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AB - Abstract The theory of Newton–Okounkov bodies attaches a convex body to a line bundle on a variety equipped with a flag of subvarieties. This convex body encodes the asymptotic properties of sections of powers of the line bundle. In this article, 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. 1 Introduction The theory of Newton polytopes [6, Chapter 6] is a bridge between algebraic geometry and polyhedral geometry. Specifically, if one is given a Laurent polynomial \[f=\sum_{\omega\in {\mathcal A}} c_\omega x^\omega\in{\mathbf{k}}\left[x_1^{\pm},\dots,x_n^{\pm}\right]\] where |${\mathcal A}\subset {\bf Z}^n$| is a finite set, |${\mathbf{k}}$| is a field, and |$c_\omega\in{\mathbf{k}}^*$|⁠, then the Newton polytope of |$f$| is the convex hull of |${\mathcal A}$|⁠. The Newton polytope can be used to understand the intersection theory of the zero locus |$Z(f)\subset ({\mathbf{k}}^*)^n$| through the Bernstein–Kushnirenko theorem. There are several ways to generalize Newton polytopes beyond hypersurfaces of the algebraic torus |$({\mathbf{k}}^*)^n$|⁠. Tropical geometry is one such way, handling higher codimensional subvarieties of tori by studying polyhedral fans instead of polytopes. Another way, the theory of Newton–Okounkov bodies [10, 14, 17] attaches a convex body to a projective |$d$|-dimensional algebraic variety |$X$| equipped with a divisor |$D$| and a flag of subvarieties \[Y_{\bullet} : X = Y_{0} \supsetneq Y_{1} \supsetneq \, \ldots \, \supsetneq Y_{d-1} \supsetneq Y_{d} = \{pt\}\] such that each |$Y_i$| is smooth at |$Y_d$|⁠. Specifically, the Newton–Okounkov body encodes the generic vanishing of sections of |${\mathcal O}(mD)$| for |$m\in{\bf Z}_{\geq 1}$| on the flag. It is convex and bounded, and it has many of the desirable properties of Newton polytopes. However, except in low dimensions, it may be non-polyhedral. Moreover, because it encodes information about the asymptotic behaviour of |${\mathcal O}(mD)$|⁠, it is much more difficult to compute. Newton polytopes and tropical geometry both have extensions to the “nonconstant coefficient case.” Specifically, if one has a valued field |${\bf K}$| with |${\rm{val}}\,\colon{\bf K}\to{\bf R}$| (say, the fraction field of a discrete valuation ring |${\mathcal O}$|⁠), one obtains an unbounded convex body by attaching to \[f=\sum_{\omega\in {\mathcal A}} c_\omega x^\omega\in {\bf K}\left[x_1^{\pm},\dots,x_n^{\pm}\right]\!,\] the upper hull which is defined to be the convex hull of the set \[\{(\omega,t)\mid \omega\in{\mathcal A},\ t\geq{\rm{val}}(a_\omega)\}\subset {\bf R}^n\times{\bf R}.\] The lower faces of this set induce a subdivision of the Newton polytope |$P(f)$| [6, 7]. This is the Newton subdivision. The lower faces of the upper hull form the graph of a piecewise linear convex function |$\psi\colon P(f)\to {\bf R}$|⁠. In greater generality, tropical geometry attaches a polyhedral complex to a subvariety |$X\subset ({\bf K}^*)^n$|⁠. The purpose of this article is to consider Newton–Okounkov bodies in the nonconstant coefficient case. To set notation, let |${\mathcal O}$| be a discrete valuation ring with fraction field |${\bf K}$| and residue field |${\mathbf{k}}$|⁠. Let |$\pi$| be a uniformizer of |${\mathcal O}$|⁠. For convenience, we shall call semistable any irreducible, regular scheme that is proper, flat, and of finite type over |${\mathcal O}$| whose generic fibre is smooth and whose closed fibre is a reduced normal crossings divisor. Let |${\mathscr X}$| be a projective semistable scheme over |${\mathcal O}$| with generic fibre |$X={\mathscr X}\times_{{\mathcal O}}{\bf K}$|⁠. Let |${\mathscr Y}_\bullet$| denote a descending flag of subschemes \[{\mathscr X}=Y_0\supsetneq Y_1\supsetneq \, \ldots \, \supsetneq Y_{d+1}\] where each |$Y_i$| is a codimension |$i$| subscheme that is either a semistable scheme over |${\mathcal O}$| smooth at |$Y_{d+1}$| or a smooth subvariety of a component of the closed fibre |${\mathscr X}\times_{\mathcal O} {\mathbf{k}}$|⁠. Let |${\mathscr D}$| be a divisor on |${\mathscr X}$| flat over |${\bf Spec}\, {\mathcal O}$|⁠. We will define a Newton–Okounkov body, |$\underline\Delta_{{\mathscr Y}_\bullet}({\mathscr D})\subset{\bf R}^{d+1}$| by looking at sections of |${\mathcal O}(m{\mathscr D})$| for |$m\in{\bf Z}_{\geq 1}$| under a valuation attached to the flag. In contrast to the classical case but similar to upper hulls, this Newton–Okounkov body will be unbounded, albeit in a single direction. This unboundedness is a consequence of the fact that if |$s\in H^0({\mathscr X},{\mathcal O}({\mathscr D}))$|⁠, then |$\pi^ks\in H^0({\mathscr X},{\mathcal O}({\mathscr D}))$| for any |$k\geq 0$|⁠. This is formalized by the following theorem: Theorem 4.5. Let |$p_{\pi}:{\bf R}^{d+1}\rightarrow {\bf R}^d$| be projection along the direction through the valuation vector of |$\pi$|⁠. Then, the image |$\Delta=p_{\pi}(\underline{\Delta}_{{\mathscr Y}_\bullet}({\mathcal O}({\mathscr D}))$| is compact. □ In fact, if |$Y_d$| is semistable over |${\mathcal O}$| and |$Y_{d+1}$| is a point of |$Y_d\times_{\mathcal O} {\mathbf{k}}$|⁠, we are in what we call the tropical case and more can be said: Theorem 4.7. In the tropical case, |$\underline{\Delta}_{{\mathscr Y}_\bullet}({\mathscr D})$| is given by the overgraph of a convex function |$\psi\colon\Delta_{Y_\bullet}(D)\to{\bf R}$| for a particular Newton–Okounkov body of the generic fibre |$X={\mathscr X}\times_{\mathcal O} {\bf K}$|⁠. □ Here, the overgraph is the set of points of |$\Delta_{Y_\bullet}(D)\times{\bf R}$| lying above the graph of |$\psi$|⁠. We will give a complete description in the special case of Newton–Okounkov bodies of toric schemes with respect to a toric flag. Here, it is an analogous to the field case and involves a particular polyhedron |$P_{\mathscr D}$| depending on |${\mathscr D}$|⁠, and |$\phi_{\bf R}$|⁠, a certain linear map depending on |${\mathscr Y}_{\bullet}$|⁠. Theorem 5.1. Let |${\mathscr D}$| be a torus-invariant divisor on a toric scheme |${\mathscr X}$| that is flat over |${\bf Spec}\, {\mathcal O}$| with generic fibre |$D$| such that |${\mathcal O}(D)$| is a big line bundle on |$X$|⁠. Then, the Newton–Okounkov body of |${\mathcal O}({\mathscr D})$| is given by \[\underline{\Delta}_{{\mathscr Y}_\bullet}({\mathscr D})=\phi_{\bf R}(P_{\mathscr D}).\] □ Newton–Okounkov bodies over |${\mathbf{k}}$| in low dimensions turn out to be particularly tractable. In the case of curves, the Newton–Okounkov body is the interval |$[0,\deg(D)]$| and so captures only the degree of the divisor. In the case of surfaces, the Newton–Okounkov body is a polytope encoding the Zariski decomposition of a family of divisors. The case of semistable curves over discrete valuation rings shares features of both of these cases. This is perhaps not surprising because such a curve is of relative dimension |$1$| and absolute dimension |$2$|⁠. We will give a fairly complete description of such Newton–Okounkov bodies in terms of the Baker–Norine theory of linear systems on graphs [2]. We will phrase our description in a language reminiscent of the Zariski decomposition of divisors. To describe the case of semistable curves, we introduce an intermediate object, the Newton–Okounkov linear system, |$L_\Delta^+({\mathscr D})$|⁠. Like Newton–Okounkov bodies, it measures the asymptotics of vanishing orders of sections of |${\mathcal O}(m{\mathscr D})$| for |$m\in{\bf Z}_{\geq 1}$|⁠. However, instead of incorporating vanishing orders on a flag, it incorporates vanishing orders on components of the closed fibre. It is a convex subset of the space of functions |$\varphi\colon V(\Sigma)\to {\bf R},$| where |$\Sigma$| is the dual graph of the closed fibre of |${\mathscr C}$|⁠. We prove that the Newton–Okounkov linear system is combinatorial by relating it to a combinatorially-defined effective linear system |$L^+(\rho({\mathscr D}))$| where |$\rho({\mathscr D})$| is the combinatorial specialization of the horizontal divisor |${\mathscr D}$| on the curve |${\mathscr C}$|⁠: Theorem 6.6. If the generic fibre of |${\mathscr D}$| has positive degree, then we have equality between the Newton–Okounkov linear system and the effective linear system: \[L^+_\Delta({\mathscr D})=L^+(\rho({\mathscr D})).\] □ By incorporating the lattice of integer-valued functions |$\varphi\colon V(\Sigma)\to{\bf Z}$|⁠, this theorem gives an algebraic geometric description of rank in the Baker–Norine theory, a problem studied in the work of Caporaso et al. [5]. However, our description depends on the asymptotics of sections, and we do not know how to use our results to give algebraic geometric proofs of combinatorial results in the Baker–Norine theory. After enhancing the Theorem 6.6 by incorporating vanishing orders of sections at a point in the closed fibre of |${\mathscr C}$|⁠, we are able to give a description of the Newton–Okounkov bodies of curves in Theorems 6.21 and 6.22. We should note that quite different connections between tropical geometry and Newton–Okounkov bodies were recently found by Kaveh and Manon [11] and Postinghel and Urbinati [16]. 