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Newton-Okounkov Bodies over Discrete Valuation Rings and Linear Systems on Graphs

The theory of Newton-Okounkov bodies attaches a convex body to a line bundle on a variety equipped with flag of subvarieties. This convex body encodes the asymptotic properties of sections of powers of the line bundle. In this paper, we study Newton-Okounkov bodies for schemes defined over discrete valuation rings. We give the basic properties and then focus on the case of toric schemes and semistable curves. We provide a description of the Newton-Okounkov bodies for semistable curves in terms of the Baker--Norine theory of linear systems on graphs, finding a connection with tropical geometry. We do this by introducing an intermediate object, the Newton-Okounkov linear system of a divisor on a curve. We prove that it is equal to the set of effective elements of the real Baker-Norine linear system of the specialization of that divisor on the dual graph of the curve. As a bonus, we obtain an asymptotic algebraic geometric description of the Baker-Norine linear system.


Introduction
Let O be a discrete valuation ring with fraction field K and residue field k. Let π be a uniformizer of O. For convenience, we shall call semistable any irreducible, regular scheme proper, flat, and of finite type over O whose generic fiber is smooth and whose closed fiber is a reduced normal crossings divisor. Let X be a projective regular semistable scheme over O with closed fiber X. Let Y • denote a descending flag of subschemes where each Y i is a codimension i subscheme that is either a semistable scheme over O or a smooth subvariety of a component of the closed fiber X × O k. Let D be a divisor on X that surjects to Spec O. Let j be the index such that Y j−1 is semistable over O and Y j is contained in the closed fiber. Then Y j is a component of Y j−1 × O k. We will give names to two special cases: when j = 1, we are said to be in the Arakelovian case; when j = d + 1, we are in the tropical case. These names are explained below.
In this paper we will define a Newton-Okounkov body, ∆ Y• (D) ⊂ R d+1 by considering sections of O(D) evaluated at the valuations attached to the flag. In contrast to the classical case, this Newton-Okounkov body will be unbounded, albeit in a single direction. This unboundedness is a consequence of the fact that if s ∈ H 0 (X , O(D)), then π k s ∈ H 0 (X , O(D)) for any k ≥ 0.
We will investigate these Newton-Okounkov bodies, determining their basic properties. We have the following result establishing their boundedness except in a single direction: Theorem 1.1. Let p π : R d+1 → R d be projection along the direction through the valuation vector of π. The image ∆ = p π (∆ Y• (O(D)) is compact.
The second author was supported by the European Commission, Seventh Framework Programme, Grant Agreement n • 600376.
We will give a description of ∆ Y• (D) in the tropical case. Recall that for ∆ ⊂ R d , a convex body and a convex function ψ : ∆ → R, we may define the upper hull in R d+1 to be the set {(x, t) | x ∈ ∆, t ≥ ψ(x)}.
If ψ is piecewise linear, its domains of linearity give the Newton subdivision of ∆ [5], an important notion of tropical geometry. A related construction occurs in the work of Yuan [14] and Witt-Nyström [13].
Theorem 1.2. In the tropical case, ∆ Y• (D) is given by the upper hull of a convex function ψ : p π (∆ Y• (D)) → R.
We will give a description of Newton-Okounkov bodies in the case of toric schemes with respect to a toric flag. Here, it is analogous to the field case and involves a particular polyhedron P D depending on D, and φ R , a certain linear map depending on Y • . We will consider the case of semistable families of curves where we can give a fairly complete description of the Newton-Okounkov bodies. In fact, they can be described in terms of the Baker-Norine theory of linear systems on graphs [2]. As an intermediate object for describing the Newton-Okounkov bodies, we introduce the Newton-Okounkov linear system, L + ∆ (D), which measures the vanishing orders on components on the special fiber of sections of O(mD) for m ∈ Z ≥1 . We prove that the Newton-Okounkov linear system is combinatorial by relating it to a combinatorially-defined effective linear system L + (ρ(D)) where ρ(D) is the combinatorial specialization of the horizontal divisor D on the family C : Theorem 1.4. If the generic fiber of D has positive degree, then we have the equality between the Newton-Okounkov linear system and the combinatorial effective linear system: ). After enriching the above theorem by considering vanishing orders of sections at a point in the closed fiber of C , we are able to give a description for the Newton-Okounkov bodies in the tropical and Arakelovian cases in Theorem 6.16 and Theorem 6.17, respectively.
