Positivity of Line Bundles

and Newton-Okounkov Bodies

Alex Küronya, Victor Lozovanu

Received: November 23, 2015 Revised: April 4, 2017

Communicated by Garvil Farkas

Abstract. The purpose of this paper is to describe asymptotic base loci of line bundles on projective varieties in terms of Newton–Okounkov bodies. As a result, we obtain equivalent characterizations of ampleness and nefness via convex geometry.

2010 Mathematics Subject Classification: 14C20, 32Qxx, 52A20 Keywords and Phrases: positivity of line bundles, ampleness, augmented and restricted base loci, Newton–Okounkov bodies

Introduction

The aim of this work is to characterize positivity (both local and global) of line bundles on complex projective varieties in terms of convex geometry via the theory of Newton–Okounkov bodies. We will provide descriptions of ample and nef divisors, and discuss the relationship between Newton–Okounkov bodies and Nakayama’s σdecomposition. Based on earlier ideas of Khovanskii’s Moscow school and motivated by the work of Okounkov [O], Kaveh–Khovanskii [KKh] and Lazarsfeld–Mustaţă [LM] introduced Newton–Okounkov bodies to projective geometry, where they have been an object of interest ever since. Essentially, a refined book-keeping device encoding the orders of vanishing along subvarieties of the ambient space X, they provide a general framework for the study of the asymptotic behaviour of line bundles on projective varieties. The construction that leads to Newton–Okounkov bodies associates to a line bundle (or more generally, an ℝ-Cartier divisor) on an n-dimensional variety a collection of compact convex bodies Δ_{Y_{•}}(D) ⊆ ℝ^{n} parametrized by certain complete flags Y_{•} of subvarieties. Basic properties of these have been determined [AKL, B1, LM], and their behaviour on surfaces [KLM, LM, LSS] and toric varieties [LM, PSU] has been discussed at length. We refer the reader to the above-mentioned sources for background information.

A distinguishing property of the notion is that it provides a set of ’universal numerical invariants’, since a result of Jow [J] shows that for Cartier divisors D and D′, D is numerically equivalent to D′precisely if the associated functions

Admissible flags Y_{•} in X −→^{Δ_{Y•}(D)} Convex bodies in ℝ^{n}

agree. Turning this principle into practice, one can expect to be able to read off all sorts of numerical invariants of Cartier divisors — among them asymptotic invariants like the volume or Seshadri constants — from the set of Newton–Okounkov bodies of D. On the other hand, questions about global properties of the divisor might arise; whether one can determine ampleness or nefness of a given divisor in terms of its Newton– Okounkov bodies. As we will see, the answer is affirmative. Localizing this train of thought, local positivity of a divisor D at a point x ∈ X will be determined by the function

Admissible flags centered at x −→^{Δ_{Y•}(D)} Convex bodies in ℝ^{n} .

In particular, one can aim at deciding containment of x in various asymptotic base loci, or compute measures of local positivity in terms of these convex sets. In fact the authors have carried out the suggested analysis in the case of smooth surfaces [KL], where the answer turned out to be surprisingly complete. The current article can be rightly considered as a higher-dimensional generalization of [KL]. In search for a possible connection between Newton–Okounkov bodies and positivity, let us start with the toy example of projective curves. For an ℝ-Cartier divisor D on a smooth projective curve C, one has

D nef ⇔ deg_{C}D ⩾ 0 ⇔ 0 ∈ Δ_{P}(D) for some/any point P ∈ C,
D ample ⇔ deg_{C}D > 0 ⇔ Δ_{λ} ⊆ Δ_{P}(D) for some/any point P ∈ C,

where Δ_{λ} := [0,λ] for some real number λ > 0. Interestingly enough, the observation just made generalizes in its entirety for smooth projective surfaces. Namely, one has the following [KL, Theorem A]: for a big ℝdivisor D on a smooth projective surface X

D is nef ⇔ for all x ∈ X there exists a flag (C,x) such that (0,0) ∈ Δ_{(C,x)}(D),
D is ample ⇔ for all x ∈ X there exists a flag (C,x) and λ > 0
such that Δ_{λ} ⊆ Δ_{(C,x)}(D)

where Δ_{λ} denotes the standard full-dimensional simplex of size λ in ℝ^{2}. In higher dimensions we will also denote by Δ_{λ} ⊆ ℝ^{n} the standard simplex of length λ. Our first results are local versions of the analogous statements in higher dimensions.

Theorem A. Let D be a big ℝ-divisor on a smooth projective variety X of dimension n, let x ∈ X. Then the following are equivalent.

(1) x ∉ B_{−} (D).

(2) There exists an admissible flag Y_{•} on X centered at x such that the origin 0 ∈ Δ_{Y_{•}}(D) ⊆ ℝ^{n}.

(3) The origin 0 ∈ Δ_{Y_{•}}(D) for every admissible flag Y_{•} on X centered at x ∈ X.

Theorem B. With notation as above, the following are equivalent.

(1) x ∉ B_{+} (D).

(2) There exists an admissible flag Y_{•} on X centered at x with Y_{1} ample such that Δ_{λ} ⊆ Δ_{Y_{•}(D)}for some positive real number λ.

(3) For every admissible flag Y_{•} on X there exists a real number λ > 0 for which Δ_{λ} ⊆ Δ_{Y_{•}(D)}.