2 Notation and Conventions For a scheme |${\mathscr X}$| over a discrete valuation ring |${\mathcal O}$|⁠, we will use |${\mathscr X}_{\bf K}$| to denote its generic fibre. If |${\mathscr C}$| is a semistable curve over |${\mathcal O}$|⁠, irreducible divisors are either horizontal or vertical; see, for example, [15, Section 8.3]. Here, horizontal divisors are those that are flat over |${\bf Spec}\, {\mathcal O}$| while vertical divisors are contained in the closed fibre. Any divisor on |${\mathscr C}$| can be decomposed into a sum of horizontal and vertical divisors. Recall that a convex cone in |${\bf R}^d$| is a convex set invariant under rescaling by elements of |${\bf R}_{\geq 0}$|⁠. For a set |$S\subset{\bf R}^d$|⁠, we write |$\text{cone}_{{\bf R}^{d}}(S)$| for the minimal convex cone containing |$S$|⁠. Unless noted otherwise, divisors on algebraic varieties will have integral coefficients. Divisors on graphs will have real coefficients. We will write zero for the empty divisor. When we speak of components of a divisor, we mean irreducible components of its support. For divisors |$D$| and |$E$|⁠, we will write |$D\leq E$| if |$E-D$| is effective. If |$D$| is a divisor on a smooth variety |$X$|⁠, sections of |${\mathcal O}(D)$| can be interpreted in two ways: as a section of a line bundle or as a rational function|$s$| whose principal divisor satisfies |$(s)+D\geq 0$|⁠. Consequently if |$D\leq E$|⁠, a section of |${\mathcal O}(D)$| can be interpreted as a section of |${\mathcal O}(E)$| although its zero locus will differ by |$E-D$|⁠. We will point out the relevant interpretation by using the words “section” or “rational function”. 3 Construction of the Newton–Okounkov Body We recall the construction of Newton–Okounkov bodies in the classical setting. Let |$X$| be a smooth irreducible projective variety of dimension |$d$| over a field |${\bf K}$|⁠. Given a divisor |$D$| on X, we want to construct a convex compact subset of |${\bf R}^d$| called the Newton–Okounkov body of |$D$|. A flag of irreducible subvarieties \[Y_{\bullet} : X = Y_{0} \supsetneq Y_{1} \supsetneq \, \ldots \, \supsetneq Y_{d-1} \supsetneq Y_{d} = \{pt\}\] is called admissible if |$Y_i$| is smooth at |$Y_d$| for all |$i$|⁠. Henceforth, we will suppose that all flags are admissible unless noted otherwise. For every nonzero section |$s$| of |${\mathcal O}(D)$|⁠, set |$s_0:=s$|⁠, and for |$i=1,\dots, n$|⁠, define \begin{equation} \nu_i(s) := {\rm ord}_{Y_i}(s_{i-1}), \qquad s_i:= \left.\frac{s_{i-1}}{g_i^{\nu_i(s)}}\right|_{Y_i}, \end{equation} (3.1) where |$g_i$| is a local equation of |$Y_i$| in |$Y_{i-1}$| near |$Y_d$|⁠. Here, |$s_0$| is considered as a section of |$L_0={\mathcal O}(D)$| while |$s_i$| is considered as a section on |$Y_i$| of |$L_i=L_{i-1}|_{Y_i}\otimes {\mathcal O}_{Y_i}(-\nu_i(s)Y_i)$|⁠. We obtain the vector |$\nu(s) = (\nu_{1}(s), \dots, \nu_{d}(s))\in{\bf Z}^d$|⁠. Define the semigroup of valuation vectors by \[\Gamma_{Y_\bullet}(D):= \left\{ (\nu(s),m) \in {\bf Z}^d\times{\bf Z}_{\geq 1} \mid s \in H^{0}(X, \mathcal{O}_{X}(mD))\right\} \] and the Newton–Okounkov body of |$D$| by \[\Delta_{Y_\bullet}(D) := \overline{\text{cone}_{{\bf R}^{d+1}}\left(\Gamma_{Y_\bullet}(D)\right)}\cap \left({\bf R}^d\times \{1\}\right)\!.\] We write |$\Gamma_{Y_\bullet}(D)_m$| for |$\Gamma_{Y_\bullet}(D)\cap ({\bf Z}^d\times\{m\})$|⁠. The same construction can be performed for noncomplete graded linear series as well, see [10]. 3.1 The case of curves Let us consider a curve |$C$| of genus |$g$| and a divisor |$D$| of degree |$d>0$|⁠. Let |$Y_{\bullet}=\{ C \supsetneq p \}$|⁠, with |$p$| a point. Then, the Newton–Okounkov body is the segment |$[0, d]$| by [14, Example 1.2] as a consequence of the Riemann–Roch theorem. This is a result that we will eventually generalize to the case of semistable curves over a discrete valuation ring. 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 [14, Section 6.2]. This description provides all the information about possible shapes of Newton–Okounkov bodies of surfaces [13]. In this subsection, all divisors will be |${\bf Q}$|-divisors. Any pseudoeffective divisor |$D$| (i.e., 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, |$P_D\cdot N_D=0$|⁠, and |$N_D$| is effective with a negative definite intersection matrix. Let us consider the rank two valuation induced by a general flag |$Y_{\bullet}=\{ X \supsetneq C \supsetneq p \}$| such that |$C \notin \mbox{supp}(N_D)$|⁠. Set |$\mu:= \sup\{x \mid D-xC \mbox{ is big}\}$|⁠, and for a big divisor |$F$|⁠, let |$\alpha(F)= \mbox{ord}_p(N_F)$|⁠, |$\beta(F)=\alpha(F)+ C\cdot P_F$|⁠. Then, we have by the recipe given in [14], \[\Delta_{Y_{\bullet}}(D)=\{(x, y) \in {\bf R}^2 |\, 0 \leq x \leq \mu \mbox{ and } \alpha(D- xC) \leq y \leq \beta(D-xC)\}.\] Example 3.2. Let |$X$| be the blow up of |${\bf P}^2$| at two points with exceptional divisors |$E_{1}$| and |$E_{2}$| and consider the flag |$Y_{\bullet}=\{X \supsetneq l \supsetneq p \}$| given by a general line and a general point on it. Let |$H$| denote the pullback of the class of a line in |${\bf P}^2$|⁠. Let |$D = 2H - E_{1} - E_{2}$|⁠. In this case, we have |$\mu=1$| and the Zariski decomposition of |$D-xH$| is the sum |$(1-x)(2H-E_1-E_2) + x(H-E_1-E_2)$|⁠, 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 Newton–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 for schemes over discrete valuation rings. Let |${\mathcal O}$| be a discrete valuation ring with fraction field |${\bf K}$|⁠, residue field |${\mathbf{k}}$|⁠, and valuation |${\rm{val}}$|⁠. Let |$\pi$| be a uniformizer of |${\mathcal O}$|⁠. Let |${\mathscr X}$| be a |$d$|-dimensional semistable scheme over |${\mathcal O}$|⁠. We will write |$X={\mathscr X}\times_{\mathcal O} {\bf K}$| for the generic fibre of |${\mathscr X}$|⁠. Let |${\mathscr Y}_\bullet$| denote a descending flag of proper subschemes \[{\mathscr X}=Y_0\supsetneq Y_1\supsetneq \, \ldots \, \supsetneq Y_{d+1}\] where each |$Y_i$| is an irreducible codimension |$i$| subscheme that is either (1) A semistable scheme over |${\mathcal O}$| that is smooth at |$Y_{d+1}$|⁠, or (2) A proper smooth subvariety of a component |$X'$| of the closed fibre |${\mathscr X}\times_{\mathcal O} {\mathbf{k}}$|⁠. We call a flag satisfying these conditions admissible. Let |${\mathscr D}$| be a divisor on |${\mathscr X}$| flat over |${\mathcal O}$|⁠. Let |$j$| be the index such that |$Y_{j-1}$| is semistable over |${\mathcal O}$| and |$Y_{j}$| is a closed subvariety of |${\mathscr X}\times_{\mathcal O} {\mathbf{k}}$|⁠. Then |$Y_j$| is a component of |$Y_{j-1}\times_{\mathcal O} {\mathbf{k}}$|⁠. 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 a construction by Yuan [19] where one studies vanishing orders of sections of a line bundle on the closed fibre of a family. The name “tropical” is motivated by a fundamental notion in tropical geometry, the Newton subdivision, to which our construction specializes in the toric case. Specifically, one studies upper hulls that project to the Newton polytope [7, Section 9.10]. So in a certain sense, our work interpolates between tropical geometry and function field Arakelov theory. The Newton–Okounkov body |$\underline{\Delta}_{{\mathscr Y}_\bullet}({\mathscr D})$| is defined as a convex body in |${\bf R}^{d+1}$| exactly as above by considering sections of |${\mathcal O}(m{\mathscr D})$| over |${\mathscr X}$| for |$m\in {\bf Z}_{\geq 1}$| evaluated at the valuations attached to the flag. For every nonzero section |$s$| of |${\mathcal O}({\mathscr D})$|⁠, set |$s_0:=s$|⁠, and for |$i=1,\dots, n$|⁠, define \begin{equation} \nu_i(s) := {\rm ord}_{Y_i}(s_{i-1}), \qquad s_i:= \left.\frac{s_{i-1}}{g_i^{\nu_i(s)}}\right|_{Y_i}, \end{equation} (4.1) where |$g_i$| is a local equation of |$Y_i$| in |$Y_{i-1}$| near |$Y_{d+1}$|⁠. Here, |$s_0$| is considered as a section of |$L_0={\mathcal O}({\mathscr D})$| while |$s_i$| is considered as a section on |$Y_i$| of |$L_i=L_{i-1}|_{Y_i}\otimes {\mathcal O}_{Y_i}(-\nu_i(s)Y_i)$|⁠. We obtain the vector |$\nu(s) = (\nu_{1}(s), \dots, \nu_{d+1}(s))\in{\bf Z}^{d+1}$|⁠. Define the semigroup of valuation vectors as \[\underline{\Gamma}_{Y_\bullet}({\mathscr D}):= \left\{ (\nu(s),m) \in {\bf Z}^{d+1}\times{\bf Z}_{\geq 1} \mid s \in H^{0}({\mathscr X}, \mathcal{O}_{{\mathscr X}}(m{\mathscr D}))\right\} \] and the Newton–Okounkov body of |${\mathscr D}$| as \[\underline{\Delta}_{Y_\bullet}({\mathscr D}) := \overline{\text{cone}_{{\bf R}^{d+2}}\left(\underline{\Gamma}_{Y_\bullet}({\mathscr D})\right)}\cap \left({\bf R}^{d+1}\times \{1\}\right)\!.