1.1. Acknowledgments. We would like to thank David Anderson for enlightening conversations.

Notation and conventions
For a scheme X over O, we will use X K to denote its generic fiber. If C is a semistable curve over O, irreducible divisors are either horizontal or vertical. Here, horizontal divisors are those that that surject onto Spec O while vertical divisors are contained in the closed fiber. Any divisor on C can be decomposed into a sum of horizontal and vertical divisors.
Recall that a convex cone in R n is a convex set invariant under rescaling by elements of R ≥0 . For a set S ⊂ R n , we write cone R n+1 (S) for the minimal convex cone containing S.
We will write 0 for the empty divisor. For divisors D, E, we will write D ≤ E if E − D is effective. If D is a divisor on a smooth variety X, sections of O(D) can be interpreted in two ways: as a section of a line bundle or as a rational function s whose principal divisor satisfies (s) + D ≥ 0. Consequently if D ≤ E, a section of O(D) can be interpreted as a section of E although its zero locus will differ by E − D. We will point out the relevant interpretation by using the words "section" or "rational function."

Construction of the Newton-Okounkov body
Let us recall the construction of Newton-Okounkov bodies in the classical setting. Let X be a smooth irreducible projective variety of dimension n over a field K. Given a divisor D on X, we want to construct a convex compact subset of R n called the Newton-Okounkov body of D. A flag of smooth irreducible subvarieties Here, s 0 is considered as a section of We obtain the vector ν(s) = (ν 1 (s), . . . , ν n (s)) ∈ Z n . Define the semi-group of valuation vectors as and the Newton-Okounkov body of D as The same construction can be performed for non-complete graded linear series as well, see [9]. This follows from the Riemann-Roch theorem [12]. Here, the Okounkov body only captures numerical data.

3.2.
The case of surfaces. In the case of surfaces, the Zariski decomposition of big divisors can be used to show that the Newton-Okounkov body lies between the graphs of two functions on an interval [12]. This description provides all the information about possible shapes of Okounkov bodies of surfaces. Any pseudoeffective divisor D (that is, a divisor in the closure of the effective cone in the Neron-Severi group) can be written as D = P D + N D , where P D is nef, P D · N D = 0, and N D is effective with a negative definite intersection matrix.
Let us consider the rank two valuation induced by a general flag , obtaining the following body: Newton-Okounkov bodies in higher dimensions can be much more complicated and there is no general strategy for writing them down. We will see that for toric schemes with respect to a torus-invariant flag, the Okounkov body is determined by combinatorics.
4. Newton-Okounkov bodies over discrete valuation rings 4.1. Definition for schemes over discrete valuation rings. We now describe the case of Newton-Okounkov bodies on schemes over discrete valuation rings. Let O be a discrete valuation ring with fraction field K, residue field k, and valuation val. Let π be a uniformizer of O. Let X be an n-dimensional semistable scheme over O. We will write X = X × O K for the generic fiber of X . Let Y • denote a descending flag of proper subschemes where each Y i is a codimension i subscheme that is either a semistable scheme over O or a proper smooth subvariety of the closed fiber X × O k. Let D be a divisor on X flat over O.
Let j be the index such that Y j−1 is semistable over O and Y j is a closed subvariety of We will give names to special cases: when j = 1, we are said to be in the Arakelovian case; when j = d + 1, we are in the tropical case.
Here, the name "Arakelovian" is motivated by some constructions by Yuan [14] of which we describe the function field analogue. The name "tropical" is motivated by a fundamental notion in tropical geometry, the Newton subdivision [6] to which our construction specializes in the toric case. So in a certain sense, our work interpolates between tropical geometry and function field Arakelov theory. Our work is closely related to Okounkov bodies of filtered linear series as in the work of Boucksom-Chen [3].