These results will be proven below as Theorem 2.1, and Theorem 3.1, respectively. Making use of the connections between augmented/restricted base loci, we obtain the expected characterizations of nef/ample divisors as in Corollary 2.2 and 3.2. An interesting recent study of local positivity on surfaces was undertaken by Roé [R], where the author introduces the concept of local numerical equivalence, based on the ideas developed in [KL]. Zariski decomposition is a basic tool in the theory of linear series on surfaces, which is largely responsible for the fact that Newton–Okounkov bodies are reasonably well understood in dimension two; the polygonality of Δ_{Y_{•}}(D) in case of a smooth surface is a consequence of variation of Zariski decomposition [BKS] for instance (see [KLM, Section 2] for a discussion). Not surprisingly, the existence and uniqueness of Zariski decompositions is one of the main tools used in [KL]. Its relationship to Newton–Okounkov polygons on surfaces is particularly simple: if D is a big ℝ-divisor with the property that the point Y_{2} in the flag Y_{•} is not contained in the support of the negative part of D, then Δ_{Y_{•}}(D) = Δ_{Y_{•}}(P_{D}), where P_{D} stands for the positive part of D. In dimensions three and above, the appropriate birational version of Zariski decomposition — the so-called CKM decomposition — only exists under fairly restrictive hypotheses, hence one needs substitutes whose existence is guaranteed while they still retain some of the favourable properties of the original notion. A widely accepted concept along these lines is Nakayama’s divisorial Zariski decomposition or σ-decomposition, which exists for an arbitrary big ℝ-divisor, but where the ’positive part’ is only guaranteed to be movable (see [N, Chapter 3] or [B2]). Extending the observation coming from dimension two, we obtain the following.

Theorem C. Let X be a smooth projective variety, D a big ℝ-divisor, Γ a prime divisor, Y_{•} : Y_{0} = X ⊇ Y_{1} = Γ ⊇ ... ⊇ Y_{n} = {x} and admissible flag on X. Then

(1) Δ_{Y_{•}}(D) ⊆ (σ_{Γ}(D),0...,0) + ℝ_{+}^{n},

(2) (σ_{Γ}(D),0...,0) ∈ Δ_{Y_{•}}(D), whenever x ∈ Γ is a very general point.

(3) Δ_{Y_{•}}(D) = ν_{Y_{•}}(N_{σ} (D)) + Δ_{Y_{•}}(P_{σ} (D)). Morever, Δ_{Y_{•}}(D) = Δ_{Y_{•}}(P_{σ} (D)), when x ∉ Supp(N_{σ} (D)).

The organization of the paper goes as follows: Section 1 fixes notation, and collects some preliminary information about asymptotic base loci and Newton–Okounkov bodies. Sections 2 and 3 are devoted to the respective proofs of Theorems A and B, while Section 4 describes the relationship between Newton–Okounkov bodies and Nakayama’s σ-decomposition. Acknowledgements. We are grateful for helpful discussions to Sébastian Boucksom, Lawrence Ein, John Christian Ottem, Mihnea Popa, and Stefano Urbinati. Parts of this work were done while the authors attended the MFO workshop on Newton– Okounkov bodies, the Summer School in Geometry at University of Milano–Bicocca, and the RTG Workshop on Newton–Okounkov bodies at the University of Illinois at Chicago. We would like to thank the organizers of these events for the opportunity (Megumi Harada, Kiumars Kaveh, Askold Khovanskii; Francesco Bastianelli, Roberto Paoletti; Izzet Coskun and Kevin Tucker). Alex Küronya was partially supported by the DFG-Forschergruppe 790 “Classification of Algebraic Surfaces and Compact Complex Manifolds”, by the DFG-Graduierten-kolleg 1821 “Cohomological Methods in Geometry”, and by the OTKA grants 77476 and 81203 of the Hungarian Academy of Sciences.

1. Notation and preliminaries

1.1. Notation. For the duration of this work let X be a smooth complex projective variety of dimension n and D be a Cartier divisor on X. An admissible flag of subvarieties

Y_{•} : X = Y_{0} ⊇ Y_{1} ⊇ ... ⊇ Y_{n−1} ⊇ Y_{n} = {pt.},

is a complete flag with the property that each Y_{i} is an irreducible subvariety of codimension i and smooth at the point Y_{n}. For an arbitrary point x ∈ X, we say that Y_{•} is centered at x whenever Y_{n} = x. The associated Newton–Okounkov body will be denoted by Δ_{Y_{•}}(D) ⊆ ℝ_{+}^{n} (for the actual construction and its basic properties we refer the reader to [KKh, LM]).

Remark 1.1. Not all of our results require X to be smooth, at points it would suffice to require X to be merely a projective variety. As a rule though, we will not keep track of minimal hypotheses.

1.2. Asymptotic base loci. Stable base loci are fundamental invariants of linear series, however, as their behaviour is somewhat erratic (they do not respect numerical equivalence of divisors for instance), other alternatives were in demand. To remedy the situation, Nakamaye came up with the idea of studying stable base loci of small perturbations. Based on this, the influential paper [ELMNP1] introduced new asymptotic notions, the restricted and augmented base loci of a big divisor D. The restricted base locus of a big ℝ-divisor D is defined as

B_{−} (D) =^{def} ⋃_{A} B(D + A) ,

where the union is over all ample ℚ-divisors A on X. This locus turns out to be a countable union of subvarieties of X (and one really needs a countable union on occasion, see [L]), and, via [ELMNP1, Proposition 1.19]

B_{−} (D) = ⋃_{m∈ℕ}B(D + α_{m}) ,

where α_{m} ↘ 0 is any sequence of ample ℝ-divisor classes such that D + α_{m} is a ℚdivisor class for all m ∈ ℕ.

The augmented base locus of an ℝ-divisor D is defined to be

B_{+} (D) =^{def} ⋂_{A} B(D − A),

where the intersection is taken over all ample ℚ-divisors A on X. It follows quickly from [ELMNP1, Proposition 1.5] that B_{+} (D) = B(D − α) for any sufficiently small ℝ-divisor class α. Augmented and restricted base loci satisfy various favorable properties; for instance both B_{+} (D) and B_{−} (D) depend only on the numerical class of D, hence are much easier to study (see [ELMNP1, Corollary 2.10] and [PAG2, Example 11.3.12]). Below we make a useful remark regarding augmented/restricted base loci. The statement must be well-known to experts, as usual, we include it with proof for the lack of a suitable reference.