\] 4.2 A family of rational curves We will consider the following example: let |${\mathscr X}$| be |${\bf P}^1_{\mathcal O}$| blown up a point on the closed fibre. Then, the generic fibre of |${\mathscr X}$| is |${\bf P}^1_{\bf K}$|⁠, and the closed fibre is two copies of |${\bf P}^1_{{\mathbf{k}}}$| called |$C_1,C_2$| joined at a point. Put homogeneous coordinates |$[X:Y]$| on |${\bf P}^1$|⁠. Write |$x=X/Y$| as an inhomogeneous coordinate. We may assume that we blew up the point |$\infty=[1:0]$| on the special fibre, and |$C_2$| is the exceptional divisor. Write |$0$| for the proper transform of |$[0:1]$|⁠. Let |${\mathscr D}$| be the proper transform of |$\infty$|⁠. We will compute |$\underline{\Delta}_{{\mathscr Y}_\bullet}({\mathscr D})$| with respect to various flags. A section of |$H^0({\mathscr X},{\mathcal O}(m{\mathscr D}))$| can be expressed as a homogeneous polynomial \[P(X,Y)=\sum_{i=0}^m a_iX^iY^{m-i}\] of degree |$m$| with coefficients in |${\bf K}$|⁠. Considered as a rational function, it is \[p(x)=\sum_{i=0}^m a_ix^i\] Near |$0$| in the closed fibre, the local coordinates are |$(x,\pi)$| and the section is expressed as |$p(x)$|⁠. Therefore, the condition that |$P(X,Y)$| is regular near |$C_1$| is that the coefficients of |$p(x)$| are in |${\mathcal O}$|⁠. Moreover, the vanishing order of |$p(x)$| on |$C_1$| is |$\min_i {\rm{val}}(a_i)$|⁠. Near |$\infty$| in the closed fibre, the coordinates are |$(y=Y/X,\pi/y)$| and |$P(X,Y)$| is given as a rational function by \[q(y)=\sum_{i=0}^m a_i\pi^{-i}\left(\frac{\pi}{y}\right)^i\] Therefore, the vanishing order of |$q(y)$| on |$C_2$| is |$\min_i ({\rm{val}}(a_i)-i)$|⁠, and so |$q(y)$| does not have a pole generically along |$C_2$| exactly when |${\rm{val}}(a_i)\geq i$| for |$0\leq i\leq m$|⁠. Example 4.2. We first consider the tropical case. Let |$Y_1$| be some effective horizontal divisor of degree |$1$| over |${\mathcal O}$| specializing to a smooth point |$Y_2$| lying on |$C_1$|⁠. We will compute |$\underline{\Delta}_{{\mathscr Y}_\bullet}({\mathscr D})$|⁠. We may suppose that the divisor |$Y_1$| corresponds to the proper transform of |$0=[0:1]$|⁠. Now, the vanishing order |$\nu_1(P)$| of |$P(X,Y)$| at |$Y_1$| is the smallest |$i$| such that |$a_i\neq 0$|⁠. Therefore, |$s_1=(p(x)/x^{\nu_1(P)})|_{Y_1}=(p(x)/x^{\nu_1(P)})|_{x=0}=a_i$|⁠. The vanishing order |$\nu_2(P)$| is therefore |${\rm{val}}(a_i)$|⁠. By the above discussion, |${\rm{val}}(a_i)$| can take any value greater than |$i$|⁠. It follows that the Newton–Okounkov body, |$\underline{\Delta}_{{\mathscr Y}_\bullet}({\mathscr D})$| is the set of points lying above the graph of |$\psi\colon[0,1]\to {\bf R}$| given by |$\psi(t)=t$|⁠. □ Example 4.3. Now, we consider the Arakelovian case. Let |$Y_1=C_1$| and let |$Y_2$| be a smooth point on |$C_1$|⁠. Because the vanishing order of |$P(x)$| on |$C_1$| is |$\nu_1(P)=\min_i {\rm{val}}(a_i)$|⁠, \[s_1(x)=(p(x)/\pi^{\nu_1(P)})\big|_{Y_1}=\sum_{i=0}^d \frac{a_i}{\pi^{\nu_1(P)}} x^i\big|_{\pi=0}=\sum_{i\mid {\rm{val}}(a_i)=\nu_1(P)} \overline{\left(\frac{a_i}{\pi^{\nu_1(P)}}\right)}x^i\] where the |$\overline{c}$| denotes the image of |$c\in{\mathcal O}$| in |${\mathbf{k}}$|⁠. For |$s_1(x)$| to have a zero of order at least |$k$| at |$p$|⁠, we must have |$\deg s_1(x)\geq k$|⁠. To obtain |$\deg s_1(x)\geq k$|⁠, the minimum of |${\rm{val}}(a_i)$| must be achieved by |$a_{j_1},a_{j_2}$| with |$j_2-j_1\geq k$|⁠. From |${\rm{val}}(a_i)\geq i$|⁠, we must have |$\nu_1(P)=\min_i {\rm{val}}(a_i)\geq k$|⁠. Since any value |$k=\deg s_1(x)\leq m$| can be achieved by some |$p(x)$| with |$\min_i {\rm{val}}(a_i)=k$|⁠, the possible pairs |$(\nu_1(P),\nu_2(P))$| are the integers |$0\leq \nu_2\leq \min(\nu_1,m)$|⁠. Therefore, the Newton–Okounkov body is given by \[\underline{\Delta}_{{\mathscr Y}_\bullet}({\mathscr D})=\left\{(a,b)\in{\bf R}^2\mid 0\leq b\leq \min(a,1)\right\}\!.\] □ We will consider Newton–Okounkov bodies of curves in greater generality in Subsection 6.5. 4.3 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. Lemma 4.4. Let |$\nu(\pi)\in {\bf R}^{d+1}$| be the valuation of the uniformizer |$\pi\in {\mathcal O}$|⁠, viewed as a rational function on |${\mathscr X}$|⁠. The Newton–Okounkov body |$\underline{\Delta}_{{\mathscr Y}_\bullet}({\mathscr D})$| is closed under positive translations in the |$\nu(\pi)$|-direction. □ Proof. It suffices to show |$\Gamma_{{\mathscr Y}_\bullet}({\mathscr D})+k(\nu(\pi),0)\subset\Gamma_{{\mathscr Y}_\bullet}({\mathscr D})$| for any |$k\in{\bf Z}_{\geq 0}$|⁠. Any point of |$\Gamma_{{\mathscr Y}_\bullet}({\mathscr D})$| is of the form |$(\nu(s),m)$| for |$s\in H^0({\mathscr X},{\mathcal O}(m{\mathscr D}))$|⁠. Now, |$\pi^ks\in H^0({\mathscr X},{\mathcal O}(m{\mathscr D}))$| and |$\nu(\pi^ks)=\nu(s)+k\nu(\pi)$|⁠. ■ The Newton–Okounkov body is bounded in other directions. Let |$p_{\pi}\colon{\bf R}^{d+1}\rightarrow{\bf R}^{d+1}/({\bf R}\nu(\pi))$| be the projection along the |$\nu(\pi)$|-direction Theorem 4.5. The image under projection, |$p_{\pi}(\underline{\Delta}_{{\mathscr Y}_\bullet}({\mathscr D}))$| is bounded. □ Proof. We will follow [14, Lemma 1.10]. It suffices to find integers |$b_1,\dots,b_{j-1},b_{j+1},\dots,b_n$| such that for any nonzero element |$s\in H^0({\mathscr X},{\mathcal O}(m{\mathscr D}))$|⁠, we have that for some |$k\in{\bf Z}$|⁠, |$\pi^ks$| obeys |$\nu_i(\pi^k s)\leq mb_i$| for |$i\neq j$|⁠. Choose an ample divisor |$H$| on |$X$| and an ample divisor |$h$| on |$Y_j$|⁠, which we recall, is a component of the projective variety |$Y_{j-1}\times_{\mathcal O} {\mathbf{k}}$|⁠. Recall that the |$H$|-degree of a divisor |$E$| on |$X$| is |$\deg_{X,H}(E)=H^{d-1}\cdot E$|⁠. For a divisor |$E$| on |$Y_i$| and |$k>i$|⁠, we will write |$E|_{Y_k}$| for |${\mathcal O}_{Y_i}(E)|_{Y_k}$|⁠. We begin with the following observation: given a divisor |$D$| and an irreducible effective divisor |$Y$| on some variety |$Z$|⁠, there exists an integer |$b$| such that for any section |$s$| of |${\mathcal O}(mD)$|⁠, the vanishing order of |$s$| on |$Y$| is at most |$mb$|⁠. Indeed, we may choose |$b\geq \deg_{Z,H}(D)/\deg_{Z,H}(Y)$|⁠. Then, |$(D-bY)\cdot H^{d-1}<0$|⁠. By multiplying this inequality by |$m$|⁠, we see that |${\mathcal O}(mD-mbY)$| cannot have any regular sections. The existence of a uniform choice of |$b$| depends on |$\deg_{Z,H}(D)$| being bounded above and |$\deg_{Z,H}(Y)$| being nonzero. Let |$s\in H^0({\mathscr X},{\mathcal O}(m{\mathscr D}))$|⁠. The vanishing orders |$\nu_1(s),\dots,\nu_{j-1}(s)$| are determined by the restriction of |$s$| to the generic fibre of |${\mathscr X}$|⁠. We define line bundles on |$Y_i\times_{{\mathcal O}} {\bf K}$|⁠. Set |$L_0={\mathcal O}(mD)$| and define |$L_i$| inductively by |$L_i=L_{i-1}|_{Y_i}\otimes {\mathcal O}_{Y_i}(-\nu_i(s)Y_i)$|⁠. We can choose |$b_{i+1}\geq \deg_{Y_i,H}(L_i)/(m\deg_{Y_i,H}(Y_{i+1}))$| for |$i\leq j-2$|⁠. The quantity |$\deg_{Y_i,H}(L_i)/m$| is bounded by induction by using the fact that |$\nu_i(s)\leq mb_i$|⁠. Now, we note that |$\pi=0$| is a local equation for |$Y_j$| in |$Y_{j-1}$| near |$Y_{d+1}$|⁠. Therefore, we may interpret |$s_j=(s_{j-1}/\pi^{\nu_j(s)})|_{Y_j}$| as a section of |$L_j=L_{i-1}|_{Y_j}\otimes {\mathcal O}_{Y_j}(-\nu_j(s)Y_j)$|⁠. But, |${\mathscr X}_0\cap Y_{j-1}$| is a principal divisor in |$Y_{j-1}$|⁠. Consequently, |$\deg_{Y_j,h}(L_j)=\deg_{Y_j,h}(L_{j-1}|_{Y_j})$|⁠. The quantity \[\frac{\deg_{Y_j,h}(L_j)}{m}=\frac{\deg_{Y_j,h}\left(m{\mathscr D}|_{Y_j}-\nu_1(s)Y_1|_{Y_j}-\dots-\nu_{j-1}(s)Y_{j-1}|_{Y_j}\right)}{m}\] is bounded because |$\nu_i(s)\leq mb_i$|⁠. Therefore, we may proceed as before, defining for |$i\geq j+1$|⁠, |$L_i=L_{i-1}|_{Y_i}\otimes {\mathcal O}_{Y_i}(-\nu_i(s)Y_i)$|⁠, noting that |$\deg_{Y_i,h}(L_i)/m$| is bounded. ■ Remark 4.6. In the tropical case, if |$Y_{d+1}$| is a smooth point of the central fibre, |$p_{\pi}$| is the projection along the |$(d+1)$|-st component. □ 4.4 The tropical case We now consider the tropical case where the admissible flag |${\mathscr Y}_\bullet$| is given by semistable schemes |$Y_1,\dots,Y_d$| in |${\mathscr X}$| and a point |$Y_{d+1}$| in the closed fibre |${\mathscr X}_{\mathbf{k}}$|⁠. Let |${\mathscr D}$| be a divisor on |${\mathscr X}$| flat over |${\bf Spec}\, {\mathcal O}$| whose generic fibre is |$D\subset X$|⁠. We will relate the Newton–Okounkov bodies to overgraphs. Recall that for a convex body |$\Delta\subset {\bf R}^d$| and a convex function |$\psi\colon\Delta\rightarrow{\bf R}$|⁠, the overgraph in |${\bf R}^{d+1}$| is the set \[\{(x,t)\mid x\in\Delta,\ t\geq\psi(x)\}.\] If |$\psi$| is piecewise linear, its domains of linearity give a subdivision of |$\Delta$|⁠. Theorem 4.7. Let |${\mathscr Y}_{\bullet}$| be an admissible tropical flag on a scheme |${\mathscr X}$|⁠. Define a flag on the generic fibre |$X={\mathscr X}\times_{{\mathcal O}} {\bf K}$| by \[Y_\bullet=\{X\supsetneq Y_1\times_{{\mathcal O}} {\bf K} \supsetneq \, \ldots \, \supsetneq Y_d \times_{{\mathcal O}} {\bf K}\}.