On the finite generation of valuation semigroups on toric surfaces

We provide a combinatorial criterion for the finite generation of a valuation semigroup associated with an ample divisor on a smooth toric surface and a non-toric valuation of maximal rank. As an application, we construct a lattice polytope such that none of the valuation semigroups of the associated polarized toric variety coming from one-parameter subgroups and centered at a non-toric point are finitely generated.


Introduction
Finite generation of semigroups or rings arising from geometric situations has been a question of interest for a long time.As a salient example, we can recall the finite generation of canonical or adjoint rings from birational geometry, which motivated the field through the minimal model program for several decades; cf.[BCHM10].The question of finite generation of valuation semigroups arising from Newton-Okounkov theory appears to be equally difficult in general, with little progress beyond the completely toric situation, but potentially great benefits such as the existence of toric degenerations, cf.[And13], and completely integrable systems, cf.[HK15], to name but a few.In this article, we take a few steps away from the situation where every participant is toric: we consider valuation semigroups associated with torus-invariant divisors on toric surfaces with respect to a non-toric valuation.
The main idea behind Newton-Okounkov theory is to attach combinatorial/convex-geometric objects to geometric situations to facilitate their analysis, in other words, to partially replicate the setup of toric geometry in settings without any useful group action.The basis for the theory was developed by Kaveh-Khovanskii [KK12] and Lazarsfeld-Mustaţȃ [LM09], building on earlier work of Okounkov [Oko96], but the subject has seen substantial growth in the last decade.By now applications of Newton-Okounkov theory range from combinatorics and representation theory through birational geometry, cf.[KL17a, KL17b, KL19, KL18a], to mirror symmetry, cf.[RW19], and geometric quantization in mathematical physics.
Given a projective variety X and a divisor D on X, Newton-Okounkov theory associates to (X, D) a semigroup S Y • (D), the valuation semigroup, and a convex body ∆ Y • (D), the Newton-Okounkov body of D. Both the valuation semigroup and the Newton-Okounkov body depend, however, on a maximal rank valuation of the function field of X coming from an admissible flag Y • of subvarieties.
The Newton-Okounkov body ∆ Y • (D) is an asymptotic version of S Y • (D) and is, accordingly, a lot easier to determine.Newton-Okounkov bodies on surfaces end up being almost rational polygons; cf.[KLM12].If the section ring of D is finitely generated, then a suitably general flag valuation will yield a rational simplex as its Newton-Okounkov body; cf.[AKL14].In the case of a toric variety X with torus-invariant D and Y • , the associated Newton-Okounkov body recovers the moment polytope of the polarized toric variety.
In this paper, we will focus on the valuation semigroup S Y • (D), more concretely, on the question of whether or not it is finitely generated.It is a classical fact that S Y • (D) is often not finitely generated even if X is a smooth projective curve.
It is known that S Y • (D) is a finitely generated semigroup if X is a toric variety, D a torus-invariant divisor, and Y • an admissible flag of torus-invariant subvarieties.We consider the next open question, namely, the case of toric surfaces and non-torus-invariant flag valuations.Although the divisorial geometry of toric surfaces themselves is not particularly complicated, things get out of control once we start blowing up non-toric points.Blowing up just one point on a toric surface can lead to surfaces with infinitely many negative curves on them, as recent research of Castravet-Laface-Tevelev-Ugaglia [CLTU23] illustrates.Blowing up many general points quickly leads to notoriously difficult situations like Nagata's conjecture.

