Newton-Okounkov Bodies over Discrete Valuation Rings and Linear Systems on Graphs
aa r X i v : . [ m a t h . AG ] N ov NEWTON–OKOUNKOV BODIES OVER DISCRETE VALUATION RINGSAND LINEAR SYSTEMS ON GRAPHS
ERIC KATZ AND STEFANO URBINATI
Abstract.
The theory of Newton–Okounkov bodies attaches a convex body to a line bun-dle on a variety equipped with flag of subvarieties. This convex body encodes the asymptoticproperties of sections of powers of the line bundle. In this paper, we study Newton–Okounkovbodies for schemes defined over discrete valuation rings. We give the basic properties andthen focus on the case of toric schemes and semistable curves. We provide a descriptionof the Newton–Okounkov bodies for semistable curves in terms of the Baker–Norine theoryof linear systems on graphs, finding a connection with tropical geometry. We do this byintroducing an intermediate object, the Newton–Okounkov linear system of a divisor on acurve. We prove that it is equal to the set of effective elements of the real Baker–Norinelinear 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
The theory of Newton polytopes [6, Ch. 6] is a bridge between algebraic geometry andpolyhedral geometry. Specifically, if one is given a Laurent polynomial f = X ω ∈A c ω x ω ∈ k [ x ± , . . . , x ± n ]where A ⊂ Z n is a finite set, k is a field, and c ω ∈ k ∗ , then the Newton polytope of f is the convex hull of A . The Newton polytope can be used to understand the intersectiontheory of the zero locus Z ( f ) ⊂ ( k ∗ ) n through Bernstein’s theorem. There are several waysto generalize Newton polytopes beyond hypersurfaces of the algebraic torus ( k ∗ ) n . Tropicalgeometry is one such way, handling higher codimensional subvarieties of tori by studyingpolyhedral fans instead of polytopes. Another way, the theory of Newton–Okounkov bodies[16, 10, 14] attaches a convex body to a smooth projective d -dimensional algebraic variety X equipped with a divisor D and a flag of subvarieties Y • : X = Y ) Y ) ... ) Y d − ) Y d = { pt } . Specifically, the Newton–Okounkov body encodes the generic vanishing of sections of O ( mD )for m ∈ Z ≥ . It is convex and bounded and has many of the desirable properties of Newtonpolytopes. However, except in low dimensions, it may be non-polyhedral. Moreover, itencodes information about the asymptotic behavior of O ( mD ) making it much more difficultto compute. The second author was supported by the European Commission, Seventh Framework Programme, GrantAgreement n ◦ ewton polytopes and tropical geometry both have extensions to the “non-constant co-efficient case.” Specifically, if one has a valued field K with val : K → R (say, the fractionfield of a discrete valuation ring O ), one obtains an unbounded convex body by attaching to f = X ω ∈A c ω x ω ∈ K [ x ± , . . . , x ± n ] , the upper hull which is defined to be the convex hull of the set { ( ω, t ) | ω ∈ A , t ≥ val( a ω ) } ⊂ R n × R . The lower faces of this set induce a subdivision of the Newton polytope P ( f ) [6, 7]. Thisis the Newton subdivision. The lower faces of the upper hull are the graph of a piecewiselinear convex function ψ : P ( f ) → R . In greater generality, tropical geometry attaches apolyhedral complex to a subvariety X ⊂ ( K ∗ ) n .The purpose of this paper is to consider Newton–Okounkov bodies in the non-constantcoefficient case. To set notation, 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 that is proper, flat, and of finite type over O whose genericfiber is smooth and whose closed fiber is a reduced normal crossings divisor. Let X be aprojective regular semistable scheme over O with closed fiber X . Let Y • denote a descendingflag of subschemes X = Y ) Y ) ... ) Y d +1 where each Y i is a codimension i subscheme that is either a semistable scheme over O or asmooth subvariety of a component of the closed fiber X × O k . Let D be a divisor on X flat over Spec O .We will define a Newton–Okounkov body, ∆ Y • ( D ) ⊂ R d +1 by considering sections of O ( m D ) for m ∈ Z ≥ evaluated at a valuation attached to the flag. In contrast to the classicalcase but similar to upper hulls, this Newton–Okounkov body will be unbounded, albeit in asingle direction. This unboundedness is a consequence of the fact that if s ∈ H ( X , O ( D )),then π k s ∈ H ( X , O ( D )) for any k ≥
0. This is formalized by the following theorem:
Theorem 1.1.
Let p π : R d +1 → R d be projection along the direction through the valuationvector of π . The image ∆ = p π (∆ Y • ( O ( D )) is compact. In fact, if Y d is semistable over O and Y d +1 is a point of Y d × O k , we are in what we callthe tropical case and more can be said: Theorem 1.2.
In the tropical case, ∆ Y • ( D ) is given by the overgraph of a convex function ψ : p π (∆ Y • ( D )) → R . Here, the overgraph is the set of points of p π (∆ Y • ( D )) × R lying above the graph of ψ .We will give a complete description in the special case of Newton–Okounkov bodies oftoric schemes with respect to a toric flag. Here, it is analogous to the field case and involvesa particular polyhedron P D depending on D , and φ R , a certain linear map depending on Y • . heorem 1.3. Let D be a torus-invariant divisor on a toric scheme X that is flat over Spec O with generic fiber D such that O ( D ) is a big line bundle on X . Then, the Newton–Okounkov body of O ( D ) is given by ∆ Y • ( D ) = φ R ( P D ) . Newton–Okounkov bodies over 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 capturesonly the degree of the divisor. In the case of surfaces, the Newton–Okounkov body is apolytope encoding the Zariski decomposition of a family of divisors. The case of semistablecurves over discrete valuation rings shares features of both of these cases. This is perhaps notsurprising because such a curve is of relative dimension 1 and absolute dimension 2. We willgive 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 languagereminiscent of the Zariski decomposition of divisors.To describe the case of semistable curves, we choose to introduce an intermediate object,the Newton–Okounkov linear system , L +∆ ( D ). Like Newton–Okounkov bodies, it measurethe asymptotics of vanishing orders of sections of O ( m D ) for m ∈ Z ≥ . However, instead ofincorporating vanishing orders on a flag, it incorporates vanishing orders on components ofthe special fiber. It is a convex subset of the space of functions ϕ : V (Σ) → R where Σ isthe dual graph of the closed fiber of C . We prove that the Newton–Okounkov linear systemis 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 curve C : Theorem 1.4.