The Newton-Okounkov body ∆ Y• (D) is defined as a convex body in R n+1 exactly as above by considering sections of O(mD) over X for m ∈ N evaluated at the valuations attached to the flag.

4.2.
Boundedness. In contrast to the field case, Newton-Okounkov bodies over discrete valuation rings are not bounded. However, the failure of boundedness can be precisely described.
The Newton-Okounkov body is bounded in other directions. Let p π : R n+1 → R n+1 /(Rν(π)) be the projection along the ν(π)-direction Proof. Choose an ample divisor H on X and an ample divisor h on Y j , which is a component of the projective variety Y j−1 × O k.
We will follow [12, Lemma 1.10], whose methods come from [4]. We begin with the following observation: give a line bundle L and a prime divisor Y on some scheme Z, there exists an integer b such that for any section s m of L ⊗m , the vanishing order of s m on Y is at most mb. Indeed, choose b sufficiently large such that (L − bY ) · H d−1 < 0. Multiplying this inequality by m, we see that mL − mbY cannot have any regular sections.
We claim that there exists positive integers b 1 , . . . , b j−1 such that for any s ∈ H 0 (X, O(mD K )), , restriction to the generic fiber gives s ∈ H 0 (X, O(mD K )). We choose b 1 as in the above paragraph. Then, v 2 is given by the vanishing order at Y 2 of the restriction of s to Y 1 considered as a section of Choose b 2 to be the max of the b's produced in the above paragraph for L 1,a for 0 ≤ a ≤ b 1 . We continue by defining b i 's inductively.
Now there are finitely many line bundles on Y j−1 whose sections we will restrict to Y j . We will take the maximum of the b's chosen for all line bundles. Suppose we have one line bundle on Y j−1 . We may restrict our attention to Y j−1 , and so it suffices to prove the result for j = 1. Let us consider elements of H 0 (X, O(D)). These are exactly rational sections of O(mD) over X that are allowed to have poles along components of the closed fiber. By multiplying such sections by a suitable power of π, we can ensure that the section is regular. Therefore, the Newton-Okounkov body associated to H 0 (X, O(D)) with respect to the flag Y 1 ... Y d is exactly the Minkowski sum of the Newton-Okounkov body of H 0 (X , O(D)) and the line Rν(π). Since we're only interested in the projection of that Newton-Okounkov body along the line through ν(π), it suffices to consider the complementary slice given by elements of H 0 (X, O(D)) that have neither poles nor zeroes generically along Y 1 . Now, the sections under consideration restrict to sections on Y 1 . We may now apply the above argument with H replaced by h to obtain b 1 , . . . , b d .
In the tropical case, fi Y d+1 is a smooth point of the central fiber, p π is the projection along the (n + 1)-st component.

4.3.
Newton subdivision in the tropical case. We now consider the tropical case where the admissible flag Y • is given by semistable schemes We have a surjection of Newton-Okounkov bodies, Moreover, ∆ Y• is given as the upper hull of a convex function Proof. Because any section s ∈ H 0 (X, O(D)) has some multiple by π satisfying π k s ∈ H 0 (X , O(D)), we have that the projection p π : is closed under positive translation by e n+1 = ν(π). From the convexity of Newton-Okounkov bodies, it follows that ∆ Y• (D) is the set of points in R n+1 lying above the graph of a convex function ψ : In circumstances where the function ψ is piecewise linear, we can consider the domains of linearity of ψ as giving a subdivision of ∆ Y• (D). This is of interest in tropical geometry [6].
We note that Newton-Okounkov bodies of schemes over discrete valuation rings need not be polyhedral. Indeed, one may take a semistable model of a variety with a flag over K that already has a non-polyhedral Newton-Okounkov body [11] and extend the flag to a tropical one.

Toric schemes
In this section, we discuss toric schemes over discrete valuation rings. See [10] for a classical source or [6] for a rigid analytic perspective.

Toric varieties.
