total transforms of the exceptional divisors on X K . To ease notation, let A i (resp. B i ) be the proper (resp. total) transform of E i ⊂ X i on X K , for i ∈ {0, . . . , k − 1}. Then:

Lemma 4.2 (i) A k = E k is the only curve with A
The sheaf π * (I P 1 |P 2 ) is invertible on X K and defines the fundamental cycle E of π.
Then, making use of Lemma 4.2, the multiplicities a i can be easily computed as follows:  On the other hand, using the partial fractions s i = p i q i of s = p q , one has the same recursive relations q j = n j q j−1 + q j−2 for j = 2, . . . , r , with q 0 = 0 and q 1 = 1. Thus, we get that a k j = q j for any j = 0, . . . , r . In particular, we have a k r = q.
In the following the pair ( p r −1 , q r −1 ) of the partial fraction s r −1 = p r−1 q r−1 will play an important role, so we fix some notation. When s is not an integer (i.e. r 2), we set p = p r −1 , q = q r −1 so that s r −1 = p q If s is an integer, i.e., r = 1, then we set p = q = 1. In order to find the flags on X K associated to v ± (C, s), we need to have a better understanding of the cycle E through its dual graph. The dual graph of E is a chain, i.e. a tree with only two end points, corresponding to A 1 and A n 1 +1 . If A is the proper transform of C on X K , then A intersects E only at one point on A n 1 +1 . Thus the dual graph of A + E is also a chain, with end points corresponding to A 1 and A. The curve A k intersects exactly two other components of A + E, precisely: (a) if s is not an integer (so that r 2), then A k intersects A k−1 and A k r−1 , whose multiplicities in the cycle A + E are a k−1 = q − q and a k r−1 = q ; (b) if s is an integer (so that s = k = n 1 ), then A k intersects A k−1 and A, both having multiplicity one.
Note that A + E − A k has two connected components, only one containing A. We denote this component by A + and the other by A − . We will denote by x ± the intersection point of A k with A ± , and by x the general point of A k .
The total transform C * on X K of C has the same support as A + E, but the multiplicities are different. In particular, denoting Fig. 2 The Enriques diagram [9, 3.9] of the cluster of centres of Example 4.6. Each vertex in the diagram corresponds to one of the points, with each vertex joined to its immediate prececessor by an edge; edges are curved for free points, and straight segments for satellites, to represent the rigidity of their position. The segments joining a sequence of satellites proximate to the same point lie on the same line, orthogonal to the immediately preceding edge Lemma 4.5 (i) The divisor C * contains A k with multiplicity p and C * − p A k passes through x + (resp. x − ) with multiplicity p (resp. p − p ). (ii) The total transform L of the line x = 0 on X K contains A k with multiplicity q and L − q A k passes through x + (resp. x − ) with multiplicity q (resp. q − q ).
Proof We prove only (i), the proof of (ii) being analogous. When s is an integer the assertion is trivial. So, assume that s is not an integer (i.e. r 2). We first show that the multiplicity of A k in C * is equal to p. This is done inductively on k = n 1 + · · · + n r . From the standard properties of continuous fractions it is worth to note that the numerator of [n 1 ; n 2 , . . . , n r − 1] is equal to p − p r −1 , where p r −1 is the numerator of the continued fraction The multiplicity of A k in C * is the same as the multiplicity of A k in B 1 + · · · + B n 1 +1 . So, using Lemma 4.2 repeatedly, the statement follows easily.
The multipliticies of C * − p A k at x + and x − equal the multiplicities in C * of A k r−1 and of A k−1 respectively in this order if r is odd, and reversed if r is even (as r 2). Arguing as before, one deduces easily also these statements.
Clusters are often represented by means of Enriques diagrams (see [9, p. 98]) as explained in Fig. 2 illustrating this example.

Proposition 4.7
In the above setting, the flags associated to the rank 2 valuations v − (C, s) and v + (C, s) are Proof The above discussion makes it clear that A k is the centre of v 1 (C, s). It remains to prove that x ± are the centres of v ± (C, s). Let η = 0 be a local equation of A k on X K around x + . Consider f 1 = f q 0 /x p as in the proof of Proposition 3.10. By Lemma 4.5, the pull-back of f 1 to X K is not divisible by η. Again by Lemma 4.5, it vanishes at x + with multiplicity p . Furthermore, by Proposition 3.10, one has v + (C, s; f 1 ) > 0. By the same token, f −1 1 is not divisible by η, it vanishes at x − and has v − (C, s; f −1 1 ) > 0, proving the assertion.
by the proof of Proposition 3.10. By Lemma 4.5, one has By standard properties of continued fractions, one has pq − qp = (−1) r . Thus

Remark 4.9
The same relations, given in Remark 4.8, hold for the corresponding Newton-Okounkov bodies. It is worth to note that both 2 × 2 matrices transform vertical line into vertical lines. Furthermore, any vertical segment in Y ± (D) is translated into a vertical segment in C,s ± whose length is multiplied by a factor of q with respect to the initial one, where D is the class of a line.