If the generic fiber of D has positive degree, then we have the equality betweenthe Newton–Okounkov linear system and the effective linear system: L +∆ ( D ) = L + ( ρ ( D )) . By incorporating the lattice of integer-valued functions ϕ : V (Σ) → Z , this theorem givesan algebraic geometric description of rank in the Baker–Norine theory, a problem studiedin work of Caporaso–Len–Melo [5]. However, our description works only by studying theasymptotics of sections, and we do not know how to use our results to give algebraic geometricproofs of combinatorial results in the Baker–Norine theory.After enriching the above theorem by incorporating vanishing orders of sections at a pointin the closed fiber of C , we are able to give a description of the Newton–Okounkov bodiesof curves in Theorem 6.17 and Theorem 6.18.We should note that recent, quite different connections between tropical geometry andNewton–Okounkov bodies were recently found by Kaveh–Manon [11].1.1. Acknowledgments.
We would like to thank David Anderson and Alex K¨uronya forenlightening conversations and helpful comments. . Notation and conventions
For a scheme X over O , we will use X K to denote its generic fiber. If C is a semistablecurve over O , irreducible divisors are either horizontal or vertical, see, for example, [15,Sec. 8.3]. Here, horizontal divisors are those that that are flat over Spec O while verticaldivisors are contained in the closed fiber. Any divisor on C can be decomposed into a sumof horizontal and vertical divisors.Recall that a convex cone in R d is a convex set invariant under rescaling by elements of R ≥ . For a set S ⊂ R d , we write cone R d +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 intwo ways: as a section of a line bundle or as a rational function s whose principal divisorsatisfies ( s ) + D ≥
0. Consequently if D ≤ E , a section of O ( D ) can be interpreted as asection of O ( E ) although its zero locus will differ by E − D . We will point out the relevantinterpretation 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 bea smooth irreducible projective variety of dimension d over a field K . Given a divisor D onX, we want to construct a convex compact subset of R d called the Newton–Okounkov bodyof D . A flag of smooth irreducible subvarieties Y • : X = Y ) Y ) ... ) Y d − ) Y d = { pt } is called full and admissible if codim Y i = i . For every non-zero section s of O ( D ), if s := s ,for i = 1 , . . . , n define(3.1) ν i ( s ) := ord Y i ( s i − ) , s i := s i − g ν i ( s ) i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Y i , where g i is the local equation of Y i in Y i − near Y n . Here, s is considered as a section of L = O ( D ) while s i is considered as a section on Y i of L i = L i − | Y i ⊗ O Y i ( − ν i ( ϕ ) Y i ). Weobtain the vector ν ( s ) = ( ν ( s ) , . . . , ν d ( s )) ∈ Z n . Define the semigroup of valuation vectorsas Γ Y • ( D ) := { ( ν ( s ) , m ) ∈ Z n × Z ≥ | s ∈ H ( X, O X ( mD )) } and the Newton–Okounkov body of D as∆ Y • ( D ) := cone R d +1 (Γ Y • ( D )) ∩ ( R n × { } ) . We write Γ Y • ( D ) m for Γ Y • ( D ) ∩ ( Z n × { m } ). The same construction can be performed fornon-complete graded linear series as well, see [10]. .1. The case of curves.
Let us consider a curve C of positive genus g and a divisor D of degree d >
0. Let Y • = { C ) p } , with p a point. Then the Newton–Okounkov body isthe segment [0 , d ] by [14, Ex. 1.2] as a consequence of the Riemann-Roch theorem. This isa result that we will eventually generalize to the case of semistable curves over a discretevaluation ring.3.2. The case of surfaces.
In the case of surfaces, the Zariski decomposition of big divi-sors can be used to show that the Newton–Okounkov body lies between the graphs of twofunctions on an interval [14, Sec. 6.2]. This description provides all the information aboutpossible shapes of Newton–Okounkov bodies of surfaces.Any pseudoeffective divisor D (that is, a divisor in the closure of the effective cone in theNeron–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 Y • = { X ) C ) p } suchthat p / ∈ supp( N D ).Let α ( D ) = ord p ( N D ), β ( D ) = α ( D ) + C · P D , and µ := sup { x | D − xC is big } . Then,we have by the recipe given in [14]∆ Y • ( D ) = { ( x, y ) ∈ R | ≤ x ≤ µ and α ( D − xC ) ≤ y ≤ β ( D − xC ) } . Example 3.2.
Let X be the blow up of P at two points with exceptional divisors E , E and consider the flag Y • = { X ) l ) p } given by a general line and a general point on it.Let H denote the pullback of the class of a line in P .Let D = 2 H − E − E . In this case we have µ = 1 and the Zariski decomposition of D − xH is the sum (1 − x )(2 H − E − E ) + x ( H − E − E ), obtaining the following body: Newton–Okounkov bodies in higher dimensions can be much more complicated and thereis no general strategy for writing them down. We will see that for toric schemes with respectto a torus-invariant flag, the Newton–Okounkov body is determined by combinatorics.4.
Newton–Okounkov bodies over discrete valuation rings
Definition for schemes over discrete valuation rings.