Newton-Okounkov bodies
Assume that X = T V (Σ) is a smooth projective toric variety of dimension n.Then ample torus-invariant divisors D or their associated line bundles O X (D) can be understood as lattice polytopes ∆(D) in the character lattice M Z n of the torus T acting on X.Using this description, the starting fan Σ can be recovered as the normal fan of these polytopes.
Moreover, it is a well-known feature of the toric theory that the vector space Γ (X, O X (D)) has the set of lattice points ∆(D) ∩ M as its distinguished basis.That is, the polytope ∆(D) gives Γ (X, O X (D)) not just a dimension but also a shape.
In [LM09] and [KK12], this concept was generalized to arbitrary projective varieties X (still of dimension n).If we are given a so-called admissible flag of nested (irreducible) varieties with dim Y i = n − i that are smooth in the special point Y n , then for every ample divisor D, there is an associated convex body ∆ Y • (D) ("Newton-Okounkov body") in R n reflecting many properties of D; see Section 2.1 for details.
Note that Z n has ceased to be a character lattice because there is no longer a torus around.While the Newton-Okounkov body ∆ Y • (D) depends not on D but only on its numerical class, the dependence on the chosen flag is striking.
If, for instance, X is toric as at the very beginning, then the construction of ∆ Y • (D) recovers the correspondence between divisors and polytopes we mentioned above.But to make this true requires a toric flag ; i.e., all subvarieties Y i are supposed to be orbit closures.
The fact that toric varieties with toric flags lead to well-known polytopes is not a one-way street.In fact, if, for a general variety X, the semigroups S Y • (D) are finitely generated, then they provide a toric degeneration of X.This was shown in [And13].Observe that in general the semigroup S Y • (D) being finitely generated is much stronger than ∆ Y • (D) being polyhedral.This finite generation of the valuation semigroup S Y • (D) is the main point of this paper.

Results
In [IM19] the finite generation was shown for complexity one T-varieties with toric flags.Note that this implies the toric case.Whenever one is only interested in the Newton-Okounkov body (instead of the semigroup), there are more results: The most general one solves the question for surfaces using Zariski decomposition; cf.[LM09,Theorem 6.4].In particular, the Newton-Okounkov bodies are polyhedral in this case.Specializing this situation, [HKW20] has provided an explicit combinatorial description of ∆ Y • (D) for toric surfaces with certain non-toric flags.
In the present paper, partially inspired by [AP20], we consider the very same setup of toric surfaces, namely with the following admissible flag: where Y 1 is the closure of a one-parameter subgroup of the torus, which is non-torus-invariant and Y 2 is a general smooth point.Then we prove that, in dependence on Y 1 , the semigroups S Y • (D) can be both finitely generated and not finitely generated.
The main result of the paper (Theorem 6.8) comes from understanding the relationship between the Newton-Okounkov body of D and the Newton polygon associated with the non-toric flag curve Y 1 .The significance of our contribution lies in the fact that we infer the finite generation of valuation semigroups from asymptotic/convex-geometric data and provide a combinatorial criterion.
In order to state our Theorem, we introduce a bit of terminology.For our flag Y • , we consider a non-torus-invariant curve Y 1 given as the closure of the one-parameter subgroup determined by a primitive vector v N ∈ N ; our flag point is going to be 0 ⊗ Z 1 on the torus T N ⊗ Z C × .For a strongly convex cone σ ⊆ N R and a lattice point u ∈ int(σ ) ∩ N , we say that u is strongly decomposable in σ if u = u ′ + u ′′ for suitable u ′ , u ′′ ∈ int(σ ) ∩ N .Given a divisor D on X, we construct strongly convex cones σ + and σ − associated with v N that are spanned by certain rays of the fan of X (see Definition 6.7).With this said, our main result goes as follows.
Theorem (Theorem 6.8).Let X be a smooth toric surface associated with a fan Σ and D an ample divisor on X.The valuation semigroup S Y • (D) is finitely generated if and only if v N is not strongly decomposable in σ + and −v N is not strongly decomposable in σ − .
To illustrate the combinatorial content, Figure 1 pictures the situation in the case of the 7-gon of [CLTU23], where the blow-up surface X = T V (Σ) accomodates infinitely many negative curves (cf. Figure 10 for a complete picture).As an application of the above theory, we construct in Example 6.11 a lattice polygon with a strong non-finite-generation property.To be more concrete, we look at the ample divisor D associated with the polytope ∆(D) given in Figure 11A on the toric variety X = T V (Σ) corresponding to the fan Σ in Figure 11B.Example 6.11 shows that the semigroup S Y • (D) will not be finitely generated for any v N ∈ N we pick.