Proposition 1.2. Let X be a projective variety, x ∈ X an arbitrary point. Then

(1) B_{+} (x) =^{def}{α ∈ N^{1}(X)_{ℝ} | x ∈ B_{+} (α)} ⊆ N^{1}(X)_{ℝ} is closed,

(2) B_{−} (x) =^{def}{α ∈ N^{1}(X)_{ℝ} | x ∈ B_{−} (α)} ⊆ N^{1}(X)_{ℝ} is open,

both with respect to the metric topology of N^{1}(X)_{ℝ}.

Remark 1.3. We point out that unlike required in [ELMNP1], one does not need the normality assumption on X for [ELMNP1, Corollary 1.6] to hold.

Proof. (i) First we deal with the case of augmented base loci. Observe that it suffices to prove that

B_{+} (x) ∩ Big(X) ⊆ Big(X) is closed,

since the big cone is open in the Néron–Severi space. We will show that whenever (α_{n})_{n∈ℕ} is a sequence of big ℝ-divisor classes in B(x) converging to α ∈ Big(X), then α ∈ B_{+} (x) as well. By [ELMNP1, Corollary 1.6], the class α has a small open neighbourhood 𝒰 in the big cone for which

β ∈ 𝒰 ⟹ B_{+} (β) ⊆ B_{+} (α) .

If x ∈ B_{+} (α_{n}) for infinitely many n ∈ ℕ, then since α_{n} ∈ U for n large, we also have x ∈ B_{+} (α). (ii) Let α ∈ N^{1}(X)_{ℝ} be arbitrary, and fix an ℝ-basis A_{1},...,A_{ρ} of N^{1}(X)_{ℝ} consisting of ample divisor classes. Observe that x ∈ B_{−} (α) implies that x ∈ B_{−} (α + t_{0}∑ _{i=1}^{ρ} A_{i}) for some t_{0} > 0 thanks to the definition of the restricted base locus. Since subtracting ample classes cannot decrease B_{−} , it follows that x ∈ B_{−} (α) yields x ∈ B_{−} (γ) for all classes of the form α + t_{0}∑ _{i=1}^{ρ} A_{i} − ∑ _{i=1}^{ρ} ℝ_{⩾0}A_{i}, which certainly contains an open neighbourhood of α ∈ N^{1}(X)_{ℝ}. □

1.3. Newton–Okounkov bodies. We start with a sligthly different definition of Newton–Okounkov bodies; it has already appeared in print in [KLM], and although it is an immediate consequence of [LM], a complete proof was first given in [B1]Proposition 4.1.

Proposition 1.4 (Equivalent definition of Newton-Okounkov bodies). Let ξ ∈ N^{1}(X)_{ℝ} be a big ℝ-class and Y_{•} be an admissible flag on X. Then

Δ_{Y_{•}}(ξ) = closed convex hull of {ν_{Y_{•}}(D) | D ∈ Div_{⩾0}(X)_{ℝ},D ≡ ξ},

where the valuation ν_{Y_{•}}(D), for an effective ℝ-divisor D, is constructed inductively as in the case of integral divisors.

Remark 1.5. Just as in the case of the original definition of Newton–Okounkov bodies, it becomes a posteriori clear that valuation vectors ν_{Y_{•}}(D) form a dense subset of

closed convex hull of {ν_{Y_{•}}(D) | D ∈ Div_{⩾0}(X)_{ℝ},D ≡ ξ} ,

hence it would suffice to take closure in Proposition 1.4.

The description of Newton–Okounkov bodies above is often more suitable to use than the original one. For example, the following statement follows immediately from it.

Proposition 1.6. Suppose ξ is a big ℝ-class and Y_{•} is an admissible flag on X. Then for any t ∈ [0,μ(ξ,Y_{1})), we have

Δ_{Y_{•}}(ξ)_{ν_{1}⩾t} = Δ_{Y_{•}}(ξ − tY_{1}) + te_{1},

where μ(ξ,Y_{1}) = sup{μ > 0|ξ − μY_{1} is big} and e_{1} = (1,0,...,0) ∈ ℝ^{n}.

This statement first appeared in [LM, Theorem 4.24] with the additional condition that Y_{1} ⊈ B_{+} (ξ). We will need a version of [AKL, Lemma 8] for real divisors.

Lemma 1.7. Let D be a big ℝ-divisor, Y_{•} an admissible flag on X. Then the following hold.

(1) For any real number ϵ > 0 and any ample ℝ-divisor A on X, we have Δ_{Y_{•}}(D) ⊆ Δ_{Y_{•}}(D + ϵA).

(2) If α is an arbitrary nef ℝ-divisor class, then Δ_{Y_{•}}(D) ⊆ Δ_{Y_{•}}(D + α).

(3) If α_{m}is any sequence of nef ℝ-divisor classes with the property that α_{m}−α_{m+1} is nef and ∥α_{m}∥ → 0 as m → ∞ with respect to some norm on N^{1}(X)_{ℝ}, then

Δ_{Y_{•}}(D) = ⋂_{m} Δ_{Y_{•}}(D + α_{m}) .

Proof. For the first claim, since A is an ample ℝ-divisor, one can find an effective ℝdivisor M ∼_{ℝ} A with Y_{n} ∉ Supp(M). Then for any arbitrary effective divisor F ∼_{ℝ} D one has F + ϵM ≡_{ℝ} D + ϵA and ν_{Y_{•}}(F + M) = ν_{Y_{•}}(F). Therefore

{ν_{Y_{•}}(ξ) | ξ ∈ Div_{⩾0}(X)_{ℝ},D ≡ ξ} ⊆ {ν_{Y_{•}}(ξ) | ξ ∈ Div_{⩾0}(X)_{ℝ},D + ϵA ≡ ξ}

and we are done by Proposition 1.4. For (ii), note first that whenever α is ample, we can write α = ∑ _{i=1}^{r} ϵ_{i}A_{i} for suitable real numbers ϵ_{i} > 0 and ample integral classes A_{i}, therefore an interated application of (i) gives the claim. The general case then follows by continuity and and approximating a nef ℝ-divisor class by a sequence of ample ones. The equality in (iii) is a consequence of (ii) and the continuity of Newton–Okounkov bodies. □

2. Restricted base loci

Our main goal here is to give a characterization of restricted base loci in the language of Newton–Okounkov bodies.