We now describe the caseof Newton–Okounkov bodies on schemes over discrete valuation rings. Let O be a discretevaluation ring with fraction field K , residue field k , and valuation val. Let π be a uniformizerof O . Let X be an d -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 X = Y ) Y ) ... ) Y d +1 here each Y i is a codimension i subscheme that is either a semistable scheme over O or aproper 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 − is semistable over O and Y j is a closed subvariety of X × O k . Then Y j is a component of Y j − × O 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 tropicalcase . Here, the name “Arakelovian” is motivated by a construction by Yuan [18]. Thename “tropical” is motivated by a fundamental notion in tropical geometry, the Newtonsubdivision [7] 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.The Newton–Okounkov body ∆ Y • ( D ) is defined as a convex body in R d +1 exactly as aboveby considering sections of O ( m D ) over X for m ∈ Z ≥ evaluated at the valuations attachedto the flag.4.2. Boundedness.
In contrast to the field case, Newton–Okounkov bodies over discretevaluation rings are not bounded. However, the failure of boundedness can be preciselydescribed.
Lemma 4.1.
Let ν ( π ) ∈ R d +1 be the valuation of the uniformizer π ∈ O , viewed as a rationalfunction on X . The Newton–Okounkov body ∆ Y • ( D ) is closed under positive translationsin the ν ( π ) -direction.Proof. It suffices to show Γ Y • ( D ) + k ( ν ( π ) , ⊂ Γ Y • ( D ) for any k ∈ Z ≥ . Any point ofΓ Y • ( D ) is of the form ( ν ( s ) , m ) for s ∈ H ( X , O ( m D )). Now, π k s ∈ H ( X , O ( m D )) and ν ( π k s ) = ν ( s ) + kν ( π ). (cid:3) The Newton–Okounkov body is bounded in other directions. Let p π : R d +1 → R d +1 / ( R ν ( π ))be the projection along the ν ( π )-direction Lemma 4.2.
The image under projection, p π (∆ Y • ( D )) is bounded.Proof. Choose an ample divisor H on X and an ample divisor h on Y j , which is a componentof the projective variety Y j − × O k .We will follow [14, Lemma 1.10]. We begin with the following observation: given a divisor D and an irreducible divisor Y on some scheme Z , there exists an integer b such that forany section s m of O ( mD ), the vanishing order of s m on Y is at most mb . Indeed, choose b sufficiently large such that ( D − bY ) · H d − <
0. Multiplying this inequality by m , we seethat O ( mD − mbY ) cannot have any regular sections.We claim that there exists positive integers b , . . . , b j − such that for any section s ∈ H ( X, O ( m D K )), we have ν i ( s ) ≤ mb i , for i = 1 , . . . , j − . Given s ∈ H ( X , O ( m D )), restriction to the generic fiber gives s ∈ H ( X, O ( m D K )). Wechoose b as in the above paragraph. Then, v is given by the vanishing order at Y of therestriction of s to Y , considered as a section of O ( D ,a ) = O ( D ) | Y ⊗ O Y ( − aY ) for some a ith 0 ≤ a ≤ b . Choose b to be the max of the b ’s produced in the above paragraph for D ,a for 0 ≤ a ≤ b . We continue by defining b i ’s inductively.Now, there are finitely many line bundles on Y j − whose sections we will restrict to Y j . Wewill take the maximum of the b ’s chosen for all line bundles. Therefore, it suffices to workwith one line bundle O ( D ) on Y j − at a time. Because we are restricting our attention to Y j − , it also suffices to prove the result for j = 1. Let us consider elements of H ( X, O ( D )).These are exactly rational sections of O ( m D ) over X that are allowed to have poles alongcomponents of the closed fiber. By multiplying such sections by a suitable power of π , wecan ensure that the section is regular. Therefore, the Newton–Okounkov body associatedto H ( X, O ( D )) with respect to the flag Y ) ... ) Y d is exactly the Minkowski sum of theNewton–Okounkov body of H ( X , O ( D )) and the line R ν ( π ). Since we’re only interestedin the projection of that Newton–Okounkov body along the line through ν ( π ), it suffices toconsider the complementary slice given by elements of H ( X, O ( D )) that have neither polesnor zeroes generically along Y .Now, the sections under consideration restrict to sections on Y . We may now apply theabove argument with H replaced by h to obtain b , . . . , b d . (cid:3) Remark . In the tropical case, if Y d +1 is a smooth point of the central fiber, p π is theprojection along the ( d + 1)-st component.4.3. The tropical case.
We now consider the tropical case where the admissible flag Y • isgiven by semistable schemes Y , . . . , Y d in X and a point Y d +1 in the closed fiber X k . Let D be a divisor on X flat over Spec O whose generic fiber is D ⊂ X .We will relate the Newton–Okounkov bodies to overgraphs. Recall that for ∆ ⊂ R d , aconvex body and a convex function ψ : ∆ → R , we define its overgraph in R d +1 to be theset { ( x, t ) | x ∈ ∆ , t ≥ ψ ( x ) } . If ψ is piecewise linear, its domains of linearity give a subdivision of ∆. Theorem 4.4.
We have a surjection of Newton–Okounkov bodies, p π : ∆ Y • ( D ) → ∆ Y • ( D ) . Moreover, ∆ Y • ( D ) is given as the overgraph of a convex function ψ : ∆ Y • ( D ) → R where ∆ Y • ( D ) is the Newton–Okounkov body of D = D × O K on X = X × O K with respectto Y • = { X ) Y × O K ) ... ) Y d × O K } . Proof.