Notation and preliminaries
Let X be a two-dimensional smooth projective variety, and assume that we are given an admissible flag as in Section 1.1, and an ample divisor D on X.

Newton-Okounkov bodies and valuation semigroups
Following [LM09], we obtain a rank two valuation-like function (or, equivalently, a rank two valuation of the function field of X, see [KMR21]) as follows: Let f be an equation for Y 1 near Y 2 .For a non-trivial section s ∈ Γ (X, L) of a line bundle L, e.g., L = O X (D), we define The valuation semigroup S Y • (D) of D (with respect to the flag Y • ) is defined as The Newton-Okounkov body of D (with respect to the flag Y • ) is defined as the set For Newton-Okounkov theory on surfaces, see [KL18b].

Toric setup
Let N Z 2 be a two-dimensional lattice with dual lattice M and Σ a smooth complete fan associated with the toric surface X = T V (Σ).We may assume that our ample divisor D is toric, hence is represented by a polytope ∆(D).Recall that T = Spec (C[M]) is our torus acting on X; hence M becomes its character lattice and N the associated lattice of one-parameter subgroups.
Within the torus T , the curve C := Y 1 is given by the binomial equation f := x v M − 1 with v M ∈ M being one of the two primitive elements of v ⊥ N ⊂ M R .The associated Newton polytope ∆ newt := newt(f ) is the line segment [0, v M ] connecting 0 and v M in M R .This way ι : P 1 C is the normalization map.
For a toric line bundle L on X, the pullback ι * : Γ (X, L) Γ (P 1 , ι * L) corresponds to the projection which we will identify with v N : M Z M. Observe that this almost fits the setup of [IM19] as Y 1 is invariant under the codimension one torus given by v N .We only deviate from [IM19] by choosing a non-invariant flag point Y 2 .

Torifying the curve
Note that C is also a prime (Cartier) divisor on X, which properly intersects all torus-invariant curves, i.e., all boundary curves of X.Therefore, C is nef and C Lemma 2.1.The polygon ∆ nef := ∆(C ′ ) is given by where the rays ρ ∈ Σ(1) are identified with their first (hence primitive) lattice points.

An alternative view on ∆ nef
Beside the explicit description of Lemma 2.1, it is possible to describe the shape of ∆ nef in the following more combinatorial way.The relation We denote by r max , r min ∈ M the vertices of ∆(D), where ⟨∆(D), v N ⟩ admits its extremal values (cf. Figure 3A).Moreover, we define σ max , σ min to be the two-dimensional cones generated by the two edges of ∆(D) that contain the vertices r max and r min , respectively.
We take the line segment ∆ newt and fit it inside the cone σ max until it hits both rays of this cone.In this way, we construct a lattice triangle ∆ max with base ∆ newt and top vertex r max .We construct ∆ min (cf.Figure 3A) in a similar way.In other words, the cones σ max and σ min are cut off along v N -constant lines producing edges of ∆ max and ∆ min , respectively, of lattice length one.Note that both cut lines are parallel translates.Gluing ∆ max and ∆ min along ∆ newt (cf. Figure 3B) yields Note that ∆ nef ⊇ ∆ newt is the smallest polytope containing ∆ newt and having Σ as a refinement of its normal fan.Actually, either ∆ nef is a quadrangle with ∆ newt serving as one of its diagonals, or it is a triangle with ∆ newt as a side.

Valuation semigroups associated with non-toric flags
In this section, we determine the valuation semigroup S Y • (D) associated with an ample (Cartier) divisor D and a non-toric flag Y • as a subset of N 3 .The main result is Theorem 3.11, where the abstract semigroup S Y • (D) is described in terms of lattice points coming from a polyhedral construction in M.
Let us fix ℓ ≥ 1 and k ≥ 0. The space of sections s ∈ Γ (X, O X (ℓD)) which have vanishing order at least