Theorem 2.1. Let D be a big ℝ-divisor on a smooth projective variety X of dimension n, let x ∈ X. Then the following are equivalent.

(1) x ∉ B_{−} (D).

(2) There exists an admissible flag Y_{•} on X centered at x such that 0 ∈ Δ_{Y_{•}}(D) ⊆ ℝ^{n}.

(3) The origin 0 ∈ Δ_{Y_{•}}(D) for every admissible flag Y_{•} on X centered at x ∈ X.

Coupled with simple properties of restricted base loci we arrive at a precise description of big and nef divisors in terms of convex geometry.

Corollary 2.2. With notation as above the following are equivalent for a big ℝdivisor D.

(1) D is nef.

(2) For every point x ∈ X there exists an admissible flag Y_{•} on X centered at x such that 0 ∈ Δ_{Y_{•}}(D) ⊆ ℝ^{n}.

(3) For every admissible flag Y_{•}, one has 0 ∈ Δ_{Y_{•}}(D).

Proof. Immediate from Theorem 2.1 and [ELMNP1, Example 1.18]. □

The essence of the proof of Theorem 2.1is to connect the asymptotic multiplicity of D at x to a certain function defined on the Newton-Okounkov body of D. Before turning to the actual proof, we will quickly recall the notion of the asymptotic multiplicity or the asymptotic order of vanishing of a ℚ-divisor F at a point x ∈ X. Let F be an effective Cartier divisor on X, defined locally by the equation f ∈ 𝒪_{X,x}. Then multiplicity of F at x is defined to be mult_{x}(F) = max{n ∈ ℕ|f ∈ 𝔪_{X,x}^{n}}, where 𝔪_{X,x}denotes the maximal ideal of the local ring 𝒪_{X,x}. If |V | is a linear series, then the multiplicity of |V | is defined to be

mult_{x}(|V |) =^{def} min_{F ∈|V |}{mult_{x}(F)} .

By semicontinuity the above expression equals the multiplicity of a general element in |V | at x. The asymptotic multiplicity of a ℚ-divisor D at x is then defined to be

mult_{x}(||D||) =^{def} lim_{p→∞} \frac{mult_{x}(|pD|)}{p} .

The multiplicity at x coincides with the order of vanishing at x, given in Definition 2.9 from [ELMNP1]. In what follows we will talk about the multiplicity of a divisor, but the order of vanishing of a section of a line bundle. An important technical ingredient of the proof of Theorem 2.1 is a result of [ELMNP1], which we now recall.

Proposition 2.3. ([ELMNP1, Proposition 2.8]) Let D be a big ℚ-divisor on a smooth projective variety X, x ∈ X an arbitrary (closed) point. Then the following are equivalent.

(1) There exists C > 0 having the following property: if |pD| ≠ ∅ for some positive integer p, then mult_{x}(|pD|) < C.

(2) mult_{x}(∥D∥) = 0.

(3) x ∉ B_{−} (D).

The connection between asymptotic multiplicity and Newton–Okounkov bodies comes from the claim below.

Lemma 2.4. Let M be an integral Cartier divisor on a projective variety X (not necessarily smooth), s ∈ H^{0}(X,𝒪_{X}(M)) a non-zero global section. Then

(2.4.1)  ord_{x}(s) ⩽ ∑_{i=1}^{i=n} ν_{i}(s),

for any admissible flag Y_{•} centered x, where ν_{Y_{•}}= (ν_{1},...,ν_{n}) is the valuation map arising from Y_{•}.

Proof. Since Y_{•} is an admissible flag and the question is local, we can assume without loss of generality that each element in the flag is smooth, thus Y_{i} ⊆ Y_{i−1} is Cartier for each 1 ⩽ i ⩽ n. As the local ring 𝒪_{X,x}is regular, order of vanishing is multiplicative. Therefore

ord_{x}(s) = ν_{1}(s) + ord_{x}(s − ν_{1}(s)Y_{1}) ⩽ ν_{1}(s) + ord_{x}((s − ν_{1}(s)Y_{1})|_{Y_{1}})

by the very definition of ν_{Y_{•}}(s), and the rest follows by induction. □

Remark 2.5. Note that the inequality in (2.4.1) is not in general an equality for the reason that the zero locus of s might not intersect an element of the flag transversally. For the simplest example of this phenomenon set X = ℙ^{2}, and take s = xz − y^{2} ∈ H^{0}(ℙ^{2},𝒪_{ℙ^{2}}(2)), Y_{1} = {x = 0} and Y_{2} = [0 : 0 : 1]. Then clearly ν_{1}(s) = 0, and ν_{2}(s) = ord_{Y_{2}}(−y^{2}) = 2, but since Y_{2} is a smooth point of (s)_{0} = {xz − y^{2} = 0}, ord_{Y_{2}}(s) = 1 and hence ord_{Y_{1}}(s) < ν_{1}(s) + ν_{2}(s).

For a compact convex body Δ ⊆ ℝ^{n}, we define the sum function σ : Δ → ℝ_{+} by σ(x_{1},...,x_{n}) = x_{1} + ...+ x_{n}. Being continuous on a compact topological space, it takes on its extremal values. If Δ_{Y_{•}}(D) ⊆ ℝ^{n} is a Newton–Okounkov body, then we write σ_{D} for the sum function on Δ_{Y_{•}}(D) (suppressing the fact that it depends on the choice of the flag Y_{•}).

Proposition 2.6. Let D be a big ℚ-divisor on a projective variety X (not necessarily smooth) and let x ∈ X a point. Then

(2.6.2)  mult_{x}(||D||) ⩽ minσ_{D}.

for any admissible flag Y_{•} centered at x.