Zariski decomposition of valuative divisors
In this subsection we will describe, with few details, some of the properties of the valuation v 1 that will be used in the next section. As before let s = p/q 1 be a rational number and K the cluster of centres associated to the rank 1 valuation v 1 (C, s), with π : X K → P 2 the sequence of blow-ups constructed in the previous section where the valuation v 1 (C, s) becomes equivalent to a valuation given by the order of vanishing along an exceptional curve on X K . We will denote by B s def = e 1 B 1 + · · · + e k B k , where as usual B i is the total transform of the exceptional divisor E i on X K and e i is the weight of the center P i , whose blow-up is the curve E i (whereas A i is the strict transform in X K of E i ). Note that the proximity equalities (10) mean that B s · A i = 0 for all 1 i k −1, and that the weights are also determined by these equalities and e k = 1/q (see section 8.2 in [9]). Knowing this divisor B s we usually know almost everything about the valuation v 1 . Using (9) one deduces the following: Lemma 4.10 For a divisor Z on X K not containing any of the exceptional curves A i , one has v 1 (C, s; π * (Z )) = B s · Z.
For the computation of Newton-Okounkov bodies, the following properties of B s will also be useful. Proof The proof of (i) is done inductively using the description of the cluster of centres obtained previously, and we leave the details to the reader. Let us prove (ii). If k = 1 then s = p = 1, B s = B k = B 1 and there is nothing to prove; so assume k > 1. Since the intersection matrix of the collection The Zariski decomposition of D x relative to π is D x = P π + N π (see [10,Sect. 8] for all m such that m D x is a Weil divisor, and the negative part of this relative Zariski decomposition is a part of the full Zariski decomposition: N π N . We claim that N π = x p B s − x B k ; to prove it, we need to show that x p B s − x B k is an effective divisor satisfying (a) and (b). A direct computation shows that the coefficient ν i of A i in x p B s −x B k is positive for i = 1, . . . , k −1 and zero for i = k, so it is an effective divisor. On the other hand,

Newton-Okounkov bodies on the tree QM
From now on we will mainly concentrate on the study of C,s + when s varies in [1, +∞).
The case of C,s − is not conceptually different and will be often left to the reader. Proof For the first inclusion, note that by the proof of Proposition 3.10 evaluating an equation of C and the variable x, forces both points (1, 0) and ( s d , ± 1 d ) to be contained in C,s ± . The origin is also contained in C,s ± since it is the valuation of any line not passing through the centre of the valuation. For the equality statement one uses that the area C,s ± is 1 2 , by Theorem 2.13 and Remark 3.12.

General facts
For the second inclusion notice first that by definition of μ(C, s) from Sect. 3.4 one has that the convex sets C,s ± sit to the left of the vertical line t = μ. To prove that C,s + also lies above the t-axis and below the line t = su, we need to show Assuming (5) holds, this follows from (6) and (8), as i + s j s j. The equality statement is again implied by the fact that the area of C,s + is equal to 1 2 . The analogous facts for C,s − are left to the reader. In particular, this implies Nagata's Conjecture and it shows how difficult it is to compute Newton-Okounkov bodies.
where D is the pull back to X d 2 of a line. Let Z := D − d A d 2 , which can be written Remark that this is actually the Zariski decomposition of Z , because A is nef, as A is irreducible and A 2 = 0, and d 2 −1 i=1 i d A i has clearly a negative definite intersection form. Also, Z sits on the boundary of the pseudo-effective cone, as A 2 = 0. Thus, Z − t A d 2 is not pseudo-effective for t > 0. Now the proof follows easily using Theorem 2.17. Alternatively, by Remark 5.4, it suffices to prove that μ(C, d 2 ) = d. Since v 1 = ord E d 2 as valuations (by Remark 4.8, noting that s is an integer) one gets from the above paragraph that μ(C, d 2 ) d. The opposite inequality follows from Lemma 5.1.