\] Let |${\mathscr D}$| be a divisor on |${\mathscr X}$| with generic fibre |$D={\mathscr D}\times_{{\mathcal O}} {\bf K}$|⁠. Let |$\underline{\Delta}_{{\mathscr Y}_\bullet}({\mathscr D})$| be the Newton–Okounkov body for |${\mathscr X}$|⁠, and let |$\Delta_{Y_\bullet}(D)$| be the Newton–Okounkov body for |$X$|⁠. Then, we have a surjection of Newton–Okounkov bodies, \[p_\pi:\underline{\Delta}_{{\mathscr Y}_\bullet}({\mathscr D})\to\Delta_{Y_\bullet}(D).\] Moreover, |$\underline{\Delta}_{{\mathscr Y}_\bullet}({\mathscr D})$| is given as the overgraph of a convex function \[\psi\colon\Delta_{Y_\bullet}(D)\to{\bf R}\] □ Proof. Because any section |$s\in H^0(X,{\mathcal O}(mD))$| has some multiple by |$\pi$| satisfying |$\pi^ks\in H^0({\mathscr X},{\mathcal O}({\mathscr D}))$|⁠, we have that the projection |$p_\pi:{\bf R}^{d+1}\to{\bf R}^d$| maps |$\underline{\Delta}_{{\mathscr Y}_\bullet}({\mathscr D})$| surjectively to |$\Delta_{Y_\bullet}(D)$|⁠. Now, |$\underline{\Delta}_{{\mathscr Y}_\bullet}({\mathscr D})$| is closed under positive translation by |$e_{d+1}=\nu(\pi)$|⁠. From the convexity of Newton–Okounkov bodies, it follows that |$\underline{\Delta}_{{\mathscr Y}_\bullet}({\mathscr D})$| is the set of points in |${\bf R}^{d+1}$| lying above the graph of a convex function |$\psi\colon\Delta_{Y_\bullet}(D)\to {\bf R}$|⁠. ■ We note that Newton–Okounkov bodies of schemes over discrete valuation rings need not be polyhedral. Indeed, given a variety |$X$|⁠, flag |$Y_\bullet$|⁠, and divisor |$D$| over |${\bf C}$| with a non-polyhedral Newton–Okounkov body, one may define an example over |${\mathcal O}={\bf C}[[t]]$| as follows: set |${\mathscr X}=X\times_{\bf C} {\mathcal O}$|⁠; define a new flag |$Y'_i$| on |${\mathscr X}$| with |$Y'_i=Y_i\times_{\bf C} {\mathcal O}$| for |$0\leq i\leq d$| and |$Y'_{d+1}=Y'_d\times_{\mathcal O} {\bf C}$|⁠; and set |${\mathscr D}=D\times_{\bf C} {\mathcal O}$|⁠. This gives a tropical flag whose attached Newton–Okounkov is non-polyhedral by the Theorem 4.7. Remark 4.8. Our construction of functions on Newton–Okounkov bodies has some relation to the filtered linear systems that have appeared in the work of Boucksom and Chen [3], Witt Nyström [18], and Yuan [20]. Let |$X$| be an algebraic variety or scheme equipped with a flag of subvarieties or subschemes. Suppose that there is a family of norms |$||s||_m$| on sections of the line bundles |$L^{\otimes m}={\mathcal O}(mD)$| (possibly as the sup-norm coming from a metric on |${\mathcal O}(D)$|⁠). The function on the Newton–Okounkov body is induced by taking the infimum for |$\frac{1}{m}\log ||s||_m$| of sections |$s$| of |${\mathcal O}(mD)$| corresponding to a point in the Newton–Okounkov body. The function, called a Chebyshev transform is related to metric and adelic volumes in Kähler and Arakelov geometry. This is similar to our work in that norms of sections of line bundles are related to vanishing orders on closed fibres in the non-Archimedean setting. In greater generality, one may consider filtrations on sections of |$L^{\otimes m}$| induced by valuations as in the study of vanishing sequences by Boucksom et al. [4]. □ 5 Toric Schemes In this section, we discuss toric schemes over discrete valuation rings. See [12] for a classical source or [7] for a rigid analytic perspective. 5.1 Toric varieties We begin by reviewing Newton–Okounkov bodies for smooth projective toric varieties [14, Section 6.1]. Let |$N$| be an |$d$|-dimensional lattice. A toric variety |$X(\Delta)$| is specified by a complete rational fan |$\Delta$| in |$N_{\bf R}=N\otimes{\bf R}\cong{\bf R}^d$|⁠. The variety |$X(\Delta)$| 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 |$T=N\otimes {\bf K}^*$| denote the |$d$|-dimensional algebraic torus acting on |$X(\Delta)$|⁠. To each |$k$|-dimensional cone |$\sigma$| of |$\Delta$|⁠, there corresponds an orbit closure |$V(\sigma)$| which is a codimension |$k$| subvariety. Any torus-invariant divisor is given by \[D=\sum_\sigma a_\sigma V(\sigma)\] where the sum is over rays |$\sigma$| in |$\Delta$| and |$a_\sigma\in{\bf Z}$|⁠. Attached to |$D$| is a polyhedron |$P_D\subset M_{\bf R}$| where |$M$| is the dual lattice of |$N$|⁠, defined by \[P_D={\rm{Conv}}(\{m\in M \mid \langle m,u_\sigma\rangle\geq -a_\sigma\}).\] where |$u_\sigma\in N$| is the primitive integer vector (with respect to |$N$|⁠) along |$\sigma$|⁠. This polyhedron arises by considering sections of |${\mathcal O}(D)$|⁠: the vector space of sections |$H^0(X,{\mathcal O}(D))$| has a decomposition into |$T$|-eigenspaces; the lattice points of |$P_D$| are exactly the characters of |$T$| that arise; for |$m\in P_D$|⁠, the character |$\chi^m$| on |$T$| extends to a section of |${\mathcal O}(D)$| on |$X$|⁠. Indeed, the vanishing order of |$\chi^m$| (considered as a section of |${\mathcal O}(D)$|⁠) on the divisor |$V(\sigma)$| is |$\langle m,u_\sigma\rangle+a_\sigma$| so the inequalities defining |$P_D$| are exactly the conditions that |$\chi^m$| is regular at the generic point of the torus-invariant divisors. Consequently, |$\dim H^0(X,{\mathcal O}(D))=|P_D\cap M|.$| The line bundle |${\mathcal O}(D)$| is big if and only if |$P_D$| is |$d$|-dimensional. Because |$X(\Delta)$| is smooth, a |$T$|-invariant flag |$Y_1, Y_2,\dots, Y_n$| can be written as \[Y_i=D_1\cap\cdots\cap D_i\] for a choice of |$T$|-invariant divisors |$D_1,\dots,D_n$| corresponding to rays |$\sigma_1,\dots,\sigma_n$|⁠. Let |$u_1,\dots,u_n$| be the primitive integer vectors along |$\sigma_1,\dots,\sigma_n$|⁠. We define a linear map |$\phi\colon M_{\bf R}\to{\bf R}^n$| by |$\phi(v)=\big(\langle v,u_{\sigma_i}\rangle+a_{\sigma_i}\big)_{1\leq i\leq n}$|⁠. We have the following equality for big line bundles |${\mathcal O}(D)$|⁠: \[\Delta_{Y_\bullet}(D)=\phi(P_D).\] 5.2 Toric schemes Complete toric schemes over a discrete valuation ring |${\mathcal O}$| are defined from complete rational fans in |$N_{{\bf R}}\times {\bf R}_{\geq 0}$| where |$N\cong {\bf Z}^d$| is a lattice [7]. Given such a fan |$\Sigma$|⁠, there is a natural morphism of toric varieties |$X(\Sigma)_{\bf Z}\rightarrow X({\bf R}_{\geq 0})_{\bf Z}={\bf A}^1_{\bf Z}$| and the toric scheme is given by |${\mathscr X}=X(\Sigma)\times_{{\bf A}^1} {\bf Spec}\,({\mathcal O})$|⁠. Here, we will map |$t$|⁠, the coordinate on |${\bf A}^1$|⁠, to the uniformizer of |${\mathcal O}$|⁠. We will suppose that |$\Sigma$| is a unimodular fan and therefore that the total space |${\mathscr X}$| is regular. If we set |$\Delta=\Sigma\cap (N_{\bf R}\times\{0\})$|⁠, then the generic fibre of |${\mathscr X}$| is the toric variety |$X=X(\Delta)$|⁠. The closed fibre of |${\mathscr X}$| is a union of toric varieties described combinatorially by the polyhedral complex |$\Sigma_1=\Sigma\cap (N_{\bf R}\times\{1\})$| in |$N_{\bf R}\times\{1\}$|⁠. The components of the closed fibre are in bijective correspondence with the vertices of |$\Sigma_1$|⁠. We will suppose that the vertices of |$\Sigma_1$| are at points of |$N\times\{1\}$| which ensures that |${\mathscr X}$| has reduced closed fibre and, therefore, is semistable. Let |$T=N\otimes{\bf K}^*$| denote the torus of |$X$|⁠. A |$T$|-invariant divisor |$D$| on |$X$| has many extensions |${\mathscr D}$| to |${\mathscr X}$|⁠. In particular, we may write |$D=\sum_\sigma a_\sigma V(\sigma)$| where |$a_\sigma\in{\bf Z}$| and |$V(\sigma)$| is the divisor of |$X$| corresponding to a ray |$\sigma$| of |$\Delta$|⁠. Any extension is of the form \[{\mathscr D}=\sum_\sigma a_\sigma V(\sigma) + \sum_v a_v V(v)\] where |$a_v\in{\bf Z}$| and |$V(v)$| are the divisors on |${\mathscr X}$| corresponding to rays in |$N_{\bf R}\times{\bf R}_{\geq 0}$| through the vertices of |$\Sigma_1$|⁠. We construct the Newton–Okounkov body. Considering the total space |$X(\Sigma)$| as an |$(d+1)$|-dimensional toric variety, we define a polyhedron |$P_{\mathscr D}\subset M_{\bf R}\times {\bf R}_{\geq 0}$| by \[P_{\mathscr D}={\rm{Conv}}\big(\big\{(m,h)\in M_{{\bf R}}\times {\bf R}_{\geq 0} \mid \langle m,u_\sigma\rangle\geq -a_\sigma,\ \langle m,v\rangle+h\geq -a_v\big\}\big).\] The second set of inequalities come from |$v\in\Sigma_1$| corresponding to vertices |$(v,1)\in\Sigma_1$|⁠. Note that |$P_{\mathscr D}$| projects onto |$P_D$| by |$(m,u)\mapsto m$|⁠. We may define a piecewise linear convex function on |$P_D$|⁠, \[\psi(m)=\max(-a_v-\langle m,v\rangle)\] where |$v$| is taken over vertices of |$\Sigma_1$|⁠. Then |$P_{\mathscr D}$| is the overgraph of |$\psi$|⁠. Now, we will explain how the polyhedron |$P_{\mathscr D}$| relates to the Newton–Okounkov body of |${\mathscr X}$| with respect to a flag |${\mathscr Y}_\bullet$| of torus-fixed subschemes. Following [14], we may choose irreducible toric divisors |$D_1,\dots,D_{d+1}$| of |$X(\Sigma)$| such that |$Y_i=D_1\,\cap \cdots\, \cap D_i$|⁠. Suppose that |$D_i$| corresponds to a ray in |$N_{\bf R}\times {\bf R}$| whose primitive integer vector is |$w_i\in N\times {\bf Z}$|⁠. Here, |$\{w_1,\dots w_{d+1}\}$| is a basis for |$N\times{\bf Z}$|⁠. Write \[{\mathscr D}=\sum a_w V(\sigma_w)\] where |$w$| runs over primitive integer vectors of rays |$\sigma_w$| of |$\Sigma$|⁠. We define \[\phi_{\bf R}\colon M_{\bf R}\times {\bf R}\to {\bf R}^{d+1},\quad \phi\colon(v,h)\mapsto\big(\big\langle(v,h),w_i\big\rangle+a_{w_i}\big)_{1\leq i\leq d+1}\] where the pairing is between |$M_{\bf R}\times{\bf R}$| and |$N_{\bf R}\times{\bf R}$|⁠. We have the following analogue of [14, Proposition 6.1]: Theorem 5.1. Let |${\mathscr D}$| be a torus-invariant divisor on |${\mathscr X}$| that surjects onto |${\bf Spec}\, {\mathcal O}$| with generic fibre |$D$| such that |${\mathcal O}(D)$| is a big line bundle on |$X$|⁠. Then, the Newton–Okounkov body of |${\mathcal O}({\mathscr D})$| is given by \[\underline{\Delta}_{{\mathscr Y}_\bullet}({\mathscr D})=\phi_{\bf R}(P_{\mathscr D}).