We begin by reviewing Newton-Okounkov bodies for smooth projective toric varieties [12, Section 6.1]. Let N be an n-dimensional lattice. A toric variety X(∆) is specified by a complete rational fan ∆ in N R = N ⊗ R ∼ = R n . The variety X(∆) is smooth if and only if the fan is unimodular, that is, the fan is simplicial and every cone is spanned by integer vectors forming a subset of a basis of N. Let T = N ⊗ K * denote the n-dimensional algebraic torus acting on X(∆). To each k-dimensional cone σ of ∆, there corresponds an orbit closure V (σ) which is a codimension k subvariety. Any torus-invariant divisor is given by where the sum is over rays σ in ∆ and a σ ∈ Z. Attached to D is a polyhedron P D ⊂ M R where M is the dual lattice of N defined by Because X(∆) is smooth, a T -invariant flag Y 1 , Y 2 , . . . , Y n can be written as Y i = D 1 ∩ · · · ∩ D i for a choice of T -invariant divisors D 1 , . . . , D n corresponding to rays σ 1 , . . . , σ n . Let u 1 , . . . , u n be the primitive integer vectors along σ 1 , . . . , σ n . We define a linear map φ : M R → R n by φ(v) = v, u σ i + a σ i 1≤i≤n . We have the following equality for big line bundles O(D):

Toric schemes.
Complete toric schemes over a discrete valuation ring O are described by complete rational fans in N R ×R ≥0 where N ∼ = Z d is a lattice. Given such a fan Σ, there is a natural morphism of toric varieties X(Σ) Z → X(R ≥0 ) Z = A 1 Z and the toric scheme is given by X = X(Σ) × A 1 Spec(O). Here, we will map t, the coordinate on A 1 , to the uniformizer of O. We will suppose that Σ is a unimodular fan and therefore that the total space X is regular. If we set ∆ = Σ ∩ (N R × {0}), then the generic fiber of X is the toric variety X = X(∆). The closed fiber of X is a union of toric varieties described combinatorially by the polyhedral complex The components of the closed fiber are in bijective correspondence with the vertices of Σ 1 . We will suppose that the vertices of Σ 1 are at points of N × {1} which ensures that X has reduced closed fiber and, therefore, is a semistable family. Let T = N ⊗ K * denote the torus of X.
A T -invariant divisor D on X has many extensions D to X . In particular, we may write D = σ a σ V (σ) where a σ ∈ Z and V (σ) is the divisor of X corresponding to a ray σ of ∆. Any extension is of the form where a v ∈ Z and V (v) are the divisors on X corresponding to rays through the vertices of Σ 1 .
We construct the Newton-Okounkov body. Considering the total space X(Σ) as an (n+1)dimensional toric variety, we define a polyhedron P D ⊂ M R × R ≥0 by The second set of inequalities come from v ∈ Σ 1 corresponding to vertices (v, 1) ∈ Σ 1 . Note that this projects onto P D by (m, u) → m. We may define a piecewise-linear convex function on P D , where v is taken over vertices of Σ 1 . Then P D is the set of points lying above the graph of ψ. This function ψ (studied in [5]) is important in tropical geometry, and its domains of linearity give the Newton subdivision of P D . Now, we will explain how the polyhedron P D relates to the Newton-Okounkov body of X with respect to a flag Y • of torus-fixed subschemes. Following [12], we may choose toric prime divisors D 1 , . . . , D n+1 of X(Σ) such that Y i = D 1 ∩ · · · ∩ D i . Suppose that D i corresponds to a ray in N R × R whose primitive integer vector is w i ∈ N × Z. Here, where w runs over primitive integer vectors of rays σ w of Σ. We define where the pairing is between M R × R and N R × R. We have the following analogue of It follows that ν(s) ∈ φ R (P D ). By considering sections of the form π h χ m , we see that ∆ Y• (D) contains φ R (P D ).

Newton-Okounkov bodies of curves
We will relate the Newton-Okounkov bodies of curves over O to the Baker-Norine theory of linear systems on graphs. We will use R-divisors on graphs whose theory is combinatorially much less rich than that of integral divisors. 6.1. Review of linear systems on graphs. We review some results on specialization of linear systems from curves to graphs due to Baker [1]. Let C be a semistable curve over Spec O such that the generic fiber C K is smooth. The semistability condition ensures that the closed fiber C 0 is reduced with only ordinary double points as singularities. A node in the closed fiber of C is formally locally parameterized in C by O[x, y]/(xy − π). Definition 6.1. The dual graph Σ of a semistable curve C is a graph Σ whose vertices V (Σ) correspond to components of the normalization π : C 0 → C 0 , whose edges E(Σ) correspond to nodes of C 0 . To each vertex v is associated the component C v of C 0 .