Return to toric geometry
Our goal is to understand the restriction of global sections via toric geometry.Therefore, we implement two changes.First, we will shift the linear series of the flag curve, which enables us to replace some of the line bundles we study with torus-invariant ones.Second, we will normalize the restriction.
We are going to use C ′ = C − div(f ) from Section 2.3.In terms of the associated sheaves, this means where f is the equation x v M − 1 of C mentioned earlier.This leads to the possibility of replacing L(ℓ, k) by the isomorphic, but torus-invariant, line bundle Accordingly, we set Recall that the nef invertible sheaves O X (ℓD) and O X (k C ′ ) correspond to the polytopes ℓ∆(D) and k∆ nef , respectively.This implies that L ′ (ℓ, k) has a monomial base provided by In particular, Θ(1, 1) = conv([0, 0], [−1, 0], [0, −1/2]) (cf. the yellow polytope in Figure 4).The projection looks like π : M M = M/Zv M .This map (cf. Figure 4) can be identified with v N : M Z, that is,

An alternative view on Θ(ℓ, k)
In general, Σ is not the normal fan of Θ(ℓ, k) as it was of ∆(D).Geometrically, this means that Θ(ℓ, k) does, in general, not encode a nef Cartier divisor on X.While Θ(ℓ, k) is defined as some kind of a difference of polytopes, it is in general not true that the inclusions become equalities (cf.Example 3.4).We present a suggestion on how to overcome this.
Recall from Section 2.4 that we had denoted by r max , r min ∈ M the vertices of ∆(D) where ⟨∆(D), v N ⟩ attains its extremal values.Similarly, we denote by r max The latter lead to the line segments r max ′ (ℓ, k) + k∆ newt and r min ′ (ℓ, k) + k∆ newt , which cut the polytope ℓ∆(D) into three subpolytopes which we call □ max (ℓ, k), ∆(D) C(ℓ,k) , and □ min (ℓ, k).
More concretely, here is how we obtain □ max (ℓ, k) and □ min (ℓ, k): We take the line segment k∆ newt and fit it inside the polytope ℓ∆(D) until it hits the boundary twice.This way, we construct the lattice polygon (not necessarily a triangle) □ max (ℓ, k) such that k∆ newt is one of its edges and r max (ℓ, k) is one of its vertices.In a similar way, we construct □ min (ℓ, k) using r min (ℓ, k).
As ℓ∆(D) before, the polytope ∆(D) C(ℓ,k) just defined still fulfills the equality however, now we also have the equality Remark 3.3.After this point, we will use the shorter notation r max Nevertheless, one should keep in mind that all of these quantities depend on ℓ, k.

Projections of polytopes
We start by pulling back the sheaf L ′ (ℓ, k).To this end, we define where width v N (•) denotes the lattice width of a polytope with respect to the linear functional v N ∈ N ; i.e., if ∆ ⊆ M R is a polytope, then width v N (∆) := max m,m ′ ∈∆ |⟨m, v N ⟩ − ⟨m ′ , v N ⟩|.Note that this equals the length of the line segment ∆ := π(∆); i.e., Proof.We obtain Remark 3.6.Altogether, this yields the sequence of inclusions Summarizing what we have done so far, we obtain the following.

Shape of the semigroup
As we did before, let us fix a pair (ℓ, k).We know from Section 2.1 that we are supposed to collect the values ord Y 2 ( s| C ) for all possible sections s, where Y 2 = {1} is a smooth point on C. In Section 3.1, we have transferred this setup to ord 1∈P 1 (ι * s ′ ), where s ′ runs through all global sections represented by the polytope Proposition 3.8 implies that the pullbacks ι * s ′ run through all e(ℓ, k) elements of π(Θ(ℓ, k) ∩ M) ⊆ M Z.Each element of Z represents a rational monomial function on P 1 .We are supposed to find the orders of vanishing at 1 ∈ P 1 of all of their linear combinations.
Proof.The definition of the valuation semigroup can be reformulated as Then everything follows from Proposition 3.8.□