\] □ Proof. By replacing |${\mathscr D}$| with a positive integer multiple, we may suppose that the vertices of |$P_{\mathscr D}$| are points of |$N\times{\bf Z}$|⁠. Write the restriction of |$s\in H^0(({\mathscr X},{\mathcal O}({\mathscr D})))$| to |$X$| as \[s=\sum_m c_m \chi^m\] for |$c_m\in {\bf K}$| where the above is a finite sum over characters. The vanishing order of |$s$| (considered as a rational function) on the divisor |$D_w$| corresponding to a ray |$\sigma_w$| of |$\Sigma$| is \[b_w=\min\big(\big\langle\big(m,{\rm{val}}(c_m)\big),w\big\rangle\big)\] where |${\rm{val}}(0)=\infty$| and the pairing is the one between |$M_{\bf R}\times {\bf R}$| and |$N_{\bf R}\times {\bf R}$|⁠. Observe that in the above, if |$\sigma_w$| is a ray of |$\Delta$|⁠, then |$\langle(m,{\rm{val}}(c_m)),w\rangle=\langle m,u_\sigma\rangle_N$| where the second pairing is the pairing between |$M_{\bf R}$| and |$N_{\bf R}$|⁠. If |$w=(v,1)$| corresponds to a vertex of |$\Sigma_1$|⁠, then |$\big\langle\big(m,{\rm{val}}(c_m)\big),w\big\rangle=\langle m,v\rangle_N+{\rm{val}}(c_m)$|⁠. On |${\mathscr X}$|⁠, we have that following formula for the principal divisor: \[(s)=\sum b_w D_w.\] The vanishing order of |$s$|⁠, considered as a section of |${\mathcal O}({\mathscr D})$|⁠, on |$D_w$| is |$b_w+a_w$|⁠. Consequently, a sum of characters like the above corresponds to an element of |$H^0(({\mathscr X},{\mathcal O}({\mathscr D}))$| if and only |$b_w\geq -a_w$| for all |$w$|⁠. In fact, |$\pi^h\chi^m\in H^0(({\mathscr X},{\mathcal O}({\mathscr D}))$| if and only if |$(m,h)\in P_{\mathscr D}$|⁠. Now, the valuation of such a section of |${\mathcal O}({\mathscr D})$| is \[\nu(s)=\left(b_{w_1}+a_{w_1},\dots,b_{w_{d+1}}+a_{w_{d+1}}\right)\!.\] It follows that |$\nu(s)\in\phi_{\bf R}(P_{\mathscr D})$|⁠. By considering sections of the form |$\pi^h\chi^m$|⁠, we see that |$\underline{\Delta}_{{\mathscr Y}_\bullet}({\mathscr D})$| contains |$\phi_{\bf R}(P_{\mathscr D})$|⁠. ■ 6 Newton–Okounkov Bodies of Curves We will relate the Newton–Okounkov bodies of curves over |${\mathcal O}$| to the Baker-Norine theory of linear systems on graphs. Throughout, we will assume that |${\mathcal O}$| is a complete local ring with algebraically closed residue field. 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 |${\mathscr C}$| be a semistable curve over |${\bf Spec}\, {\mathcal O}$|⁠. The semistability condition ensures that the closed fibre |${\mathscr C}_0$| is reduced with only ordinary double points as singularities. A node in the closed fibre of |${\mathscr C}$| is formally locally described in |${\mathscr C}$| by |${\mathcal O}[x,y]/(xy-\pi)$|⁠. Definition 6.1. The dual graph|$\Sigma$| of a semistable curve |${\mathscr C}$| is a graph |$\Sigma$| whose vertices |$V(\Sigma)$| correspond to components of the normalization |$p\colon\widetilde{{\mathscr C}}_0\rightarrow{\mathscr C}_0$| and whose edges |$E(\Sigma)$| correspond to nodes of |${\mathscr C}_0$|⁠. For each vertex |$v\in V(\Sigma)$|⁠, we write |$C_v$| for the corresponding component of |$\widetilde{{\mathscr C}}_0$|⁠. □ We will denote the edges of |$E(\Sigma)$| by |$e=vw$| even though |$\Sigma$| 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 and sum over |$v$| itself. A divisor on |$\Sigma$| is an element of the real vector space with basis |$V(\Sigma)$|⁠. We write a divisor as |$D=\sum_{v\in V(\Sigma)} a_v(v)$| with |$a_v\in{\bf R}$|⁠. We may write |$D(v)=a_v$|⁠. The vector space of all divisors is denoted by |${\rm{Div}}(\Sigma)$|⁠. We say a divisor |$D$| is effective and write |$D\geq 0$| if |$a_v\geq 0$| for all |$v\in V(\Sigma)$|⁠. We write |$D\geq D'$| if |$D-D'\geq 0$|⁠. The degree of a divisor is given by \[\deg(D)=\sum_v a_v.\] We will study functions |$\varphi\colon V(\Sigma)\to {\bf R}$|⁠. The Laplacian of |$\varphi$|⁠, |$\Delta(\varphi)$| is the divisor on |$\Sigma$| given by \[\Delta(\varphi)=\sum_{v\in V(\Sigma)} \sum_{e\in E(\Sigma) \mid e=vw} (\varphi(v)-\varphi(w))(v).\] Note that |$\Delta(\varphi)$| is of degree |$0$|⁠. The specialization map |$\rho\colon{\rm{Div}}({\mathscr C})\rightarrow{\rm{Div}}(\Sigma)$| is defined by, for |${\mathcal{D}}\in{\rm{Div}}({\mathscr C})$|⁠, \[\rho({\mathcal{D}})=\sum_{v\in\Gamma}\deg(p^*{\mathcal O}({\mathcal{D}})|_{C_v})(v).\] The specialization of a vertical divisor |$\sum_v \varphi(v)C_v$| satisfies \[\rho\left(\sum_v \varphi(v) C_v\right)=-\Delta(\varphi).\] For a divisor |$H$| on |$C_{\bf K}$|⁠, we will write |$\rho(H)$| to mean the specialization of its closure in |${\mathscr C}$|⁠. Observe that for |$H$|⁠, horizontal and effective, we have |$\rho(H)\geq 0$|⁠. Definition 6.2. Let |$\Lambda$| be a divisor on |$\Sigma$|⁠. We define the linear system|$L(\Lambda)$| to be the set of functions |$\varphi\colon V(\Sigma)\to {\bf R}$| on |$\Sigma$| with |$\Delta(\varphi)+\Lambda\geq 0$|⁠. The effective linear system|$L^+(\Lambda)$| is the subset of |$L(\Lambda)$| consisting of everywhere nonnegative functions |$\varphi$|⁠. □ Let |${\mathscr D}$| be a divisor on |${\mathscr C}$|⁠. Then we will interpret a global section of |${\mathcal O}({\mathscr D})$| as a rational function |$s$| on |${\mathscr C}$| such that |$(s)+{\mathscr D}\geq 0$|⁠. If we write |$(s)={\mathscr H}+V$| where |${\mathscr H}$| is a horizontal divisor over |${\mathcal O}$| and |$V$| is a vertical divisor contained in the closed fibre, we may decompose |$V$| as |$V=\sum_v \varphi_s(v) C_v$| where we call |$\varphi_s\colon V(\Sigma)\rightarrow {\bf Z}$| 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 |${\mathscr C}$|⁠. The following lemma is standard and we include the proof only for completeness. Lemma 6.3. Let |$D$| be a divisor on |$C_{\bf K}$| whose closure |${\mathscr D}$| has specialization |$\Lambda=\rho({\mathscr D})$|⁠. For a rational function |$s$| on |$C$| corresponding to a section of |${\mathcal O}(D)$| with vanishing function |$\varphi$|⁠, we have \[\Delta(\varphi)+\Lambda\geq 0\] or, in other words, |$\varphi\in L(\Lambda)$|⁠. □ Proof. Because |$s$| is principal, we have |$(s)\cdot C_v=0$| for all components of the closed fibre. If we write |$(s)={\mathscr H}+\sum_v \varphi(v) C_v$|⁠, we have \[0=\rho\left((s)\right)=\rho\big({\mathscr H}+\sum_v \varphi(v) C_v\big)=\rho({\mathscr H})-\Delta(\varphi).\] Since |${\mathscr H}+{\mathscr D} \geq 0$| is horizontal, we have \[0\leq \rho({\mathscr H})+\rho({\mathscr D})=\Delta(\varphi)+\Lambda.\] ■ The linear system |$L(\Lambda)$| has a tropical semigroup structure as noted in [8]: Lemma 6.4. For |$\varphi_1,\varphi_2\in L(\Lambda)$|⁠, let |$\varphi\colon V(\Sigma)\to {\bf R}$| be the pointwise minimum of |$\varphi_1,\varphi_2\colon V(\Sigma)\to {\bf R}$|⁠. Then |$\varphi\in L(\Lambda)$|⁠. □ Proof. Let |$v\in V(\Sigma)$|⁠. Without loss of generality, suppose that |$\varphi(v)=\varphi_1(v)$|⁠. Then \begin{align*} \Delta(\varphi)(v)+\Lambda(v)&=\sum_{e=vw} (\varphi(v)-\varphi(w))+\Lambda(v)\\ &\geq \sum_{e=vw} (\varphi_1(v)-\varphi_1(w))+\Lambda(v)\\ &\geq 0. \end{align*} ■ 6.2 Geometric and tropical linear systems Definition 6.5. Now, let |${\mathscr D}$| be a horizontal divisor on |${\mathscr C}$|⁠. Let |$m\in {\bf Z}_{\geq 1}$|⁠. There is a natural map \begin{eqnarray*} \varrho_m\colon H^0({\mathscr C},{\mathcal O}(m{\mathscr D}))&\to &L^+(\rho({\mathscr D}))\\ s&\mapsto& \frac{1}{m}\varphi_s \end{eqnarray*} where |$s\in H^0({\mathscr C},m{\mathscr D})$| is interpreted as a rational function |$s$| with |$(s)+m{\mathscr D}\geq 0$| and |$\varphi_s$| is the vanishing function of |$s$|⁠. □ We define the Newton–Okounkov linear system|$L^+_\Delta({\mathscr D})$| to be the subset of |$L^+(\rho({\mathscr D}))$| given by the closure of the union of the convex hulls of the images of |$\varrho_m$| for |$m$| ranging over |${\bf Z}_{\geq 1}$|⁠. This definition does not depend on the choice of a flag. Instead, we are looking at vanishing orders with respect to all components of the closed fibre. We may extend this definition to horizontal |${\bf Q}$|-divisors by defining |$L^+_\Delta({\mathscr D})$| to be |$\frac{1}{m}L^+_\Delta(m{\mathscr D})$| where |$m$| is chosen sufficiently divisible. Theorem 6.6. Let |${\mathscr D}$| be a horizontal divisor. If the generic fibre of |${\mathscr D}$| has positive degree, then we have equality between the Newton–Okounkov linear system and the effective linear system: \[L^+_\Delta({\mathscr D})=L^+(\rho({\mathscr D})).