Observe that each node of an irreducible component C v of C 0 corresponds to a loop at v. We will denote the edges of E(Σ) by e = vw even though Σ may not be a simple graph. Thus, when we sum over edges adjacent to v, we may need to sum over certain vertices more than once.
A divisor on Σ is an element of the free abelian group on V (Σ). We write a divisor as D = v∈V (Σ) a v (v). The group of all divisors is denoted by Div(Σ). We say a divisor D is The degree of a divisor is given by We will study functions ϕ : V (Σ) → R. The Laplacian of ϕ, ∆(ϕ) is the divisor on Σ given by Note that ∆(ϕ) is of degree 0.
The specialization map ρ : Div(C ) → Div(Σ) is defined by, for D ∈ Div(C ), For a divisor H on C K , we will write ρ(H) to mean the specialization of its closure in C .
Observe that for H, horizontal and effective, we have ρ(H) ≥ 0. Let D be a divisor on C . Then we will interpret a global section of O(D) as a rational function s on C such that (s) + D ≥ 0. If we write (s) = H + V where H is a horizontal divisor over O and V is a vertical divisor contained in the closed fiber, we may decompose V as where we call ϕ s : V (Σ) → Z the vanishing function of s. For s, a rational function on C, we will abuse notation and take the vanishing function of s to be vanishing function of the extension of s to C .
The following lemma is standard and we include the proof only for completeness. Proof. Because s is principal, we have (s) · C v = 0 for all components of the closed fiber. If we write (s) = H + v ϕ(v)C v , we have Since H + D ≥ 0 is horizontal, we have 0 ≤ ρ(H ) + ρ(D) = ∆(ϕ) + ρ(D).
The linear system L(∆) has a tropical semi-group structure as noted in [7]: Proof. Let v ∈ V (Σ). Suppose without loss of generality that ϕ(v) = ϕ 1 (v). Then We define the Newton-Okounkov linear system L + ∆ (D) to be the subset of L + (ρ(D)) given by the closure of the union of the convex hulls of the images of ̺ m for m ranging over Z ≥1 . We may extend this definition to horizontal Q-divisors by defining L + ∆ (D) to be 1 m L + ∆ (mD) where m is chosen arbitrarily divisible. Theorem 6.6. If the generic fiber of D has positive degree, then we have the equality between the Newton-Okounkov linear system and the combinatorial effective linear system: ). Before proving the proposition, we need a preparatory lemma adapted from [8]. Proof. It suffices to show that for any edge e = vw in Σ, |ϕ(w) − ϕ(v)| ≤ M(∆(ϕ)). Indeed, let v 0 , v 1 be the vertices where the minimum and maximum of ϕ are achieved, respectively. By picking a path from v 0 to v 1 of length at most diam(Σ) and comparing the values of ϕ along that path, we achieve the desired conclusion.
. Let O(Σ ≤t ) be the set of outgoing edges, that is, the edges e = vw ∈ E(Σ) with v ∈ Σ ≤t and w ∈ Σ ≤t . Observe that for such edges e = vw, we have ϕ(w) − ϕ(v) > 0. Now, because the contribution from edges contained in Σ ≤t cancel in pairs. From this we conclude that for any outgoing edge e = vw, ϕ(w) − ϕ(v) ≤ M(∆(ϕ)).
We have the following immediate corollary: Corollary 6.9. Let ϕ, ϑ : V (Σ) → R ≥0 be functions such that where F and G are effective divisors of degree at most d. Suppose that there are (not necessarily distinct) vertices v, w such that ϕ(v) = ϑ(w) = 0 Then Interchanging the roles of ϕ and ϑ, we get the conclusion.
We now prove Proposition 6.6.