Proof. Since both sides of (2.6.2) are homogeneous of degree one in D, we can assume without loss of generality that D is integral. Fix a natural number p ⩾ 1 such that |pD| ≠ ∅, and let s ∈ H^{0}(X,𝒪_{X}(pD)) be a non-zero global section. Then

\frac{1}{p} mult_{x}(|pD|) ⩽ \frac{1}{p} ord_{x}(s) ⩽ \frac{1}{p}( ∑_{i=1}^{i=n} ν_{i}(s))

by Lemma 2.4. Multiplication of sections and the definition of the multiplicity of a linear series then yields mult_{x}(|qpD|) ⩽ qmult_{x}(|pD|) for any q ⩾ 1, which, after taking limits leads to

mult_{x}(||D||) ⩽ \frac{1}{p} mult_{x}(|pD|) ⩽ \frac{1}{p}( ∑_{i=1}^{i=n} ν_{i}(s)).

Varying the section s and taking into account that Δ_{Y_{•}}(D) is the closure of the set of normalized valuation vectors of sections, we deduce the required statement. □

Example 2.7. The inequality in (2.6.2) is usually strict. For a concrete example take X = Bl_{P}(ℙ^{2}), D = π^{∗}(H) + E and the flag Y_{•} = (C,x), where C ∈ |3π^{∗}(H) − 2E| is the proper transform of a rational curve with a single cusp at P, and {x} = C ∩ E, i.e. the point where E and C are tangent to each other. Then

mult_{x}(||D||) = lim_{p→∞} ( \frac{mult_{x}(|pD|)}{p}) = lim_{p→∞} ( \frac{mult_{x}(|pE|)}{p}) = 1 .

On the other hand, a direct computation using [LM, Theorem 6.4] shows that

Δ_{Y_{•}}(D) = \{(t,y) ∈ ℝ^{2} | 0 ⩽ t ⩽ \frac{1}{3}, and 2 + 4t ⩽ y ⩽ 5 − 5t\} .

As a result, minσ_{D} = 2 > 1. For more on this phenomenon, see Proposition 2.10 below.

Remark 2.8. We note here a connection with functions on Okounkov bodies coming from divisorial valuations. With the notation of [BKMS], our Lemma 2.4 says that ϕord_{x} ⩽ σ_{D}, and a quick computation shows that we obtain equality in the case of projective spaces, hyperplane bundles, and linear flags. Meanwhile, Example 2.7 illustrates that minϕord_{x} ≠ mult_{x}∥D∥ in general.

Proof of Theorem 2.1. (1) ⇒ (3) We are assuming x ∉ B_{−} (D); let us fix a sequence of ample ℝ-divisor classes α_{m} with the properties that α_{m}− α_{m+1} is ample for all m ⩾ 1, ∥α_{m}∥ → 0 as m → ∞, and such that D + α_{m} is a ℚ-divisor. Let Y_{•} be an arbitrary admissible flag centered at x. Then x ∉ B(D + α_{m}) for every m ⩾ 1, furthermore, Lemma 1.7 yields

(2.8.3)  Δ_{Y_{•}}(D) = ⋂_{m=1}^{∞} Δ_{Y_{•}}(D + α_{m}) .

Because x ∉ B(D + α_{m}) holds for any m ⩾ 1, there must exist a sequence of natural numbers n_{m}⩾ 1 and a sequence of global sections s_{m}∈ H^{0}(X,𝒪_{X}(n_{m}(D+α_{m}))) such that s_{m}(x) ≠ 0. This implies that ν_{Y_{•}}(s_{m}) = 0 for each m ⩾ 1. In particular, 0 ∈ Δ_{Y_{•}}(D + α_{m}) for each m ⩾ 1. By (2.8.3) we deduce that Δ_{Y_{•}}(D) contains the origin as well. The implication (3) ⇒ (2) being trivial, we will now take care of (2) ⇒ (1). To this end assume that Y_{•} is an admissible flag centered at x having the property that 0 ∈ Δ_{Y_{•}}(D), α_{m} a sequence of ample ℝ-divisor classes such that α_{m}− α_{m+1} is nef, ∥α_{m}∥ → 0 as m → ∞, and D + α_{m} is a ℚ-divisor for all m ⩾ 1.

By Lemma 1.7,

0 ∈ Δ_{Y_{•}}(D) ⊆ Δ_{Y_{•}}(D + α_{m})

for all m ⩾ 0, whence minσ_{D+α_{m}} = 0 for all sum functions σ_{D+α_{m}} : Δ_{Y_{•}}(D + α_{m}) → ℝ_{+} . By Proposition 2.6 this forces mult_{x}(||D + α_{m}||) = 0 for all m ⩾ 1, hence [ELMNP1, Proposition 2.8] leads to x ∉ B_{−} (D + α_{m}) for all m ⩾ 1. As

B_{−} (D) = ⋃_{m} B_{−} (D + α_{m}) = ⋃_{m} B(D + α_{m})

according to [ELMNP1, Proposition 1.19], we are done. □

Remark 2.9. A closer inspection of the above proof reveals that the implication (1) ⇒ (3) holds on an arbitrary projective variety both in Theorem 2.1 and Corollary 2.2.

We finish with a precise version of Proposition 2.6 in the surface case, which also provides a complete answer to the question of where the Newton-Okounkov body starts in the plane. Note that unlike Theorem 4.2, it gives a full description for an arbitrary flag.

Proposition 2.10. Let X be a smooth projective surface, (C,x) an admissible flag, D a big ℚ-divisor on X with Zariski decomposition D = P(D) + N(D). Then

(1) minσ_{D} = a + b, where a = mult_{C}(N(D)) and b = mult_{x}(N(D − aC)|_{C}),

(2) mult_{x}(||D||) = a + b′, where b′= mult_{x}(N(D − aC)). Moreover, (a,b) ∈ Δ_{(C,x)}(D) and Δ_{(C,x)}(D) ⊆ (a,b) + ℝ_{+}^{2} .