\] □ Before proving the theorem, we need a preparatory lemma adapted from [9]. Definition 6.7. Let |$f\colon V(\Sigma)\rightarrow {\bf R}$| be a function. Set \[M(f)=\max_{S\subseteq V(\Sigma)} \left\{ \left| \sum_{v\in S} f(v) \right| \right\}\!.\] □ Lemma 6.8. Let |$\varphi\colon V(\Sigma)\rightarrow {\bf R}$| be a function. Then, \[\max \varphi-\min \varphi\leq M(\Delta(\varphi))\operatorname{diam}(\Sigma).\] □ Proof. It suffices to show that for any edge |$e=vw$| in |$\Sigma$|⁠, |$|\varphi(w)-\varphi(v)|\leq M(\Delta(\varphi))$|⁠. Indeed, let |$v_0,v_1$| be the vertices where the minimum and maximum of |$\varphi$| are achieved, respectively. By picking a path from |$v_0$| to |$v_1$| of length at most |$\operatorname{diam}(\Sigma)$| and comparing the values of |$\varphi$| along that path, we achieve the desired conclusion. Let |$e=vw$| with |$t=\varphi(v)<\varphi(w)$|⁠. Set |$\Sigma_{\leq t}$| be the subgraph of |$\Sigma$| induced by |$\varphi^{-1}((-\infty,t])$|⁠. Let |$O(\Sigma_{\leq t})$| be the set of outgoing edges, that is, the edges |$e'=v'w'\in E(\Sigma)$| with |$v'\in\Sigma_{\leq t}$| and |$w'\not\in\Sigma_{\leq t}$|⁠. In particular, |$e\in O(\Sigma_{\leq t})$|⁠. Observe that for such edges |$e'=v'w'$|⁠, we have |$\varphi(w')-\varphi(v')>0$|⁠. Now, \begin{eqnarray*} M(\Delta(\varphi))&\geq& \left|\sum_{v\in V(\Sigma_{\leq t})} \Delta(\varphi)(v)\right|\\ &=&\left|\sum_{v\in V(\Sigma_{\leq t})} \left(\sum_{e'=v''w} \left(\varphi(v')-\varphi(w')\right)\right)\right|\\ &=&\sum_{e'=v'w'\in O(\Sigma_{\leq t})} \left(\varphi\left(w'\right)-\varphi\left(v'\right)\right) \end{eqnarray*} where the last equality holds because the contribution from edges contained in |$\Sigma_{\leq t}$| cancel in pairs. From this, we conclude that for |$e=vw$|⁠, |$\varphi(w)-\varphi(v)\leq M(\Delta(\varphi))$|⁠. ■ We have the following corollary: Corollary 6.9. Let |$\varphi,\vartheta:V(\Sigma)\rightarrow{\bf R}_{\geq 0}$| be functions such that \[\Delta(\varphi)-\Delta(\vartheta)=F-G\] where |$F$| and |$G$| are effective divisors of degree at most |$d$|⁠. Suppose that there are (not necessarily distinct) vertices |$v,w$| such that |$\varphi(v)=\vartheta(w)=0$|⁠. Then \[\max(|\varphi-\vartheta|)\leq d\operatorname{diam}(\Sigma).\] □ Proof. By hypothesis, |$\min(\varphi-\vartheta)\leq \varphi(v)-\vartheta(v)\leq 0$|⁠. We note that |$M(\Delta(\varphi-\vartheta))\leq d$|⁠. Consequently, \[\max(\varphi-\vartheta)\leq \max(\varphi-\vartheta)-\min(\varphi-\vartheta)\leq d\operatorname{diam}(\Sigma).\] Interchanging the roles of |$\varphi$| and |$\vartheta$|⁠, we get the conclusion. ■ We will make use of the following lemma. For a horizontal divisor |${\mathscr E}$| and |$p$| a smooth point of |${\mathscr C}_0$|⁠, let |${\rm{mult}}_p({\mathscr E})$| be the multiplicity of |${\mathscr E}$| at |$p$| Lemma 6.10. There exists a positive integer |$M_0\geq g$| such that for any effective divisor |$E$| on |$\Sigma$|⁠, there exists an effective horizontal divisor |${\mathscr E}$| such that |$\rho({\mathscr E})=M_0E$|⁠. Moreover, we can choose |${\mathscr E}$| (possibly after picking another |$M_0$|⁠) in such a way that (1) |${\mathscr E}$| has no components in common with any given horizontal divisor |${\mathscr F}$|⁠, and (2) Given any |$0\leq u \leq 1$|⁠, |$\varepsilon>0$|⁠, |$w\in V(\Sigma)$|⁠, and |$p$|⁠, any smooth |${\mathbf{k}}$|-point on |$C_w$|⁠, we have \[\left| \frac{{\rm{mult}}_p({\mathscr E})}{M_0} - uE(w)\right| < \varepsilon.\] □ The second condition needs some explanation. By degree considerations, the multiplicity of the |${\bf Q}$|-divisor |$(1/M_0){\mathscr E}$| at |$p$| is between |$0$| and |$E(w)$|⁠. By making a judicious choice of |$M_0$| and |${\mathscr E}$|⁠, we may ensure that the multiplicity is arbitrarily close to any value in that range. Proof. For each component |$C_v$| of |${\mathscr C}_0$|⁠, pick a smooth point |$p_v\in C_v({\mathbf{k}})$|⁠. By [15, Lemma 8.3.35], there is a effective horizontal divisor such |${\mathscr E}_v$| such that |${\mathscr C}_0\cap {\rm{Supp}}({\mathscr E}_v)=\{p_v\}$|⁠. Therefore, |$\rho({\mathscr E}_v)=m_v(v)$| for |$m_v={\rm{mult}}_{p_v}({\mathscr E}_v)$|⁠. Set |$M_0=\prod_v m_v$| and \[{\mathscr E}=\sum_{v\in V(\Sigma)} E(v)\frac{M_0}{m_v} {\mathscr E}_v.\] Then, we have \[\rho({\mathscr E})=\sum_{v\in V(\Sigma)} (M_0E(v))(v)=M_0E.\] By replacing |${\mathscr E}$| and |$M_0$| by integer multiples, we may suppose that |$M_0\geq g$|⁠. By examining the proof of [15, Lemma 8.3.35], we see we have the freedom to choose |${\mathscr E}$| to not contain any component of |${\mathscr F}$|⁠. For the multiplicity condition, choose |$p_w$| distinct from |$p$| and pick an additional horizontal divisor |${\mathscr E}'_{w}$| such that |${\mathscr C}_0\cap {\rm{Supp}}({\mathscr E}'_{w})=\{p\}$|⁠. Let |$m'_w={\rm{mult}}_{p}({\mathscr E}'_w)$|⁠. By replacing |${\mathscr E}'_w$| by |$m_w{\mathscr E}'_w$| and replacing |${\mathscr E}_w$| by |$m'_w{\mathscr E}_w$|⁠, we may ensure that |${\rm{mult}}_p({\mathscr E}'_w)={\rm{mult}}_{p_w}({\mathscr E}_w)$|⁠. Call this common multiplicity |$m_w$|⁠. Let |$M_0=(\prod m_v).$| By replacing |$M_0$| by a positive integer multiple, we may ensure |$m_w/M_0<\varepsilon$|⁠. Now, pick |$k$| to be an integer closest to |$uE(w)M_0/m_w$|⁠, and set \[{\mathscr E}=\left(\sum_{v\neq w} E(v)\frac{M_0}{m_v} {\mathscr E}_v\right)+k {\mathscr E}'_w+ \left(E(w)\frac{M_0}{m_w}-k\right){\mathscr E}_w.\] We have |$\rho({\mathscr E})=M_0E$| and \[{\rm{mult}}_p({\mathscr E})=km_w.\] But, |$|k-uE(w)M_0/m_w|<1$| implies \[\left|\frac{km_w}{M_0}-uE(w)\right|<\frac{m_w}{M_0}<\varepsilon.\] ■ We now prove Theorem 6.6. Proof. Set |$\Lambda=\rho({\mathscr D})$|⁠. From Lemma 6.3 and the fact that |${\mathscr D}$| is horizontal, it follows that |$L^+_\Delta({\mathscr D})\subseteq L^+(\Lambda)$|⁠. Therefore, we must show that any |$\vartheta\in L^+(\Lambda)$| can be approximated by some element in the image of |$\rho_M$| for some positive integer |$M$|⁠. First, we may suppose that |$\vartheta$| takes rational values. Let |$\varepsilon>0$|⁠. We will choose a positive integer |$M$| such that there exists |$s\in H^0({\mathscr C},{\mathscr O}(M{\mathscr D}))$| such that \[ \left|\frac{\varphi_s}{M}-\vartheta\right|<\varepsilon.\] Pick a positive integer |$m$| such that (1) It is sufficiently divisible that |$m\vartheta$| takes integer values, and (2) We have the inequality |$\operatorname{diam}(\Sigma)/m<\varepsilon.$| Because |$\varrho_m(\pi^k s)=\varrho_m(s)+\frac{k}{m}$|⁠, we can replace |$\vartheta$| by |$\vartheta-\min_v \vartheta(v)$| and suppose that |$\vartheta\geq 0$| with |$\vartheta(v)=0$| for some vertex |$v$|⁠. We have that |$\Delta(m\vartheta)+m\Lambda\geq 0$|⁠. Pick an effective divisor |$E$| on |$\Sigma$| such that |$E(v)\in{\bf Z}$| for all |$v\in V(\Sigma)$| and |$\Delta(m\vartheta)+m\Lambda-E$| has degree equal to |$1$|⁠. By Lemma 6.10, we can choose an effective horizontal divisor |${\mathscr E}$| such that |$\rho({\mathscr E})=M_0E$|⁠. We will set |$M=mM_0$|⁠. Because |$\deg(mM_0{\mathscr D}_{\bf K}-{\mathscr E}_{\bf K})=M_0\geq g$|⁠, the line bundle |${\mathcal O}(mM_0{\mathscr D}_{\bf K}-{\mathscr E}_{\bf K})$| on |${\mathscr C}_{\bf K}$| has a regular section by the Riemann–Roch theorem. Therefore, there is a rational function |$s$| on |${\mathscr C}_{\bf K}$| with \[(s)+mM_0{\mathscr D}_{\bf K}-{\mathscr E}_{\bf K}\geq 0.\] By multiplying |$s$| by some power of |$\pi$|⁠, we may ensure that |$s$| (considered as a rational function on |${\mathscr C}$|⁠) is regular on the generic points of the components of the closed fibre and does not vanish identically on all of them. Consequently, |$s$|’s vanishing function, |$\varphi_s$| is nonnegative and takes the value |$0$| at some vertex |$w$|⁠. Now, \begin{eqnarray*} \Delta(\varphi_s)-\Delta(mM_0\vartheta)&=&(\Delta(\varphi_s)+mM_0\Lambda-M_0E)-(\Delta(mM_0\vartheta)+mM_0\Lambda-M_0E)\\ &=& (\Delta(\varphi_s)+\rho(mM_0{\mathscr D}-{\mathscr E}))-(\Delta(mM_0\vartheta)+mM_0\Lambda-M_0E) \end{eqnarray*} is the difference of two effective degree |$M_0$| divisors by the proof of Lemma 6.3. Consequently, by Corollary 6.9, we have \[ \left|\frac{\varphi_s}{mM_0}-\vartheta\right|\leq \frac{M_0}{mM_0}\operatorname{diam}(\Sigma)<\varepsilon.\] Because |$\varrho_{mM_0}(s)=\frac{\varphi_s}{mM_0}$|⁠, the conclusion follows. ■ To handle the case where the divisor |${\mathscr D}$| is of degree |$0$|⁠, we may employ the following result. Corollary 6.11. Let |${\mathscr D}$| be a horizontal divisor on |${\mathscr C}$| of nonnegative degree. Let |${\mathscr E}$| be an effective, non-empty horizontal divisor. Then, we have the equality \[L^+(\rho({\mathscr D}))=\bigcap_{\varepsilon>0} L^+_\Delta({\mathscr D}+\varepsilon{\mathscr E})\] where the intersection is taken over rational |$\varepsilon>0$|⁠. □ Proof. It suffices to prove that \[L^+(\rho({\mathscr D}))=\bigcap_{\varepsilon>0} L^+(\rho({\mathscr D}+\varepsilon {\mathscr E})).