Shape of the Newton-Okounkov body
Building on Section 3, we determine the Newton-Okounkov body ∆ Y • (D) in Theorem 4.3.Consider the assignment We extend this definition to all ℓ, k ∈ R ≥0 using the convention width v N (∅) = −∞.This becomes necessary when k • ∆ nef does not fit inside ℓ • ∆(D), which will happen for k ≫ ℓ.
Moreover, we observe the following.
Proof.The width function is linear in its polyhedral argument.□ Note that the same statement holds true for d(ℓ, k) but not for e(ℓ, k).
From now on, we return to (ℓ, k) ∈ N 2 .
Moreover, d(q) is a decreasing piecewise linear function with d(q) = −∞ for q ≫ 0.
Proof.Let ϵ > 0 and denote by A, B the vertices of π(Θ(ℓ, k)) = Θ(ℓ, k).First assume that the dimension of Θ(ℓ, k) equals two.Then the two fibers π −1 (T ) ∩ Θ(ℓ, k) with T = A + ϵ or B − ϵ have positive lengths greater than (or equal to) some µ > 0. In particular, all fibers in between do so as well.Hence, setting λ = 1/µ, the fibers have at least length one for λ(A + ϵ) ≤ T ′ ≤ λ(B − ϵ).If in addition T ′ ∈ M, then all of these fibers have to contain lattice points in M. Thus, we obtain Keeping q = k/ℓ constant and using Lemma 4.2, we see that e(ℓ, k) behaves like d(ℓ, k) asymptotically with respect to dilations.The result then follows by recalling the fact that Newton-Okounkov bodies are closed.
It remains to consider the pathological case dim(Θ(1, q)) = 1.Here, we can approximate [q, t] by [q − ϵ, t] so that the resulting Θ(1, q − ϵ) is full-dimensional and t ≤ d(q) ≤ d(q − ϵ).Then the previous argument shows that [q − ϵ, t] ∈ ∆ Y • (D).As Newton-Okounkov bodies are closed by definition, [q, t] ∈ ∆ Y • (D).□ We remark that the case dim(Θ(1, q)) = 1 from the previous proof requires special v N and a unique q 0 = k/ℓ.This configuration is characterized by the fact that (a shift of) q 0 ∆ newt connects two parallel edges of ∆(D).Note that Θ(ℓ, k) is also parallel to these edges.In contrast to the general case, for k/ℓ = q 0 , the number e(ℓ, k) behaves asymptotically like 1/g • d(ℓ, k), where Despite that e(ℓ, k) for k/ℓ = q 0 does not approach d(ℓ, k) at all, this does not cause a problem: as we have seen in the proof, for q ≤ q 0 , the general case applies, and for q > q 0 , we have d(q) = −∞ anyway.

Criterion for the finite generation of certain valuation semigroups
We provide a criterion for the finite generation of strictly positive (with respect to their height functions) semigroups in terms of their limit polyhedra.

Semigroups with polyhedral limit
We start with a free abelian group M of rank n, i.e., M Z n , and a linear form h : M Z which we call a height function.This induces h R : R, which we will often denote by h as well.Let S ⊆ h −1 (N) be a semigroup that is strictly positive; i.e., S ∩ ker(h) = {0}.In order to refer to the individual layers of a given height, we will write This setup allows us to define the enveloping cone as well as the convex limit figure Z is the projection on ℓ, which then leads to the Newton-Okounkov body 1) R 2 .Definition 5.1.We say that S has a polyhedral limit if ∆ S is a polytope, i.e., if ∆ S equals the convex hull of its (finitely many) vertices.This property is fulfilled whenever the semigroup S is finitely generated.In this case, ∆ S even has rational vertices; it is a rational polytope.However, the following standard example shows that the converse implication does not hold in general.