Proof. (1) This is an immediate consequence of [LM, Theorem 6.4] in the light of the fact that α is an increasing function, hence minσ_{D} is taken up at the point (a,α(a)).

(2) Since x is a smooth point, it will suffice to check that mult_{x}(||D||) = mult_{x}(N(D)). As asymptotic multiplicity is homogeneity of degree one (see [ELMNP1, Remark 2.3]), we can safely assume that D,P(D) and N(D) are all integral. As one has isomorphisms H^{0}(X,𝒪_{X}(mP(D))) → H^{0}(X,𝒪_{X}(mD)) for all m ⩾ 1 by [PAG1, Proposition 2.3.21], the definition of asymptotic multiplicity yields

mult_{x}(||D||) = mult_{x}(||P(D)||) + mult_{x}(N(D)) .

Observe that P(D) is big and nef therefore [PAG1, Proposition 2.3.12] implies mult_{x}(||P(D)||) = 0. This completes the proof. □

3. Augmented base loci

As explained in [ELMNP1, Example 1.16], one has inclusions B_{−} (D) ⊆ B(D) ⊆ B_{+} (D), consequently, we expect that whenever x ∉ B_{+} (D), Newton–Okounkov bodies attached to D should contain more than just the origin. As we shall see below, it will turn out that under the condition above they in fact contain small simplices. We will write

Δ_{ϵ} =^{def} {(x_{1},...,x_{n}) ∈ ℝ_{+}^{n} | x_{1}+ ...+ x_{n} ⩽ ϵ}

for the standard ϵ-simplex. Our main statement is the following.

Theorem 3.1. Let D be a big ℝ-divisor on a smooth projective variety X, x ∈ X be an arbitrary (closed) point. Then the following are equivalent.

(1) x ∉ B_{+} (D).

(2) There exists an admissible flag Y_{•} centered at x with Y_{1} ample such that Δ_{ϵ_{0}} ⊆ Δ_{Y_{•}}(D) for some ϵ_{0} > 0.

(3) For every admissible flag Y_{•} centered at x there exists ϵ > 0 (possibly depending on Y_{•}) such that Δ_{ϵ} ⊆ Δ_{Y_{•}}(D).

Corollary 3.2. Let X be a smooth projective variety, D a big ℝ-divisor on X. Then the following are equivalent.

(1) D is ample.

(2) For every point x ∈ X there exists an admissible flag Y_{•} centered at x with Y_{1} ample such that Δ_{ϵ_{0}} ⊆ Δ_{Y_{•}}(D) for some ϵ_{0} > 0.

(3) For every admissible flag Y_{•} there exists ϵ > 0 (possibly depending on Y_{•}) such that Δ_{ϵ} ⊆ Δ_{Y_{•}}(D).

Proof of Corollary 3.2. Follows immediately from Theorem 3.1 and [ELMNP1, Example 1.7]. □

One can see Corollary 3.2 as a variant of Seshadri’s criterion for ampleness in the language of convex geometry.

Remark 3.3. It is shown in [KL, Theorem 2.4] and [KL, Theorem A] that in dimension two one can in fact discard the condition above that Y_{1} should be ample. Note that the proofs of the cited results rely heavily on surface-specific tools and in general follow a line of thought different from the present one.

We first prove a helpful lemma.

Lemma 3.4. Let X be a projective variety (not necessarily smooth), A an ample Cartier divisor, Y_{•} an admissible flag on X. Then for all m >> 0 there exist global sections s_{0},...,s_{n} ∈ H^{0}(X,𝒪_{X}(mA)) for which

ν_{Y_{•}}(s_{0}) = 0 and ν_{Y_{•}}(s_{i}) = e_{i}, for each i = 1,...,n,

where {e_{1},...,e_{n}} ⊆ ℝ^{n} denotes the standard basis.

Proof. First, we point out that by the admissibility of the flag Y_{•}, we know that there is an open neighbourhood 𝒰 of x such that Y_{i}|_{𝒰} is smooth for all 0 ⩽ i ⩽ n. Since A is ample, 𝒪_{X}(mA) becomes globally generated for m >> 0. For all such m there exists a non-zero section s_{0} ∈ H^{0}(X,𝒪_{X}(mA)) with s_{0}(Y_{n}) ≠ 0, in particular, ν_{Y_{•}}(s_{0}) = 0, as required. It remains to show that for all m >> 0 and i = 1 ⩽ i ⩽ n we can find non-zero sections s_{i} ∈ H^{0}(X,𝒪_{X}(mA)) with ν_{Y_{•}}(s_{i}) = e_{i}. To this end, fix i and let y ∈ Y_{i} \ Y_{i+1} be a smooth point. Having chosen m large enough, Serre vanishing yields H^{1}(X,ℐ_{Y_{i}|X}⊗ 𝒪_{X}(mA)) = 0, hence the map ϕ_{m} in the diagram

\xymatrix{ &  & H^{0}(X,𝒪_{X}(mA)) \ar[d]_{ϕ_{m}} \\ 0 \ar[r] & H^{0}(Y_{i},𝒪_{Y_{i}}(m(A|_{Y_{i}}) − Y_{i+1} )) \ar[r]^{ψ_{m}} & H^{0}(Y_{i},𝒪_{Y_{i}}(mA))}

is surjective. Again, by making m high enough, we can assume |m(A|_{Y_{i}})−Y_{i+1} | to be very ample on Y_{i}, thus, there will exist 0 ≠ s̃_{i} ∈ H^{0}(Y_{i},𝒪_{Y_{i}}(mA) ⊗ 𝒪_{Y_{i}}(−Y_{i+1} )) not vanishing at x or y. Since s̃_{i}(x) ≠ 0, the section s̃_{i} does not vanish along Y_{j} for all j = i + 1,...,n. Also, the image ψ_{m}(s̃_{i}) ∈ H^{0}(Y_{i},𝒪_{Y_{i}}(mA)) of s̃_{i} vanishes at x, but not at the point y. By the surjectivity of the map ϕ_{m} there exists a section s_{i} ∈ H^{0}(X,𝒪_{X}(mA)) such that s|_{Y_{i}}= ψ_{m}(s̃_{i}) and s(y) ≠ 0. In particular, s_{i} does not vanish along any of the Y_{j}’s for 1 ⩽ i ⩽ j, therefore ν_{Y_{•}}(s) = e_{i}, as promised. □