Because any section s ∈ H ( X, O ( D )) has some multiple by π satisfying π k s ∈ H ( X , O ( D )), we have that the projection p π : R d +1 → R d maps ∆ Y • ( D ) surjectively to∆ Y • ( D ). ow, ∆ Y • ( D ) is closed under positive translation by e d +1 = ν ( π ). From the convexity ofNewton–Okounkov bodies, it follows that ∆ Y • ( D ) is the set of points in R d +1 lying abovethe graph of a convex function ψ : ∆ Y • ( D ) → R . (cid:3) We note that Newton–Okounkov bodies of schemes over discrete valuation rings need notbe polyhedral. Indeed, one may take a semistable model of a variety with a flag over K thatalready has a non-polyhedral Newton–Okounkov body [13] and extend the flag to a tropicalone. Remark . Our construction of functions on Newton–Okounkov bodies has some relationto the filtered linear systems that have appeared in the work of Boucksom–Chen [3], WittNystrom [17], and Yuan [19]. Let X be an algebraic variety or scheme equipped with a flag ofsubvarieties or subschemes. Suppose that there is a family of norms || s || m on sections of theline bundles L ⊗ m = O ( mD ) (possibly as the sup-norm coming from a metric on O ( D )). Thefunction on the Newton–Okounkov body is induced by considering the infimum of m log || s || m of sections s of O ( mD ) corresponding to a point m ν ( s ) in the Newton–Okounkov body. Theinduced function, called a Chebyshev transform is related to metric and adelic volumes inK¨ahler and Arakelov geometry. More generally, one may even consider filtrations on sectionsof mL induced by valuations as in the study vanishing sequences by Boucksom–K¨uronya–Maclean–Szemberg [4]. 5. Toric schemes
In this section, we discuss toric schemes over discrete valuation rings. See [12] for a classicalsource or [7] for a rigid analytic perspective.5.1.
Toric varieties.
We begin by reviewing Newton–Okounkov bodies for smooth projec-tive toric varieties [14, Section 6.1]. Let N be an d -dimensional lattice. A toric variety X (∆)is specified by a complete rational fan ∆ in N R = N ⊗ R ∼ = R d . The variety X (∆) is smoothif and only if the fan is unimodular, that is, the fan is simplicial and every cone is spanned byinteger vectors forming a subset of a basis of N . Let T = N ⊗ K ∗ denote the d -dimensionalalgebraic torus acting on X (∆). To each k -dimensional cone σ of ∆, there corresponds anorbit closure V ( σ ) which is a codimension k subvariety. Any torus-invariant divisor is givenby D = X σ a σ V ( σ )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 P D = Conv( { m ∈ M | h m, u σ i ≥ − a σ } ) . where u σ ∈ N is the primitive integer vector (with respect to N ) along σ . This polyhedronarises by considering sections of O ( D ): the vector space of sections H ( X, O ( D )) has adecomposition into T -eigenspaces; the lattice points of P D are exactly the characters of T that arise; for m ∈ P D , the character χ m on T extends to a section of O ( D ) on X . Indeed, the anishing order of χ m (considered as a section of O ( D )) on the divisor V ( σ ) is h m, u σ i + a σ so the inequalities defining P D are exactly the conditions that χ m is regular at the genericpoint of the torus-invariant divisors. Consequently, dim H ( X, O ( D )) = | P D ∩ M | . The linebundle O ( D ) is big if and only if P D is d -dimensional.Because X (∆) is smooth, a T -invariant flag Y , Y , . . . , Y n can be written as Y i = D ∩ · · · ∩ D i for a choice of T -invariant divisors D , . . . , D n corresponding to rays σ , . . . , σ n . Let u , . . . , u n be the primitive integer vectors along σ , . . . , σ n . We define a linear map φ : M R → R n by φ ( v ) = (cid:0) h v, u σ i i + a σ i (cid:1) ≤ i ≤ n . We have the following equality for big line bundles O ( D ):∆ Y • ( D ) = φ ( P D ) . Toric schemes.
Complete toric schemes over a discrete valuation ring O are describedby complete rational fans in N R × R ≥ where N ∼ = Z d is a lattice. Given such a fan Σ, there isa natural morphism of toric varieties X (Σ) Z → X ( R ≥ ) Z = A Z and the toric scheme is givenby X = X (Σ) × A Spec( O ). Here, we will map t , the coordinate on A , to the uniformizerof 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 × { } ), then the generic fiber of X is the toric variety X = X (∆). The closed fiber of X is a union of toric varieties described combinatoriallyby the polyhedral complex Σ = Σ ∩ ( N R × { } ) in N R × { } . The components of theclosed fiber are in bijective correspondence with the vertices of Σ . We will suppose that thevertices of Σ are at points of N × { } which ensures that X has reduced closed fiber and,therefore, is semistable. 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 = P 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 D = X σ a σ V ( σ ) + X v a v V ( v )where a v ∈ Z and V ( v ) are the divisors on X corresponding to rays in N R × R ≥ throughthe vertices of Σ .We construct the Newton–Okounkov body. Considering the total space X (Σ) as an ( d +1)-dimensional toric variety, we define a polyhedron P D ⊂ M R × R ≥ by P D = Conv (cid:16)(cid:8) ( m, h ) ∈ M R × R ≥ | h m, u σ i ≥ − a σ , h m, v i + h ≥ − a v (cid:9)(cid:17) . The second set of inequalities come from v ∈ Σ corresponding to vertices ( v, ∈ Σ . Notethat this projects onto P D by ( m, u ) m . We may define a piecewise linear convex functionon P D , ψ ( m ) = max( − a v − h m, v i )where v is taken over vertices of Σ . Then P D is the overgraph of ψ .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 [14], we may choose rreducible toric divisors D , . . . , D d +1 of X (Σ) such that Y i = D ∩ · · · ∩ D i . Suppose that D i corresponds to a ray in N R × R whose primitive integer vector is w i ∈ N × Z . Here, { w , . . . w d +1 } is a basis for N × Z . Write D = X a w V ( σ w )where w runs over primitive integer vectors of rays σ w of Σ.We define φ R : M R × R → R d +1 , φ : ( v, h ) (cid:16)(cid:10) ( v, h ) , w i (cid:11) + a w i (cid:17) ≤ i ≤ d +1 where the pairing is between M R × R and N R × R . We have the following analogue of [14,Prop. 6.1]: Theorem 5.1.