\] This follows immediately from the definitions. ■ This comparison between the purely combinatorial Baker–Norine linear system and the algebraically-defined Newton–Okounkov linear system was surprising to the authors. However, it does not capture the combinatorial richness of the Baker–Norine theory as it involves real-valued, rather than integer-valued functions |$\varphi$| on graphs. The integer-valued functions can be incorporated into our work by hand. Within the vector space of functions |$\varphi\colon V(\Sigma)\to{\bf R}$|⁠, there is a lattice |$\varphi\colon V(\Sigma)\to{\bf Z}$|⁠. In [2], a divisor |$\Lambda$| on a graph |$\Sigma$| is said to have nonnegative rank if there exists |$\varphi\colon V(\Sigma)\to{\bf Z}$| such that |$\Delta(\varphi)+\Lambda\geq 0$|⁠. From this concept, a Riemann–Roch theory for divisors on graphs is developed. Because we may add a constant to |$\varphi$| without affecting |$\Delta(\varphi)+\Lambda\geq 0$|⁠, we may suppose that |$\varphi$| is nonnegative in the above definition. From this, we can give a asymptotic formulation of nonnegative rank. Corollary 6.12. Let |${\mathscr D}$| be a horizontal divisor on |${\mathscr C}$| of nonnegative degree. Let |${\mathscr E}$| be an effective, non-empty horizontal divisor. Then, the specialization |$\rho({\mathscr D})$| has nonnegative rank if and only if there exists |$\varphi\colon V(\Sigma)\to{\bf Z}_{\geq 0}$| such that for all |$\varepsilon>0$|⁠, |$\varphi\in L^+_\Delta({\mathscr D}+\varepsilon{\mathscr E})$|⁠. □ From this, one may reformulate the Baker–Norine theory in terms of lattice points in Newton–Okounkov linear systems. It is unknown at this point whether 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 |$\Sigma$| analogous to the Zariski decomposition to use in our description of Newton–Okounkov bodies of curves. Recall that the Zariski decomposition of a big |${\bf Q}$|-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 \[H^0(X,mP)\to H^0(X,mD).\] Definition 6.13. Let |$\Lambda$| be a divisor on |$\Sigma$| such that |$L^+(\Lambda)$| is non-empty. The minimal element of |$L^+(\Lambda)$|⁠, |$\varpi\colon V(\Sigma)\to {\bf R}$| is defined by \[\varpi(v)=\min(\varphi(v)\mid \varphi\in L^+(\Lambda)).\] □ By Lemma 6.4, |$\varpi$| is indeed an element of |$L^+(\Lambda)$|⁠. The following lemma follows from the definitions: Lemma 6.14. Let |$\varpi$| be the minimal element of |$L^+(\Lambda)$|⁠. Addition of |$\varpi$| gives an isomorphism \[L^+(\Lambda+\Delta(\varpi))\to L^+(\Lambda).\] □ We can interpret |$\Lambda=(\Lambda+\Delta(\varpi))+(-\Delta(\varpi))$| as a sort of Zariski decomposition. Moreover, if |$L\subseteq L^+(\Lambda)$| is a sub-semigroup, we may define |$\varpi_L$| to be the pointwise minimum of |$\varphi\in L$|⁠. 6.4 Enhanced Newton–Okounkov linear systems We will connect the Newton–Okounkov linear systems to the Newton–Okounkov bodies of curves over discrete valuation rings. Such bodies must take into account the vanishing of sections along a flag \[{\mathscr Y}_\bullet=\{{\mathscr C}=Y_0\supsetneq Y_1\supsetneq Y_2\}\] where |$Y_2=\{p\}$| is a smooth point of the closed fibre. Consequently, we will enhance the above theory by considering such vanishing. We will also need to consider elements of |$H^0({\mathscr C},m{\mathscr D})$| whose horizontal components do not have any components in common with a fixed horizontal divisor in order to gain control over the vanishing at |$Y_1$| in the tropical case. Let |${\mathscr D},{\mathscr F}$| be horizontal divisors on |${\mathscr C}$|⁠. Let |$m\in {\bf Z}_{\geq 1}$|⁠. Definition 6.15. The |${\mathscr F}$|-controlled linear system|$H^0({\mathscr C},m{\mathscr D})_{({\mathscr F},\varepsilon)}$| is the set of all |$s\in H^0({\mathscr C},{\mathcal O}(m{\mathscr D}))$| which, when considered as regular sections of |${\mathcal O}(m{\mathscr D})$|⁠, have the property that their zero loci contain no component of |${\mathscr F}$| with multiplicity greater than |$m\varepsilon$|⁠. □ The zero locus of |$s$| as a section of a line bundle is |$(s)+m{\mathscr D}$| where |$(s)$| is the principal divisor of |$s$| considered as a rational function. Let |$p$| be a smooth point on a component |$C_v$| of the closed fibre of |${\mathscr C}$|⁠. For a divisor |${\mathscr G}$| on |${\mathscr C}$|⁠, write |$v_p({\mathscr G})$| to be the multiplicity of |$p$| in |${\mathscr H}\cap C_v$| where |${\mathscr H}$| is the horizontal part of |${\mathscr G}$|⁠. We consider the natural map \begin{eqnarray*} \varrho_{m,p}\colon H^0({\mathscr C},m{\mathscr D})_{({\mathscr F},\varepsilon)}&\to &L^+(\rho({\mathscr D}))\times {\bf R}\\ s&\mapsto& \left(\frac{1}{m}\varphi_s,\frac{1}{m}v_p\big((s)+m{\mathscr D}\big)\right) \end{eqnarray*} where |$s\in H^0({\mathscr C},m{\mathscr D})_{({\mathscr F},\varepsilon)}$|⁠. Observe that the second component of |$\varrho_{m,p}(s)$| is the vanishing at |$p$| of the horizontal component of the zero locus of |$s$|⁠, considered as a section of |${\mathcal O}({\mathscr D})$|⁠. Definition 6.16. The |${\mathscr F}$|-controlled |$p$|-enhanced Newton–Okounkov linear system|$L^+_{\Delta,p}({\mathscr D})_{({\mathscr F},\varepsilon)}$| is the subset of |$L^+(\rho({\mathscr D}))\times {\bf R}$| given by the closure of the union of the convex hulls of the images of |$H^0({\mathscr C},m{\mathscr D})_{({\mathscr F},\varepsilon)}$| under |$\varrho_{m,p}$| for |$m\in{\bf Z}_{\geq 1}$|⁠. □ For a divisor |$\Lambda$| on |$\Sigma$| and vertex |$v\in V(\Sigma)$|⁠, let the v-enhanced effective linear system|$L^+_v(\Lambda)$| be the subset of |$L^+(\Lambda)\times {\bf R}$| given by \[L^+_v(\Lambda)=\{(\varphi,u)\mid \varphi\in L^+(\Lambda),\ 0\leq u\leq \Delta(\varphi)(v)+\Lambda(v)\}.\] Observe that if |$\Lambda=\rho({\mathscr D})$| for a horizontal divisor |${\mathscr D}$| on |${\mathscr C}$|⁠, the quantity |$\Delta(\varphi)(v)+\Lambda(v)$| is exactly the degree of the divisor |$\sum_w \varphi(w)C_v+{\mathscr D}$| restricted to |$C_v$|⁠. So, the second component measures the degree-theoretic lower and upper bounds of the multiplicity of the horizontal part of the zero locus of |$s$| at a smooth point |$p$| of |$C_v$|⁠. We have the following extension of Theorem 6.6. Theorem 6.17. Let |$\varepsilon'>0$|⁠. Let |${\mathscr C}$| be a semistable curve over |${\bf Spec}\, {\mathcal O}$| with horizontal divisors |${\mathscr D}$| and |${\mathscr F}$| such that the generic fibre of |${\mathscr D}$| has positive degree. Let |$p$| be a smooth point on a component of |$C_v$| of the closed fibre of |${\mathscr C}$|⁠. We have the equality between the |${\mathscr F}$|-controlled |$p$|-enhanced Newton–Okounkov linear system and the |$v$|-enhanced effective linear system: \[L^+_{\Delta,p}({\mathscr D})_{({\mathscr F},\varepsilon')}=L_v^+(\rho({\mathscr D})).\] □ Proof. This is proved by the same method as Theorem 6.6. Set |$\Lambda=\rho({\mathscr D})$|⁠. Because the multiplicity at |$p$| of the horizontal components of the zero locus of a section |$s\in H^0({\mathscr C},{\mathcal O}(m{\mathscr D}))$| is nonnegative and bounded above by |$\Delta(\varphi_s)(v)+m\Lambda(v)$|⁠, it follows that |$L^+_{\Delta,p}({\mathscr D})_{({\mathscr F},\varepsilon')}\subseteq L_v^+(\rho({\mathscr D})).$| Now, we show that any |$(\vartheta,u')\in L_v^+(\rho({\mathscr D}))$| can be approximated within |$2\varepsilon$| by some |$\varrho_{M,p}(s)$| for some |$s\in H^0({\mathscr C},M{\mathscr D})_{({\mathscr F},\varepsilon')}$|⁠. Write |$u'=u(\Delta(\vartheta)(v)+\Lambda(v))$| for some |$u$| with |$0\leq u \leq 1$|⁠. Pick a positive integer |$m$| such that (1) It is sufficiently divisible that |$m\vartheta$| takes integer values, and (2) We have the inequalities |$\operatorname{diam}(\Sigma)/m<\varepsilon$|⁠, |$3/m<\varepsilon$|⁠, and |$1/m<\varepsilon'$|⁠. As before, we choose an integral effective divisor |$E$| on |$\Sigma$| such that |$\Delta(m\vartheta)+m\Lambda-E$| is an effective divisor of degree equal to |$1$|⁠. We pick |${\mathscr E}$| such that (1) We have |$\rho({\mathscr E})=M_0 E$|⁠, (2) The divisor |${\mathscr E}$| has no components in common with |${\mathscr F}$|⁠, and (3) We have that |$\left|{\rm{mult}}_p({\mathscr E})/M_0-uE(w)\right|<1.$| Again, the line bundle |${\mathcal O}(mM_0{\mathscr D}_{\bf K}-{\mathscr E}_{\bf K})$| on |${\mathscr C}_{\bf K}$| has a regular section by the Riemann–Roch theorem. Therefore, there is a rational function |$s$| on |${\mathscr C}_{\bf K}$| with \[(s)+mM_0{\mathscr D}_{\bf K}-{\mathscr E}_{\bf K}\geq 0.\] By replacing |$s$| by |$\pi^ks$| for some integer |$k$|⁠, we may ensure that |$s$| is regular on the generic points of the components of the closed fibre and does not vanish identically on all of them. We will show that for |$M=mM_0$|⁠, |$\left|\varrho_M(s)-\vartheta\right|<\varepsilon.