Proof of Theorem 3.1. (1) ⇒ (3). First we treat the case when D is ℚ-Cartier. Assume that x ∉ B_{+} (D), which implies by definition that x ∉ B(D−A) for some small ample ℚ-Cartier divisor A. Choose a positive integer m large and divisible enough such that mA becomes integral, and satisfies the conclusions of Lemma 3.4. Assume furthermore that B(D − A) = Bs(m(D − A)) set-theoretically. Since x ∉ Bs(m(D−A)), there exists a section s ∈ H^{0}(X,𝒪_{X}(mD−mA)) not vanishing at x, and in particular ν_{Y_{•}}(s) = 0. At the same time, Lemma 3.4 provides the existence of global sections s_{0},...,s_{n} ∈ H^{0}(X,𝒪_{X}(mA)) with the property that ν_{Y_{•}}(s_{0}) = 0 and ν_{Y_{•}}(s_{i}) = e_{i} for all 1 ⩽ i ⩽ n. But then the multiplicativity of the valuation map ν_{Y_{•}}gives

ν_{Y_{•}}(s ⊗ s_{0}) = 0, and ν_{Y_{•}}(s ⊗ s_{i}) = e_{i} for all 1 ⩽ i ⩽ n.

By the construction of Newton–Okounkov bodies, then Δ_{\frac{1}{m}} ⊆ Δ_{Y_{•}}(D). Next, let D be a big ℝ-divisor for which x ∉ B_{+} (D), and let A be an ample ℝ-divisor with the property that D − A is a ℚ-divisor and B_{+} (D) = B_{+} (D − A). Then we have x ∉ B_{+} (D − A), therefore

Δ_{ϵ} ⊆ Δ_{Y_{•}}(D − A) ⊆ Δ_{Y_{•}}(D)

according to the ℚ-Cartier case and Lemma 1.7. Again, the implication (3) ⇒ (2) is trivial, hence we only need to take care of (2) ⇒ (1). As Y_{1} is ample, [ELMNP1, Proposition 1.21] gives the equality B_{−} (D − ϵY_{1}) = B_{+} (D). for all 0 < ϵ << 1. Fix an ϵ as above, subject to the additional condition that D − ϵY_{1} is a big ℝ-divisor. Then, according to Proposition 1.6, we have

Δ_{Y_{•}}(D)_{ν_{1}⩾ϵ} = Δ_{Y_{•}}(D − ϵY_{1}) + ϵe_{1} ,

which yields 0 ∈ Δ_{Y_{•}}(D − ϵY_{1}). By Theorem 2.1, this means that x ∉ B_{−} (D − ϵY_{1}) = B_{+} (D), which completes the proof. □

Remark 3.5. The condition that X be smooth can again be dropped for the implication (1) ⇒ (3) both in Theorem 3.1and Corollary 3.2(cf. Remark 2.9). This way, one obtains the statement that whenever A is an ample ℝ-Cartier divisor on a projective variety X, then every Newton–Okounkov body of A contains a small simplex.

As a consequence, we can extend [KL, Definition 4.5] to all dimensions.

Definition 3.6 (Largest simplex constant). Let X be an arbitrary projective variety, x ∈ X a smooth point, A an ample ℝ-divisor on X. For an admissible flag Y_{•} on X centered at x, we set

λ_{Y_{•}}(A;x) =^{def} sup{λ > 0 | Δ_{λ} ⊆ Δ_{Y_{•}}(A)} .

Then the largest simplex constant λ(A;x) is defined as

λ(A;x) =^{def} sup{λ_{Y_{•}}(A;x) | Y_{•} is an admissible flag centered at x}.

Remark 3.7. It follows from Remark 3.5 that λ(A;x) > 0. The largest simplex constant is a measure of local positivity, and it is known in dimension two that λ(A;x) ⩽ ϵ(A;x) (where the right-hand side denotes the appropriate Seshadri constant) with strict inequality in general (cf. [KL, Proposition 4.7] and [KL, Remark 4.9]).

We end this section with a different characterization of B_{+} (D) which puts no restriction on the flags. In what follows X is again assumed to be smooth.

Lemma 3.8. Let D be a big ℚ-divisor on X. For a point x ∈ X, the condition x ∉ B_{+} (D) holds if and only if

(3.8.4)  lim_{p→∞} mult_{x}(||pD − A||) = 0

for some ample divisor A.

Proof. Assuming (3.8.4), x ∉ B_{+} (D) follows from [ELMNP2, Lemma 5.2]. For the converse implication, consider the equalities

B_{+} (D) = B_{−} (D − \frac{1}{p}A) = B(pD − A)

which hold for integers p ≫ 0. Hence, if x ∉ B_{+} (D), then x ∉ B(pD−A) for all p ≫ 0. But this latter condition implies mult_{x}(||pD − A||) = 0 for all p ≫ 0. □

Proposition 3.9. A point x ∉ B_{+} (D) if and only if there exists an admissible flag Y_{•} based at x satisfying the property that for any ϵ > 0 there exists a natural number p_{ϵ} > 0 such that

Δ_{ϵ} ⋂ Δ_{Y_{•}}(pD − A) ≠ ∅

for any p ⩾ p_{ϵ}.