Proof. Set Λ = ρ(D). From Lemma 6.3, it follows that L + ∆ (D) ⊆ L + (Λ). Therefore, we must show that any ϑ ∈ L + (Λ) can be approximated by some element in the image of ρ m for some positive integer m. First, we may suppose that ϑ takes rational values. Pick a sufficiently divisible m such that mϑ takes integer values. Because ̺ m (π k s) = ̺ m (s) + k m , we can replace ϑ by ϑ − min v ϑ(v) and suppose that ϑ ≥ 0 with ϑ(v) = 0 for some vertex v.
We have that ∆(mϑ)+mΛ ≥ 0. Pick an effective divisor E on Σ such that ∆(mϑ)+mΛ−E is effective of degree exactly g, the genus of C . Choose a horizontal divisor E such that ρ(E ) = E. Because deg(mD K − E K ) = g, the invertible sheaf O(mD K − E K ) on C K has a regular section by the Riemann-Roch theorem. Therefore, there is a rational function s on From s we will find a section whose image under ρ m approximates ϑ. By multiplying s by some power of π, we may ensure that s (considered as a rational function on C ) is regular on the generic points of the components of the closed fiber and does not vanish identically on all of them. Consequently, s's vanishing function ϕ s is nonnegative and takes the value 0 at some vertex w. Now, is the difference of two effective degree g divisors by Lemma 6.3. Consequently, by Corollary 6.9, we have Because ̺ m (s) = ϕs m , the conclusion follows by choosing large m. To handle the case where the divisor D is of degree 0, we may employee the following result. Proof. It suffices to prove that This follows immediately from definitions.
The rank of a divisor on a graph as defined by Baker-Norine involves an additonal lattice structure. Within the vector space set of functions ϕ : V (Σ) → R, there is a lattice ϕ : V (Σ) → Z. In [2], a divisor Λ on a graph Σ is said to have non-negative rank if there exists ϕ : V (Σ) → Z such that ∆(ϕ)+Λ ≥ 0. From this concept, they build a Riemann-Roch theory for divisors on graphs. One may reformulate this theory in terms of lattice points in Newton-Okounkov linear systems. The authors do not know if this view leads to any new proofs of known results in the Baker-Norine theory. 6.3. Horizontal-Vertical decomposition. Now, we will define a decomposition of divisors on Σ analogous to the Zariski decomposition. We will use this in our description of Newton-Okounkov bodies of curves. Recall that the Zariski decomposition of a big Q-divisor D on a smooth projective surface X is a particular decomposition of the linear equivalence class of D, D = P + N where P is nef and N is effective. It has the property that for m such that mD and mN are integral divisors, multiplication by mN gives an isomorphism H 0 (X, mP ) → H 0 (X, mD). Definition 6.11. Let Λ be a divisor on Σ such that L + (Λ) is non-empty. The minimal element of L + (Λ), ̟ : V (Σ) → R is defined by Here, we interpret ̟ as a vertical divisor. We have the following straightforward lemma following from Lemma 6.4. Lemma 6.12. Let ̟ be the minimal element of L + (Λ). Addition of ̟ gives an isomorphism We can interpret Λ = (Λ − ∆(̟)) + ∆(̟) as a sort of Zariski decomposition. Moreover, if L ⊂ L + (Λ) is a sub-semigroup, we may define ̟ L to be the pointwise minimum of ϕ ∈ L.
6.4. Enriched Newton-Okounkov linear systems. We will connect the Newton-Okounkov linear systems with the Newton-Okounkov bodies of curves over discrete valuation rings. Such bodies must take into account the vanishing of sections along a flag Y • = {Y 1 Y 2 } where Y 2 = {p} is a smooth point of the closed fiber. Consequently, we will enrich the above theory by considering such vanishing. We will also need to consider elements of H 0 (C , mD) whose horizontal components do not have any components in common with a fixed horizontal divisor in order to gain control over the vansihing at Y 1 in the tropical case.
Let D, F be horizontal divisors on C . Let m ∈ Z ≥1 .