Let D be a torus-invariant divisor on X that surjects onto Spec O withgeneric fiber D such that O ( D ) is a big line bundle on X . Then,the Newton–Okounkov bodyof O ( D ) is given by ∆ Y • ( O ( D )) = φ R ( P D ) . Proof.
By replacing D with a positive integer multiple, we may suppose that the vertices of P D are points of N × Z .Write the restriction of s ∈ H (( X , O ( D ))) to X as s = X m c m χ m for c m ∈ 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 σ w of Σ is b w = min (cid:16)D(cid:0) m, val( c m ) (cid:1) , w E(cid:17) where val(0) = ∞ and the pairing is the one between M R × R and N R × R . Observethat in the above, if σ w is a ray of ∆, then h ( m, val( c m )) , w i = h m, u σ i N where the secondpairing is the pairing between M R and N R . If w = ( v,
1) corresponds to a vertex of Σ ,then D(cid:0) m, val( c m ) (cid:1) , w E = h m, v i N + val( c m ). On X , we have that following formula for theprincipal divisor: ( s ) = X b w D w . Note that the vanishing order of s , considered as a section of O ( D ), on D w is b w + a w .Consequently, a sum of characters like the above corresponds to an element of H (( X , O ( D ))if and only b w ≥ − a w for all w . In fact, π h χ m ∈ H (( X , O ( D )) if and only if ( m, h ) ∈ P D .Now, the valuation of such a section of O ( D ) is ν ( s ) = ( b w + a w , . . . , b w d +1 + a w d +1 ) . 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 ). (cid:3) . Newton–Okounkov bodies of curves
We will relate the Newton–Okounkov bodies of curves over O to the Baker-Norine theoryof linear systems on graphs.6.1. Review of linear systems on graphs.
We review some results on specialization oflinear systems from curves to graphs due to Baker [1]. Let C be a semistable curve overSpec O . The semistability condition ensures that the closed fiber C is reduced with onlyordinary double points as singularities. A node in the closed fiber of C is formally locallydescribed 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 : e C → C and whose edges E (Σ)correspond to nodes of C . For each vertex v ∈ V (Σ), we write C v for the correspondingcomponent of e C .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 morethan once and sum over v itself.A divisor on Σ is an element of the real vector space with basis V (Σ). We write a divisoras D = P v ∈ V (Σ) a v ( v ) with a v ∈ R . We may write D ( v ) = a v . The vector space of alldivisors is denoted by Div(Σ). We say a divisor D is effective and write D ≥ a v ≥ v ∈ V (Σ). We write D ≥ D ′ if D − D ′ ≥
0. The degree of a divisor is given bydeg( D ) = X v a v . We will study functions ϕ : V (Σ) → R . The Laplacian of ϕ , ∆( ϕ ) is the divisor on Σ givenby ∆( ϕ ) = X v ∈ V (Σ) X e ∈ E (Σ) | e = vw ( ϕ ( v ) − ϕ ( w ))( v ) . Note that ∆( ϕ ) is of degree 0.The specialization map ρ : Div( C ) → Div(Σ) is defined by, for
D ∈
Div( C ), ρ ( D ) = X v ∈ Γ deg( π ∗ O ( D ) | C v )( v ) . The specialization of a vertical divisor P v ϕ ( v ) C v satisfies ρ X v ϕ ( v ) C v ! = − ∆( ϕ ) . 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 ) ≥ Definition 6.2.
Let Λ be a divisor on Σ. We define the linear system L (Λ) to be the set offunctions ϕ : V (Σ) → R on Σ with ∆( ϕ ) + Λ ≥
0. The effective linear system L + (Λ) is thesubset of L (Λ) consisting of everywhere non-negative functions ϕ . et D be a divisor on C . Then we will interpret a global section of O ( D ) as a rationalfunction s on C such that ( s ) + D ≥
0. If we write ( s ) = H + V where H is a horizontaldivisor over O and V is a vertical divisor contained in the closed fiber, we may decompose V as V = P v ϕ s ( v ) C v where we call ϕ s : V (Σ) → Z the vanishing function of s . For s , arational function on C , we will abuse notation and take the vanishing function of s to bevanishing function of the extension of s to C .The following lemma is standard and we include the proof only for completeness. Lemma 6.3.
Let D be a divisor on C K whose closure D has specialization Λ = ρ ( D ) . For arational function s on C corresponding to a section of O ( D ) with vanishing function ϕ , wehave ∆( ϕ ) + Λ ≥ or, in other words, ϕ ∈ L (Λ) .Proof. Because s is principal, we have ( s ) · C v = 0 for all components of the closed fiber. Ifwe write ( s ) = H + P v ϕ ( v ) C v , we have0 = ρ (( s )) = ρ (cid:0) H + X v ϕ ( v ) C v (cid:1) = ρ ( H ) − ∆( ϕ ) . Since H + D ≥ ≤ ρ ( H ) + ρ ( D ) = ∆( ϕ ) + ρ ( D ) . (cid:3) The linear system L (Λ) has a tropical semigroup structure as noted in [8]: Lemma 6.4.