Corollary 5.6
Let C be a plane curve of degree d. For every > 0, there exists a non-zero f ∈ K [x, y] whose C-expansion f (x, y) = a i j x i (y − ξ(x)) j satisfies: By Corollary 5.1, there exists a real number λ > 0 such that This can be seen as an infinitesimal counter-part of Theorem 2.22 for X = P 2 . When s = 1 and X is any smooth projective surface, these ideas were also developed in [19] along with Theorem 2.22. The largest λ turned out to be the Seshadri constant of the divisor. This connection can be seen clearly in the following proposition, where the notation comes from Sect. 4.4. Proof Let's check first that ( s α , 0) ∈ C,s + . By Remark 4.8, this is equivalent to showing that ( p α , 0) ∈ Y + (D). Since α D − B s is nef, then there exists a sequence of effective ample divisors H n , n 1, where x + / ∈ Supp(H n ), so that D is the limit of 1 α B s + H n . So, the point By Remark 4.9, it remains to show that the height of the slice of Y + (D) with first coordinate t = p α is equal to 1 qα . For this, we apply Theorem 2.17 for t = p α . Let N t + P t be the Zariski decomposition of D − t B k . By Lemma 4.11, we know that Thus, one has P t D − (1/α)B s ; but the latter Q-divisor is nef by hypothesis, therefore Based on the previous statement, it is natural to introduce the following constant As mentioned before, when s = 1, the constant λ(C, s) is nothing else than the Seshadri constant of D, the class of a line, at the origin O. So, one expects λ(C, s) to encode plenty of geometry also for s > 1. Note that we have the inequalities where the left-hand side is an equality if and only if the right-hand side is also such. From Conjecture 3.13 this is expected to happen when s is large enough and C a sufficiently generic choice of a curve. Thus it becomes natural to ask about the shape of the convex set C,s + when μ(C, s) = λ(C, s) = √ s does not happen.
Corollary 5.8 Under the assumptions above, one of the following happens: Proof By definition of μ(C, s) and Lemma 4.10, for any effective divisor Z in X K , not containing in its support any of the exceptional curves A i , one has So, ( μ(C, s)D − B s ) · Z 0 for any such cycle Z . By Lemma 4.11, we already know that ( μ(C, s)D − B s ) · A i 0 for any i = 1, . . . , k. Thus, the divisor μ(C, s)D − B s is nef and by Proposition 5.7 we get that λ(C, s) s/ μ(C, s). When μ(C, s) = √ s we land in case (1). Otherwise, if ( μ(C, s), c) ∈ C,s + for some c 0, then this latter condition implies that C,s + contains the convex hull of the points , ( μ(C, s), c).
Note that such a c must exist, since the projection of C,s + to the first axis is [0, μ(C, s)] as noted before. Since the area of this convex hull is 1/2, it coincides with C,s + , and one has λ(C, s) = s/ μ(C, s). Remark 5.9 It is worth to note that Corollary 5.8 takes place only because our ambient space is P 2 , especially due to properties like the Picard group of P 2 is generated by a single class, whose associated line bundle is globally generated with self-intersection equal to 1. One does not expect these phenomena to happen when we consider the valuations v ± on any smooth projective surface X . But we do expect that some parts of the considerations about the infinitesimal picture developed in [19] to be true in this more general setup. For example, the constant λ(C, s) should in some ways encode many interesting local positivity properties of the divisor class we are studying, as partially seen in Proposition 5.7.
If C,s + is not continuous for some s 0 ∈ (1, d 2 ), then we say that C,s + presents a mutation at s 0 (or mutates at s 0 ). Also, C,s + depends linearly on s in an interval I ⊆ (1, d 2 ), if the number of proper vertices of C,s + is the same for all s ∈ I and the coordinates of the vertices of C,s + are affine functions of s in I . as a function of s ∈ [1, +∞), is a tropical polynomial (see Proposition 3.7), certainly there is some point σ ∈ (a, b) of discontinuity for its derivative. Moreover v 1 (C; f )(s) = deg( f )· √ s for s = a, b. The rest of (ii) follows from the discussion in §3.2.
To show (iii) note that the mutation of C,s + at the points s ∈ (a, b) as in (ii) depends on the discontinuity of ∂v 1 (C; f ) there (see Remark 5.18 Since μ(C, s) is a continuous function of s, it follows that for every > 0 there is δ > 0 such that for |s − s 0 | < δ, The mutations described in Theorem 5.20(iii), can be called supraminimality mutations.

Corollary 5.22
Any mutation of C,s + is supraminimal.
For general choices of O and C, all known supraminimal curves are (−1)-curves. It would be interesting to explore the behavior of C,s ± on surfaces different from P 2 , or for non-quasimonomial valuations (i.e., allowing singular C, and following arcs in the whole valuative tree V rather than only QM). It is tempting to conjecture that mutations in general should be supraminimal and related to extremal rays in (some) Mori cone.

Explicit computations
In this section we compute the Newton-Okounkov bodies in the range in which μ(C, s) is known (see [11] and Sect. 3.4).