Definition 6.13. The F -controlled linear system H 0 (C , mD) F ,ε is the set of all elements s of H 0 (C , mD) which when considered as rational functions have the property that their principal divisors (s) contains no component of F with multiplicity greater than mε.
Let p be a smooth point on a component C v of the closed fiber of C . Suppose that p is chosen to be disjoint from D and F . For a divisor G on C , write v p (G ) to be the multiplicty of p in H ∩ C v where H is the horizontal part of G . We consider the natural map Definition 6.14. The p-enriched Newton-Okounkov linear system L + ∆,p (D) (F ,ε) is the subset of L + (Λ) × R given by the closure of the union of the convex hulls of the images of ̺ m,p for m ∈ Z ≥1 . When the subscript (F , ε) is suppressed, this means that we consider the empty divisor.
For a divisor Λ on Σ, let L + p (Λ) be the subset of L + (Λ) × R given by Observe that if Λ = ρ(D) for a horizontal divisor D on C , the quantity ∆(ϕ)(v) + Λ(v) is exactly the degree of the divisor v ϕ(v)C v + D restricted to C v . So, the second component measures the possible multiplicities of p in the horizontal part of (s) varying from 0 to the maximal possible degree.
We have the following extension of Proposition 6.6.
Theorem 6.15. We have the equality of sets: This is proven by the same method as Proposition 6.6. We wish to find a section s ∈ H 0 (C , mD) such that ̺ m (s) is close to some (ϕ, u) We modify the proof to choose E to intersect C v with a multiplicity m ′ close to mu at p and for E not to contain any component of F . The section s produced by the Riemann-Roch theorem has (s) + mD K − E K equal to a degree g divisor. Then, the horizontal component of (s) + mD K intersects C v with a multiplicity between m ′ and m ′ + g. Moreover, the horizontal components of (s) + mD supported on components of F are of degree at most g. By picking a sufficiently large m, we obtain a close approximation. 6.5. Newton-Okounkov bodies of curves. In this section, we give a combinatorial description of the Newton-Okounkov bodies of curves.
We first consider the tropical case of a flag X ⊃ Y 1 ⊃ Y 2 where Y 1 is a horizontal divisor and Y 2 = {p} is a smooth point of the closed fiber. Let D be a horizontal divisor. Theorem 6.16. Suppose that Y 1 is a horizontal divisor intersecting the closed fiber in smooth points and that Y 2 is a point on the component C v . Moreover, suppose that p is not contained in D. For t ∈ R, let The Newton-Okounkov body is the set of points in R 2 lying above the graph of Proof. We first show that the Newton-Okounkov body lies above the graph of a. We observe that for s ∈ H 0 (C , mD), ν 2 (s) is always greater than or equal to the multiplicity of C v in the principal divisor (s). Consequently, if t ∈ R is such that mt is an integer and s vanishes to order at least mt on Y 1 , then ̺ m (s) ∈ L t . Therefore, For a fixed rational t with 0 ≤ t < deg(D), let us find an m ∈ Z ≥1 and a section s ∈ H 0 (C , O(mD)) such that 1 m ν(s) is close to (t, ̟ Lt (v)). Set F = Y 1 . For a small ε > 0, by Theorem 6.15, we may find m large enough such that there exists s ∈ H 0 (C , m(D −tF )) (F ,ε) (considered as a rational function) such that for t ≥ 0.
The Newton-Okounkov body is the image of L + ∆,p (D) = L + p (ρ(D)) under the map (ϕ, u) → (ϕ(v), u). The conclusion follows from Theorem 6.15. Example 6.18. We conclude by giving an example of the Newton-Okounkov body for curves in the tropical and in the Arakelovian cases for the same linear system. Let us consider the example in [1, Section 4.4] of a smooth plane quartic curve of X with genus g(X) = 3. This is a plane quartic degenerating into a conic C and two lines ℓ 1 , ℓ 2 . To make the model semistable, one must blow up the intersection point of ℓ 1 and ℓ 2 , introducing a new component E of the degeneration. For this curve, the special fiber and the dual graph are given in the figure where the vertex P corresponds to the conic, Q 1 , Q 2 corresponds to the lines, and P ′ corresponds to the curve E.