For ϕ , ϕ ∈ L (Λ) , let ϕ : V (Σ) → R be the pointwise minimum of ϕ , ϕ : V (Σ) → R . Then ϕ ∈ L (Λ) .Proof. Let v ∈ V (Σ). Without loss of generality, suppose that ϕ ( v ) = ϕ ( v ). Then∆( ϕ )( v ) + Λ( v ) = X e = vw ( ϕ ( v ) − ϕ ( w )) + Λ( v ) ≥ X e = vw ( ϕ ( v ) − ϕ ( w )) + Λ( v ) ≥ . (cid:3) Geometric and tropical linear systems.Definition 6.5.
Now, let D be a horizontal divisor on C . Let m ∈ Z ≥ . There is a naturalmap ̺ m : H ( C , O ( m D )) → L + ( ρ ( D )) s m ϕ s here s ∈ H ( C , m D ) is interpreted as a rational function s with ( s ) + m D ≥ ϕ s isthe vanishing function of s .We define the Newton–Okounkov linear system L +∆ ( D ) to be the subset of L + ( ρ ( D )) givenby the closure of the union of the convex hulls of the images of ̺ m for m ranging over Z ≥ .We may extend this definition to horizontal Q -divisors by defining L +∆ ( D ) to be m L +∆ ( m D )where m is chosen arbitrarily divisible. Theorem 6.6.
If the generic fiber of D has positive degree, then we have the equality betweenthe Newton–Okounkov linear system and the effective linear system: L +∆ ( D ) = L + ( ρ ( D )) . Before proving the proposition, we need a preparatory lemma adapted from [9].
Definition 6.7.
Let f : V (Σ) → R be a function. Set M ( f ) = max S ⊆ V (Σ) n(cid:12)(cid:12) X v ∈ S f ( v ) (cid:12)(cid:12)o . Lemma 6.8.
Let ϕ : V (Σ) → R be a function. Then, max ϕ − min ϕ ≤ M (∆( ϕ )) diam(Σ) . Proof.
It suffices to show that for any edge e = vw in Σ, | ϕ ( w ) − ϕ ( v ) | ≤ M (∆( ϕ )). Indeed,let v , v be the vertices where the minimum and maximum of ϕ are achieved, respectively.By picking a path from v to v of length at most diam(Σ) and comparing the values of ϕ along that path, we achieve the desired conclusion.Let e = v ′ w ′ with t = ϕ ( v ′ ) < ϕ ( w ′ ). Set Σ ≤ t be the subgraph of Σ induced by ϕ − ([0 , t ]).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, M (∆( ϕ )) ≥ (cid:12)(cid:12)(cid:12) X v ∈ V (Σ ≤ t ) ∆( ϕ )( v ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) X v ∈ V (Σ ≤ t ) (cid:16) X e = vw (cid:0) ϕ ( v ) − ϕ ( w ) (cid:1)(cid:17)(cid:12)(cid:12)(cid:12) = X e = vw ∈ O (Σ ≤ t ) ( ϕ ( w ) − ϕ ( v ))where the last equality holds because the contribution from edges contained in Σ ≤ t cancel inpairs. From this we conclude that for any outgoing edge e = vw , ϕ ( w ) − ϕ ( v ) ≤ M (∆( ϕ )). (cid:3) We have the following corollary:
Corollary 6.9.
Let ϕ, ϑ : V (Σ) → R ≥ be functions such that ∆( ϕ ) − ∆( ϑ ) = F − G here F and G are effective divisors of degree at most d . Suppose that there are (not neces-sarily distinct) vertices v, w such that ϕ ( v ) = ϑ ( w ) = 0 Then max( | ϕ − ϑ | ) ≤ d diam(Σ) . Proof.
By hypothesis, min( ϕ − ϑ ) ≤ ϕ ( v ) − ϑ ( v ) ≤
0. We note that M (∆( ϕ − ϑ )) ≤ d .Consequently, max( ϕ − ϑ ) ≤ max( ϕ − ϑ ) − min( ϕ − ϑ ) ≤ d diam(Σ) . Interchanging the roles of ϕ and ϑ , we get the conclusion. (cid:3) 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 inthe image of ρ m for some positive integer m . First, we may suppose that ϑ takes rationalvalues. Pick a sufficiently divisible m such that mϑ takes integer values. Because ̺ m ( π k s ) = ̺ m ( s ) + km , we can replace ϑ by ϑ − min v ϑ ( v ) and suppose that ϑ ≥ ϑ ( v ) = 0 forsome vertex v .We have that ∆( mϑ ) + m Λ ≥
0. Pick an effective divisor E on Σ such that E ( v ) ∈ Z forall v ∈ V (Σ) and ∆( mϑ ) + m Λ − E is effective of degree exactly g , the genus of C . Choosea horizontal divisor E such that ρ ( E ) = E . Because deg( m D K − E K ) = g , the line bundle O ( m D K − E K ) on C K has a regular section by the Riemann-Roch theorem. Therefore, thereis a rational function s on C K with( s ) + m D K − E K ≥ . From s we will find a section whose image under ρ m approximates ϑ . By multiplying s bysome power of π , we may ensure that s (considered as a rational function on C ) is regularon the generic points of the components of the closed fiber and does not vanish identicallyon all of them. Consequently, s ’s vanishing function ϕ s is non-negative and takes the value0 at some vertex w . Now,∆( ϕ s ) − ∆( mϑ ) = (∆( ϕ s ) + m Λ − E ) − (∆( mϑ ) + m Λ − E )= (∆( ϕ s ) + ρ ( m D − E )) − (∆( mϑ ) + m Λ − E )is the difference of two effective degree g divisors by Lemma 6.3. Consequently, by Corol-lary 6.9, we have (cid:12)(cid:12)(cid:12) ϕ s m − ϑ (cid:12)(cid:12)(cid:12) ≤ gm diam(Σ) . Because ̺ m ( s ) = ϕ s m , the conclusion follows by choosing large m . (cid:3) To handle the case where the divisor D is of degree 0, we may employee the followingresult. Corollary 6.10.