$| Considering |$s$| as a rational function on |${\mathscr C}_{\bf K}$|⁠, we have \begin{eqnarray} \label{e:splitting} (s)&=&(-M{\mathscr D}_{\bf K}+{\mathscr E}_{\bf K})+((s)+M{\mathscr D}_{\bf K}-{\mathscr E}_{\bf K}). \end{eqnarray} (6.18) Therefore, the zero locus of |$s$|⁠, considered as a regular section of |${\mathcal O}(M{\mathscr D}_{\bf K})$| on |${\mathscr C}_{\bf K}$| is |$(s)+M{\mathscr D}_{\bf K}=({\mathscr E}_{\bf K})+((s)+M{\mathscr D}_{\bf K}-{\mathscr E}_{\bf K}).$| By construction, the first term in parentheses on the right has no component in common with |${\mathscr F}$|⁠. The second term in parentheses on the right has degree equal to |$M_0$|⁠. Therefore, the multiplicity of |${\mathscr F}$| in the zero locus of |$s$| is at most |$M_0$|⁠. Consequently, as long as |$M_0/(mM_0)<\varepsilon'$|⁠, |$s\in H^0({\mathscr C},m{\mathscr D})_{({\mathscr F},\varepsilon')}$|⁠. The multiplicity at |$p$| of the first term in (6.18) is |$-M{\rm{mult}}_p({\mathscr D})+{\rm{mult}}_p({\mathscr E})$| while the multiplicity at |$p$| of the second term is at most |$M_0$|⁠. By construction of |$E$|⁠, we have \begin{eqnarray} \label{e:constrofE} \left|M\Delta(\vartheta)(v)+M\Lambda(v)-M_0E(v)\right|&\leq&M_0. \end{eqnarray} (6.19) From (6.18), we have \begin{equation}\label{e:decompofv} \frac{1}{M}v_p\big((s)+M{\mathscr D}\big)=\frac{{\rm{mult}}_p({\mathscr E})}{M}+\frac{1}{M}v_p\big((s)+M{\mathscr D}_{\bf K}-{\mathscr E}_{\bf K}\big). \end{equation} (6.20) From (6.19), the construction of |${\mathscr E}$|⁠, (6.20), and then the bound on the second term in (6.18), \begin{eqnarray*} \left|\frac{1}{M}v_p\big((s)+M{\mathscr D}\big)-u\left(\Delta(\vartheta)(v)+\rho({\mathscr D})(v)\right)\right|&\leq&\frac{1}{M}\left|v_p\big((s)+M{\mathscr D}\big)-uM_0E(v)\right|+\frac{M_0}{M}\\ &\leq&\frac{1}{M}\left|v_p\big((s)+M{\mathscr D}\big)-{\rm{mult}}_p({\mathscr E})\right|+\frac{M_0}{M}+\frac{M_0}{M}\\ &\leq&\frac{1}{M}\left|v_p\big((s)+M{\mathscr D}_{\bf K}-{\mathscr E}_{\bf K}\big)\right|+2\frac{M_0}{M}\\ &\leq&3\frac{M_0}{M}\\ &=&\frac{3}{m}\\ &<&\varepsilon. \end{eqnarray*} It follows that |$\varrho_{M,p}(s)$| is within |$2\varepsilon$| of |$(\vartheta,u')$|⁠. ■ 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 |$\{{\mathscr C}\supsetneq Y_1 \supsetneq Y_2\}$| where |$Y_1$| is a horizontal divisor and |$Y_2=\{p\}$| is a smooth point of the closed fibre. Theorem 6.21. Suppose that |$Y_1$| is a horizontal divisor intersecting the closed fibre in smooth points and that |$Y_2=\{p\}$| is on the component |$C_v$|⁠. Let |${\mathscr D}$| be a horizontal divisor whose generic fibre has positive degree. Moreover, suppose that |$p$| is not contained in |${\mathscr D}$|⁠. For |$t\in {\bf R}$|⁠, let \[L_t=L^+(\rho({\mathscr D}-t Y_1)).\] The Newton–Okounkov body is the overgraph in |${\bf R}^2$| of \begin{align*} a\colon [0,\deg({\mathscr D})/\deg(Y_1)]&\rightarrow {\bf R}\\ t&\mapsto\varpi_{L_t}(v). \end{align*} □ Proof. We first show that the Newton–Okounkov body lies above the graph of |$a$|⁠. If |$s\in H^0({\mathscr C},m{\mathscr D})$| is a section vanishing to order |$mt$| along |$Y$| (with |$t\in{\bf Q}$|⁠), then |$\varrho_m(s)\in L_t$|⁠. Furthermore, |$\nu_2(s)$| is always greater than or equal to the multiplicity of |$C_v$| in the principal divisor |$(s)$|⁠. Therefore, \[ \frac{1}{m}\nu_2(s)\geq \frac{1}{m}\varphi_s(v)\geq \min(\varphi(v)\mid \varphi\in L_t)=\varpi_{L_t}(v). \] To show that the Newton–Okounkov body is exactly the overgraph of |$a$|⁠, we will show that for fixed rational |$t$| with |$0< t<\deg({\mathscr D})/\deg(Y_1)$|⁠, there is an |$m\in{\bf Z}_{\geq 1}$| (with |$mt\in{\bf Z}$|⁠) and a section |$s\in H^0({\mathscr C},{\mathcal O}(m{\mathscr D}))$| such that |$\frac{1}{m}\nu(s)$| is close to |$(t,\varpi_{L_t}(v))$|⁠. Set |${\mathscr F}=Y_1$| and pick |$\varepsilon>0$|⁠. By definition, a section |$s\in H^0({\mathscr C},m({\mathscr D}-t{\mathscr F}))_{({\mathscr F},\varepsilon)}$| vanishes to order at most |$m\varepsilon$| along |$Y_1$|⁠. Therefore, considering |$s$| as a rational function, we have |$t\leq \frac{1}{m}\nu_1(s)\leq t+\varepsilon$|⁠. The point |$(\varpi_{L_t},0)$| is in |$L_v^+(\rho({\mathscr D}-tY_1))$| and by Theorem 6.17, we may find |$m$| large and divisible and |$s\in H^0({\mathscr C},m({\mathscr D}-t{\mathscr F}))_{({\mathscr F},\varepsilon)}$| such that |$\varrho_{m,p}(s)$| is close to |$(\varpi_{L_t},0)$|⁠. Being close in the second coordinate means that the multiplicity of |$p$| in the horizontal component of the zero locus of the section |$s$| is small. In particular, there are not many horizontal components of |$s$| passing through |$p$|⁠. Restricting |$s$| to |$Y_1$|⁠, this means that the order of vanishing at |$p\in Y_1\cap C_v$| comes mostly from vanishing along the vertical component |$C_v$|⁠. However, being close in the first coordinate means that |$\frac{1}{m}\varphi_s$| is close to |$\varpi_{L_t}$|⁠, and thus the vanishing order is close to |$\varpi_{L_t}(v)$|⁠, that is, |$\frac{1}{m}\nu_2(s)$| is close to |$\varpi_{L_t}(v)$|⁠. ■ Now, we consider the Arakelovian case. Theorem 6.22. Let |${\mathscr D}$| be a horizontal divisor. Suppose that |$Y_1$| is a component |$C_v$| of the closed fibre and |$Y_2=\{p\}$| is a smooth point on |$C_v$| not contained in |${\mathscr D}$|⁠. Let |$L_t\subset L^+(\rho(D))$| be the tropical sub-semigroup of elements |$\varphi$| with |$\varphi(v)=t$|⁠. Then the Newton–Okounkov body of |${\mathcal O}(D)$| is the set of points between the graphs of |$a(t)=0$| and \[b(t)=\rho({\mathscr D})(v)+\max(\Delta(\varphi)(v)\mid \varphi\in L_t)\] for |$t\geq 0$|⁠. □ Proof. Observe that for |$s\in H^0({\mathscr C},m{\mathscr D})$|⁠, |$\nu_1(s)=\varphi_s(v)$| and \[\nu_2(s)=v_p\big((s)+m{\mathscr D}\big).\] The Newton–Okounkov body is, therefore, the image of |$L^+_{\Delta,p}({\mathscr D})=L_v^+(\rho({\mathscr D}))$| under the map |$(\varphi,u)\mapsto (\varphi(v),u)$|⁠. The conclusion then follows from Theorem 6.17. ■ Example 6.23. We conclude by giving an example of the Newton–Okounkov body for curves in the tropical and Arakelovian cases for the same linear system. Let us consider the example in [1, Section 4D] of a particular family |$X$| of plane quartic curves. This is a plane quartic degenerating into a conic |$C$| and two lines |$\ell_1,\ell_2$|⁠. To make the model semistable, one must blow up the intersection point of |$\ell_1$| and |$\ell_2$|⁠, introducing a new component |$E$| of the degeneration. For this curve, the closed fibre and the dual graph are given in the figure: the vertex |$P$| corresponds to the conic; |$Q_1,Q_2$| corresponds to the lines; and |$P'$| corresponds to the curve |$E$|⁠. We will compute the Newton–Okounkov body for a general hyperplane section |${\mathscr D}$|⁠, whose specialization is given by |$\rho({{\mathscr D}})=\Lambda=2(P)+(Q_1)+(Q_2)$|⁠. For any |$\varphi \in L(\Lambda)$| we have \begin{eqnarray*} \Delta(\varphi)&=&(4\varphi(P) -2\varphi(Q_1)-2\varphi(Q_2))(P)\\[-4pt] &+&(3\varphi(Q_1) -\varphi(P')-2\varphi(P))(Q_1)\\[-4pt] &+&(3\varphi(Q_2) -\varphi(P')-2\varphi(P))(Q_2)\\[-4pt] &+&(2\varphi(P') -\varphi(Q_1)-\varphi(Q_2))(P'). \end{eqnarray*} Tropical case: we will pick as a flag |$\{{\mathscr C}=Y_0\supsetneq Y_1 \supsetneq Y_2\}$|⁠, with |$Y_1$| a degree one horizontal divisor, intersecting the generic fibre |${\mathscr C}_{\bf K}$| in a general point and intersecting the closed fibre in a generic point |$Y_2$| of the conic |$C$|⁠. It is straightforward to compute |$\varpi_{L_t}$| for |$t\in [0,4]$| as follows: – for |$t\in [0,2]$|⁠, |$\varpi_{L_t}=0$|⁠, – for |$t\in [2,4]$|⁠, |$\varpi_{L_t}(P)=(t-2)/4$|⁠, |$\varpi_{L_t}(Q_1)=\varpi_{L_t}(Q_2)=\varpi_{L_t}(P')=0.$| This gives the following Newton–Okounkov body: Arakelovian case: we will pick as a flag |$\{{\mathscr C}=Y_0\supsetneq Y_1 \supsetneq Y_2\}$| where |$Y_1$| is the conic |$C$| and |$Y_2$| is a general point of |$C$|⁠. The function |$b(t)$| is achieved by the following choices for |$\varphi$|⁠: – for |$t\in [0,1/2]$|⁠, |$\varphi(P)=t,\ \varphi(Q_1)=\varphi(Q_2)=\varphi(P')=0$|⁠, – for |$t\in [1/2,\infty)$|⁠, |$\varphi(P)=t,\ \varphi(Q_1)=\varphi(Q_2)=\varphi(P')=t-1/2.$| Therefore, the Newton–Okounkov body is as follows: □ The two examples are reflections of each other because in the tropical case, we chose |$Y_1$| with |$\rho(Y_1)=P$|⁠, the vertex corresponding to the conic while in the Arakelovian case, we chose |$Y_1=C$|⁠, the conic. 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TI - Newton–Okounkov Bodies over Discrete Valuation Rings and Linear Systems on Graphs
JO - International Mathematics Research Notices
DO - 10.1093/imrn/rnx248
DA - 2019-07-22
UR - https://www.deepdyve.com/lp/oxford-university-press/newton-okounkov-bodies-over-discrete-valuation-rings-and-linear-0TK3sJyZBc
SP - 4516
VL - 2019
IS - 14
DP - DeepDyve
ER -