Equivalent conditions for finite generation
We assume that S ⊆ M is a strictly positive (with respect to h) semigroup that has polyhedral limit ∆ S .
Definition 5.3.We say that a point p ∈ ∆ S lifts to the semigroup S (i.e., is a valuation point) if there exists some scalar c ∈ R >0 with c • p ∈ S.
Note that in this case, both p and c have to be rational; i.e., p ∈ ∆ S ∩ M Q and c ∈ Q >0 .Hence, it is not a surprise that the assumption of the next lemma is automatically fulfilled if the semigroup S is finitely generated.
Lemma 5.4.If all vertices of ∆ S lift to S, then they are rational (i.e., ∆ S is a rational polytope) and every rational point p ∈ ∆ S ∩ M Q lifts to S.
Proof.Let v 1 , . . ., v d ∈ ∆ S be linearly independent (rational) vertices such that p is contained in their convex hull.Then the unique coefficients λ i in the representation p = d i=1 λ i v i have to be rational, too.Thus, we may choose an integer µ such that µ • λ i ∈ N. On the other hand, there is a joint factor c ∈ Z ≥1 such that all multiples c • v i belong to S. This implies Next, we formulate the main point of this section.
Proposition 5.5.A semigroup S with a polyhedral limit ∆ S is finitely generated if and only if all vertices of ∆ S lift to S.
We have already seen that this condition is necessary for the finite generation.Now we will show that it is sufficient, too.Note that, in Example 5.2, the two vertices of the line segment ∆ S indeed do not lift to the semigroup.
Let S ⊆ M be a subsemigroup with rational polyhedral limit ∆(S) (with respect to some height function h : M Z).Assume that the vertices and thus, by Lemma 5.4, all rational points of ∆ S lift to S.Moreover, we may assume that C S is a full-dimensional cone.
Proof of Proposition 5.5.Assume that S is not finitely generated.Then S has infinitely many indecomposable elements; i.e., By taking a simplicial subdivision we may, w.l.o.g., assume that the cone C S is simplicial and given as C S = cone(s 1 , . . ., s n ).Consider the lattice Λ generated by s 1 , . . ., s n .As M/Λ is finite, there must be a coset m + Λ which contains infinitely many elements of H.Here we may choose m to be a minimal representative in C S : m ∈ C S ∩ M so that m − s i C S for i = 1, . . ., n.
As the elements in C S ∩ H were indecomposable in S, they certainly are indecomposable in C S ∩ S. In particular, if we identify (m + Λ) ∩ C S with N n , we obtain an infinite set of pairwise incomparable elements, in contradiction to Dickson's lemma [CLO15, Chapter 4, Theorem 5].□

Characterising the lifting property
The following theorem gives a purely combinatorial criterion to check if our valuation semigroup S Y • (D) (in the language of Section 2.2) is finitely generated.Recall the definition from Section 4 on page 12. Theorem 6.1.The point (1, k/ℓ, d(k/ℓ)) = (1, q, d(q)) is a valuation point (i.e., a multiple of it lies in S Y • (D)) if and only if there exists a λ ∈ N such that and π(λΘ(1, q)) has endpoints in M.
Proof.If (1, q, d(q)) is a valuation point, then according to Theorem 6.1, there exists a λ ∈ N such that π(λΘ(1, q)) has vertices in M and such that π : λΘ(1, q) ∩ M π(λΘ(1, q)) ∩ M is surjective.Hence, 1 ∈ π (λ(Θ(1, q) − r min ′ ) ∩ M) ⊂ π(σ min ′ ∩M) and similarly for σ max ′ .For the converse, assume 1 ∈ π(σ min ′ ∩M) and −1 ∈ π(σ max ′ ∩M).We will construct a suitable scaling factor λ ∈ N for which π : λΘ(1, q) ∩ M − π(λΘ(1, q)) ∩ M is surjective and π(λΘ(1, q)) is a lattice polytope, in order to again apply Theorem 6.1.To this end, choose levels δ min ∈ Q >0 and δ max ∈ Q >0 such that We set Choose a λ ∈ N with λ > 1 ε min and λ > 1 ε max such that λ r min ′ and λ r max ′ are lattice points in M. Then the corresponding projection π is surjective.To show that, we divide the image π(λΘ(1, q)) into three parts.The first λδ min lattice points in π(λΘ(1, q)) ∩ M are in the image of π because π restricted to σ min ′ is surjective by assumption.The same argument holds for the last λδ max points since π is surjective on σ max ′ .The points in between are hit by projecting the lattice points in λΘ(1, q) because all the respective fibers have length greater than length(v M ) = 1, by construction.Thus π is surjective and (1, q, d(q)) is a valuation point, according to Theorem 6.1.□ Remark 6.3.The case where v N takes its minimum (or maximum) not at a vertex but at an edge is actually easier to handle.As soon as λΘ(1, q) is a lattice polytope, the edge is an integral multiple of v M .So we can omit the first (or last) of the three parts in the above proof.