Let D be a horizontal divisor on C of non-negative degree. Let E be aneffective, non-empty horizontal divisor. Then, we have the equality L + ( ρ ( D )) = \ ε> L +∆ ( D + ε E ) here the intersection is taken over rational ε > .Proof. It suffices to prove that L + ( ρ ( D )) = \ ε> L + ( ρ ( D + ε E )) . This follows immediately from definitions. (cid:3)
This comparison between the purely combinatorial Baker–Norine linear system and thealgebraically-defined Newton–Okounkov linear system was surprising to these authors. How-ever, it does not capture the combinatorial richness of the Baker–Norine theory as it involvesreal-valued, rather than integer-valued functions ϕ on graphs. The integer-valued func-tions can be incorporated into our work by hand. Within the vector space 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 tohave non-negative rank if there exists ϕ : V (Σ) → Z such that ∆( ϕ ) + Λ ≥
0. From thisconcept, a Riemann–Roch theory for divisors on graphs is developed. Because we may adda constant to ϕ without affecting ∆( ϕ ) + Λ ≥
0, we may suppose that ϕ is non-negative inthe above definition. From this, we can give a asymptotic formulation of non-negative rank. Corollary 6.11.
Let D be a horizontal divisor on C of non-negative degree. Let E be aneffective, non-empty horizontal divisor. Then, the specialization ρ ( D ) has non-negative rankif and only if there exists ϕ : V (Σ) → Z ≥ such that for all ε > , ϕ ∈ L +∆ ( D + ε E ) . From this, one may reformulate the Baker–Norine theory in terms of lattice points inNewton–Okounkov linear systems. It is unknown at this point whether this view leads toany new proofs of known results in the Baker–Norine theory.6.3.
Horizontal-Vertical decomposition.
Now, we will define a decomposition of divisorson Σ analogous to the Zariski decomposition to use in our description of Newton–Okounkovbodies of curves. Recall that the Zariski decomposition of a big Q -divisor D on a smoothprojective 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 ( X, mP ) → H ( X, mD ) . Definition 6.12.
Let Λ be a divisor on Σ such that L + (Λ) is non-empty. The minimalelement of L + (Λ), ̟ : V (Σ) → R is defined by ̟ ( v ) = min( ϕ ( v ) | ϕ ∈ L + (Λ)) . We have the following straightforward lemma following from Lemma 6.4.
Lemma 6.13.
Let ̟ be the minimal element of L + (Λ) . Addition of ̟ gives an isomorphism L + (Λ − ∆( ̟ )) → L + (Λ) . 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 mini-mum of ϕ ∈ L . .4. Enriched Newton–Okounkov linear systems.
We will connect the Newton–Okounkovlinear systems to the Newton–Okounkov bodies of curves over discrete valuation rings. Suchbodies must take into account the vanishing of sections along a flag Y • = { C = Y ) Y ) Y } where Y = { p } is a smooth point of the closed fiber. Consequently, we will enrich the abovetheory by considering such vanishing. We will also need to consider elements of H ( C , m D )whose horizontal components do not have any components in common with a fixed horizontaldivisor in order to gain control over the vanishing at Y in the tropical case.Let D , F be horizontal divisors on C . Let m ∈ Z ≥ . Definition 6.14.
The F -controlled linear system H ( C , m D ) ( F ,ε ) is the set of all s ∈ H ( C , m D ) which, when considered as rational functions, have the property that their prin-cipal divisor ( 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 . For a divisor G on C , write v p ( G ) to be the multiplicity of p in H ∩ C v where H is the horizontal part of G .We consider the natural map ̺ m,p : H ( C , m D ) ( F ,ε ) → L + ( ρ ( D )) × R s (cid:18) m ϕ s , m v p (cid:0) ( s ) + m D (cid:1)(cid:19) where s ∈ H ( C , m D ) F ,ε . Observe that the second component of ̺ m,p ( s ) is the vanishing at p of the horizontal component of the zero locus of s , considered as a section of O ( D ). Definition 6.15.
The F -controlled 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 imagesof H ( C , m D ) ( F ,ε ) under ̺ m,p for m ∈ Z ≥ . When the subscript ( F , ε ) is suppressed, thismeans that we consider the empty divisor.For a divisor Λ on Σ, let the p-enriched effective linear system L + p (Λ) be the subset of L + (Λ) × R given by L + p (Λ) = { ( ϕ, u ) | ϕ ∈ L + (Λ) , ≤ u ≤ ∆( ϕ )( v ) + Λ( v ) } . Observe that if Λ = ρ ( D ) for a horizontal divisor D on C , the quantity ∆( ϕ )( v ) + Λ( v ) isexactly the degree of the divisor P v ϕ ( v ) C v + D restricted to C v . So, the second componentmeasures the possible multiplicities of p in the horizontal part of ( s ) varying from 0 to themaximal possible degree.We have the following extension of Proposition 6.6. Theorem 6.16.
Let C be a semistable curve over Spec O with horizontal divisors D and F such that the generic fiber of D has positive degree. Let p be a smooth point on a componentof C v of the closed fiber of C . We have the equality between the F -controlled p -enrichedNewton–Okounkov linear system and the p -enriched effective linear system: L +∆ ,p ( D ) ( F ,ε ) = L + p ( ρ ( D )) . his is proved by the same method as Proposition 6.6. We wish to find a section s ∈ H ( C , m D ) such that ̺ m,p ( s ) is close to some ( ϑ, u ) We modify the proof to choose E tointersect C v with a multiplicity m ′ close to mu at p and for E not to contain any componentof D or F . The section s produced by the Riemann-Roch theorem has ( s ) + m D K − E K equal to a degree g divisor. Then, the horizontal component of ( s ) + m D K intersects C v witha multiplicity between m ′ and m ′ + g . Moreover, the horizontal components of ( s ) + m D 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 de-scription of the Newton–Okounkov bodies of curves.We first consider the tropical case of a flag { X ) Y ) Y } where Y is a horizontal divisorand Y = { p } is a smooth point of the closed fiber. Theorem 6.17.