The line case
This case is an immediate consequence of Theorem 5.11, which we state here for completeness. Proof The case s = 1 is trivial. So, assume first that s ∈ (1, 2) ∩ Q. By Remark 4.8, we know that (1, 0) ∈ C,s + . It remains to show that (s, 1) ∈ C,s + . Note that the only free points in the cluster K = K v + (C,s) are P 1 = O and P 2 . The line through P 1 and P 2 , i.e., the tangent line to C, has equation y = 0 and behaves exactly like C with respect to K. Thus, again by Remark 4.8, we have v + (C, s; y) deg(y) = v + (C, s; y) = (s, 1) , implying that (s, 1) ∈ C,s + . Take now s ∈ [2, 4) ∩ Q and use the notation of Sect. 4. By Remark 4.8, we have that ( s 2 , 1 2 ) ∈ C,s + , given by the valuation of a local equation of C. It remains to show that (2, 0) ∈ C,s + . Let L on X K be the total transform of the tangent line to C at O. Note that L − B 1 − B 2 does not contain any of the other exceptional curves. Thus, arguing as in Lemma 4.5(ii), it is easy to see that L contains A k with multipliticy 2q and L − 2q A k passes through x + with multiplicity 2q . Hence (2q, 2q ) ∈ Y + and, by Remark 4.8, this implies that (2, 0) ∈ C,s + .
Finally, by Theorem 5.11, the assertion holds for s 4.

The higher degree case
The case in which deg(C) 3 is more interesting, since it gives rise to infinitely many mutations of the Newton-Okounkov body. Recall the notation {F i } i∈Z 0 ∪{−1} for the sequence of the Fibonacci numbers, and φ for the golden ratio (see Sect. 3.4).

Proposition 5.25
For each odd integer i 5, there exists a rational curve C i ⊆ P 2 of degree F i with a single cuspidal (i.e., unibranch) singularity at O and characteristic exponent F i+2 F i−2 ∈ (6, 7), whose six free points infinitely near O are in general position. Let C 1 be a line (of degree F 1 with characteristic exponent F 3 F −1 = 2) and C 3 be a conic (of degree F 3 with characteristic exponent F 5 F 1 = 5). All these curves are (−1)-curves in their embedded resolution (i.e., after blowing up the appropriate points of the cluster of centres determined by the characteristic exponent).
If C is a general curve with deg(C) 3 through the origin O, then the curve C i , with equation f i = 0, through O and the first six infinitely near points to O along C, satisfies Thus C i is supraminimal for v 1 (C, s) for s ∈ hence C,s + mutates at s = 2, s = 5, and depends linearly on s between mutations; (ii) for s ∈ [6 + 1 4 , φ 4 ) one has i.e., C,s + mutates at F i+2 F i−2 , for all odd integers i 5 (these mutations agree with part (ii) of Theorem 5.20), and depends linearly on s between mutations; (iii) for s ∈ (φ 4 , 7), the Newton-Okounkov body is the quadrilateral with vertices First, assume that ( 3s 1+s , 3 1+s ) is valuative for some s < 7. This means that there is a polynomial f of degree d with v 1 (C, s; f ) = 3s 1 + s d, and ∂ + v 1 (C; f )(s) = 3 1 + s d.
In particular, this implies that But this contradicts that for s + < 7, the Newton-Okounkov body C,s + lies in the half-plane t + u 3. So, ( 3s 1+s , 3 1+s ) is a non-valuative vertex. Finally, ( 3s 8 , 3 8 ) is valuative because there is a unique curve V of degree 24 whose Newton polygon with respect to C has vertices (0, 9) and (64, 0). Indeed, let K be the cluster of centres of v 1 (C, 64 9 ), which consists of 8 free points followed by 8 satellites, each proximate to its predecessor and to P 7 (the continued fraction of 64 9 is [7; 9]). Then V has multiplicity 9 at each of P 1 , . . . , P 7 and multiplicity 1 at P 8 , . . . , P 16 .
The curve V has genus 1 and is obtained in this way. Consider the Cremona transformation ω determined by the homaloidal system of curves of degree 8 with triple points at a cluster C of seven general infinitely near, free, base points (this Cremona transformation appears in the construction of the curves C i in Proposition 5.25, see [26, proof of Theorem C]).
There is a unique cubic curve with a double point at the first point of C and passing simply through the remaining six points of C. This curve is contracted to a point by ω.
Let x ∈ be a general point. There is a pencil P of cubics having intersection multiplicity 8 with at x. Then P has 9 base points, 8 are given by the cluster formed by x and by the 7 points infinitely near to x along , and there is a further base point y ∈ . The general curve of P is irreducible, and its image via ω is the required curve V , which has genus 1.

Remark 5.29
It is somewhat mysterious that in the case (ii) of Corollary 5.28 one has a vertex of C,s + that is not valuative, taking into account that for s < 7, s ∈ Q, the Mori cone of X K is polyhedral (see [11]).