Strong decomposability
We have seen that it is important to decide the surjectivity of the projection of lattice points in a cone in M R .Next, we will translate this surjectivity into a statement in N .To this end, we introduce the following notion.Definition 6.4.Let σ ⊆ N R be a cone.A lattice point u ∈ int(σ ) ∩ N is strongly decomposable in σ if u = u ′ + u ′′ for suitable u ′ , u ′′ ∈ int(σ ) ∩ N .Lemma 6.5.Let σ ⊆ N R be a cone and u ∈ int(σ ) ∩ N a direction.Then the following statements are equivalent: (i) We have 1 ⟨H σ ∨ , u⟩, where H σ ∨ denotes the Hilbert basis of σ ∨ .(ii) The direction u is strongly decomposable in σ .(iii) The closure of the one-parameter subgroup λ u (C * ) ⊆ T in T V (σ ) is singular.
Proof.(i) and (iii) are equivalent: The 1-parameter subgroup represented by u can always be extended to λ u : C TV (σ ).On the dual level of regular functions, however, this corresponds to The latter map is surjective if and only if 1 ∈ ⟨σ ∨ ∩M, u⟩.
(i) ⇒ (ii): By assumption, there exist primitive lattice points s 0 , s 1 ∈ M \ σ ∨ such that the line segment between them lying on [u = 1] contains no interior lattice point but intersects σ ∨ .Note that {s 0 , s 1 } is a Z-basis because the lattice triangle conv(0, s 0 , s 1 ) is unimodular.Then int cone s 0 , s 1 ⊃ σ ∨ and thus cone t 0 , t 1 ⊂ int(σ ), where {t 0 , t 1 } denotes the basis dual to {s 0 , s 1 }.By the definition of the dual basis, this yields ⟨s i , u⟩ = 1 = ⟨s i , t 0 + t 1 ⟩ for all i; i.e., the two linear functionals coincide on the basis.Therefore, u = t 0 + t 1 .
g., the (i = 0)-case of [AP20, Theorem 2].Remark 3.1.In toric geometry, there is a well-known way to associate to every divisor D a polytope P D reflecting the global sections of O X (D).If (and only if) D is nef, this mapping D P D allows one to recover D, i.e., creates a one-to-one correspondence.Therefore, we introduce another polyhedral gadget ∆(D) providing a complete characterization for more general divisors.If D is nef, then ∆(D) = P D coincides with the former construction.However, for general divisors D = A − B (A, B nef), ∆(D) becomes a virtual polytope, i.e., a formal difference of two true polyhedra that is Minkowski-additive in its arguments.More concretely, we have ∆(A − B) = P A − P B .The global section polytope P D for arbitrary divisors D = A − B can be recovered as P D = (∆(A) : ∆(B)).If D was nef, then we could choose B as the zero divisor, i.e., D = A, and obtain again P A = ∆(A).Example 3.2.Continuing Example 2.2, we obtain
+ 1 the part of the edge of ∆(D) with vertex v 1 lying in the half-plane [v N ≥ c], and by e + 2 the part of the edge of ∆(D) with vertex v 2 also lying in the half-plane [v N ≥ c].The cone σ − ⊆ N R is the cone generated by the inner normal vectors of e + 1 and e + 2 .In the same manner, we define the line segments e − 1 and e − 2 contained in [v N ≤ c], which yield the cone σ + .
strongly decomposable in (σ min ′ ) ∨ for some q if and only if it is strongly decomposable in σ + , and correspondingly for −v N .□The valuation semigroup S Y • (D) is finitely generated if and only if the morphism P 1 X ′ given by v N is a smooth embedding, where X ′ is the toric variety associated with the fan generated by σ + and σ − .Example 4.8], which is a good polytope in the language of op.cit..