Suppose that Y is a horizontal divisor intersecting the closed fiber in smoothpoints and that Y = { p } is a point on the component C v . Let D be a horizontal divisor whosegeneric fiber has positive degree. Moreover, suppose that p is not contained in D . For t ∈ R ,let L t = L + ( ρ ( D − tY )) . The Newton–Okounkov body is the overgraph in R of a : [0 , deg( D ) / deg( Y )] → R t ̟ L t ( v ) . Proof.
We first show that the Newton–Okounkov body lies above the graph of a . We observethat for s ∈ H ( C , m D ), ν ( s ) is always greater than or equal to the multiplicity of C v inthe principal divisor ( s ). Consequently, if t ∈ R is such that mt is an integer and s vanishesto order at least mt on Y , then ̺ m ( s ) ∈ L t . Therefore,1 m ν ( s ) ≥ m ϕ s ( v ) ≥ min( ϕ ( v ) | ϕ ∈ L t ) = ̟ L t ( v ) . For a fixed rational t with 0 ≤ t < deg( D ), let us find an m ∈ Z ≥ and a section s ∈ H ( C , O ( m D )) such that m ν ( s ) is close to ( t, ̟ L t ( v )). Set F = Y . For a small ε >
0, byTheorem 6.16, we may find m large enough such that there exists s ∈ H ( C , m ( D − t F )) ( F ,ε ) (considered as a rational function) such that1 m v p (cid:0) ( s ) + m ( D − t F ) (cid:1) < ε and m ϕ s is within ε of ̟ L t . It follows that m ν ( s ) is close to t . From the fact that ν ( s ) = 1 m (cid:0) ϕ s ( v ) + v p (( s ) + m D − ν ( s ) F )) (cid:1) , it follows that all but a small part of v ( s ) comes from the vertical component of ( s ) along C v . Thus v ( s ) can be made arbitrarily close to ̟ L t ( v ). (cid:3) Now, we consider the Arakelovian case. heorem 6.18. Let D be a horizontal divisor. Suppose that Y is a component C v of theclosed fiber and Y = { p } is a smooth point on C v not contained in D . Let L t ⊂ L + ( ρ ( D )) be the tropical sub-semigroup of elements ϕ with ϕ ( v ) = t . Then the Newton–Okounkov bodyof O ( D ) is the set of points between the graphs of a ( t ) = 0 and b ( t ) = ρ ( D )( v ) + max(∆( ϕ )( v ) | ϕ ∈ L t ) for t ≥ .Proof. Observe that for s ∈ H ( C , m D ), ν ( s ) = ϕ s ( v ) and ν ( s ) = v p (cid:0) ( s ) + m D (cid:1) . 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 6.16. (cid:3) Example 6.19.
We conclude by giving an example of the Newton–Okounkov body for curvesin the tropical and 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 withgenus g ( X ) = 3. This is a plane quartic degenerating into a conic C and two lines ℓ , ℓ . Tomake the model semistable, one must blow up the intersection point of ℓ and ℓ , introducinga new component E of the degeneration. For this curve, the special fiber and the dual graphare given in the figure: the vertex P corresponds to the conic; Q , Q corresponds to thelines; and P ′ corresponds to the curve E . ℓ ℓ CE Q Q PP ′ We will compute the Newton–Okounkov body for a general hyperplane section D , whosespecialization is given by ρ ( D ) = Λ = 2( P ) + ( Q ) + ( Q ). Note that for any ϕ ∈ L (Λ) wehave ∆( ϕ ) = (4 ϕ ( P ) − ϕ ( Q ) − ϕ ( Q ))( P )+ (3 ϕ ( Q ) − ϕ ( P ′ ) − ϕ ( P ))( Q )+ (3 ϕ ( Q ) − ϕ ( P ′ ) − ϕ ( P ))( Q )+ (2 ϕ ( P ′ ) − ϕ ( Q ) − ϕ ( Q ))( P ′ ) . • Tropical case: we will pick as a flag { C = Y ) Y ) Y } , with Y a degree onehorizontal divisor, intersecting the generic fiber X K in a general point and intersectingthe closed fiber in a generic point Y of the conic C . It is straightforward to compute ̟ L t for t ∈ [0 ,
4] as follows: – for t ∈ [0 , ̟ L t = 0, – for t ∈ [2 , ̟ L t ( P ) = ( t − / ̟ L t ( Q ) = ̟ L t ( Q ) = ̟ L t ( P ′ ) = 0 . his gives the following Newton–Okounkov body: • Arakelovian case: we will pick as a flag { C = Y ) Y ) Y } where Y is theconic C and Y is a general point of C . The function b ( t ) is achieved by the followingchoices for ϕ : – for t ∈ [0 , / ϕ ( P ) = t, ϕ ( Q ) = ϕ ( Q ) = ϕ ( P ′ ) = 0, – for t ∈ [1 / , ∞ ), ϕ ( P ) = t, ϕ ( Q ) = ϕ ( Q ) = ϕ ( P ′ ) = t − / . Therefore, the Newton–Okounkov body is as follows: The two examples are reflections of each other because in the tropical case, we chose Y with ρ ( Y ) = P , the vertex corresponding to the conic while in the Arakelovian case, wechose Y = C , the conic. In general, the Newton–Okounkov bodies in the two cases will nothave such an obvious relation to each other. References [1] M. Baker,
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Department of Mathematics, The Ohio State University, 231 W. 18th Avenue, Columbus,OH, 43210, USA
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