Proof. Assume first that x ∉ B_{+} (D). Again, by [ELMNP1, Proposition 1.21], we have B_{+} (D) = B_{−} (D − \frac{1}{p}A) = B_{−} (pD − A) for all p ≫ 0. Then x ∉ B_{−} (pD − A) for all p ≫ 0, hence 0 ∈ Δ_{Y_{•}}(pD − A) for all p ≫ 0 by Theorem 2.1, which implies Δ_{ϵ} ∩ Δ_{Y_{•}}(pD − A) ≠ ∅ for all p ≫ 0. As far as the converse implication goes, Proposition 2.6 shows that

mult_{x}(||pD − A||) ⩽ minσ_{pD−A}. ,

hence the condition in the statement implies lim_{p→∞} mult_{x}(||pD − A||) = 0. But then we are done by Lemma 3.8. □

4. Nakayama’s divisorial Zariski decomposition and Newton–Okounkov bodies

In the previous sections we saw the basic connections between Newton–Okounkov bodies associated to a big line bundle D and the asymptotic base loci B_{+} (D) and B_{−} (D). In [N], Nakayama performes a deep study of these loci, he shows for instance that B_{−} (D) can only have finitely many divisororial components. Along the way he introduces his σ-invariant, which measures the asymptotic multiplicity of divisorial components of B_{−} (D). The goal of this section is to study the connection between divisorial Zariski decompositions and Newton–Okounkov bodies. First, we briefly recall the divisorial Zariski decomposition or σ-decomposition introduced by Nakayama [N] and Boucksom [B2]. Let X be a smooth projective variety, D a pseudo-effective ℝ-divisor on X. Although B_{−} (D) is a countable union of closed subvarieties, [N, Theorem 3.1] shows that it only has finitely many divisorial components. Let A be an ample divisor. Following Nakayama, for each prime divisor Γ on X we set

σ_{Γ} =^{def} lim_{ϵ→0^{+}} inf{mult_{Γ}(D′) | D′∼_{ℝ} D + ϵA and D′⩾ 0} .

In [N, Theorem III.1.5], Nakayama shows that these numbers do not depend on the choice of A and that there are only finitely many prime divisors Γ with σ_{Γ}(D) > 0. Write

N_{σ} (Γ) =^{def} ∑_{Γ} σ_{Γ}(D)Γ and P_{σ} (D) = D − N_{σ} (D) ,

and we call D = P_{σ} (D) + N_{σ} (D) the divisorial Zariski decomposition or σdecomposition of D. In dimension two the divisorial Zariski decomposition coincides with the usual Fujita–Zariski decomposition for pseudo-effective divisors. The main properties are captured in the following statement.

Theorem 4.1. [N, III.1.4, III.1.9, V.1.3] Let D be a pseduo-effective ℝ-disivor. Then (1) N_{σ} (D) is effective and Supp(N_{σ} (D)) coincides with the divisorial part of B_{−} (D).

(2) For all m ⩾ 0, H^{0}(X,𝒪_{X}(⌊mP_{σ} (D)⌋)) ≃ H^{0}(X,𝒪_{X}(⌊mD⌋)).

As Theorem 2.1 describes how to determine B_{−} (D) from the Newton–Okounkov bodies associated to D, it is natural to wonder how we can compute the numbers σ_{Γ}(D) and N_{σ} (D) in terms of convex geometry. Relying on Theorem 2.1 and Nakayama’s work, we are able to come up with a reasonable answer.

Theorem 4.2. Let D be a big ℝ-divisor, Γ a prime divisor on X, Y_{•} : Y_{0} = X ⊇ Y_{1} = Γ ⊇ ... ⊇ Y_{n} = {x} an admissible flag on X. Then

(1) Δ_{Y_{•}}(D) ⊆ (σ_{Γ}(D),0...,0) + ℝ_{+}^{n},

(2) (σ_{Γ}(D),0...,0) ∈ Δ_{Y_{•}}(D), whenever x ∈ Γ is a very general point.

(3) Δ_{Y_{•}}(D) = ν_{Y_{•}}(N_{σ} (D)) + Δ_{Y_{•}}(P_{σ} (D)). Morever, Δ_{Y_{•}}(D) = Δ_{Y_{•}}(P_{σ} (D)), when x ∉ Supp(N_{σ} (D)).

Proof of Theorem 4.2. For the duration of this proof we fix an ample divisor A.

(1) This is equivalent to σ_{Γ}(D) ⩽ ν_{1}(D′) for every effective ℝ-divisor D′≡ D. Fix a real number ϵ > 0, let D′′∼_{ℝ} D + ϵA is an effective ℝ-divisor. Then

inf{mult_{Γ}(D′)|D′∼_{ℝ} D + ϵA} ⩽ mult_{Γ}(D′′) = ν_{1}(D′′) .

By Proposition 1.4, this implies the inclusion

Δ_{Y_{•}}(D + ϵA) ⊆ (σ′(D + ϵA),0,...0) + ℝ_{+}^{n} .

Then Lemma 1.7 and the definition of σ_{Γ}(D) imply the claim.

(2) By [N, Lemma 2.1.5] we have σ_{Γ}(D − σ_{Γ}(D)Γ) = 0. Consequently, we obtain Γ ⊈ B_{−} (D − σ(D)Γ). Because B_{−} (D − σ(D)Γ) is a countable union of subvarieties of X, a very general point x ∈ Γ lies outside B_{−} (D − σ_{Γ}(D)Γ). Theorem 2.1 yields 0 ∈ Δ_{Y_{•}}(D − σ_{Γ}(D)Γ), therefore the point (σ_{Γ}(D),0...,0) is contained in Δ_{Y_{•}}(D).

(3) Let D_{σ} ∼_{ℝ} P_{σ} (D) be an effective ℝ-divisor, then D_{σ} + N_{σ} (D) ∼_{ℝ} D is also an effective divisor for which

ν_{Y_{•}}(D_{σ} + N_{σ} (D)) = ν_{Y_{•}}(D_{σ}) + ν_{Y_{•}}(N_{σ} (D)) .

This implies the inclusion ν_{Y_{•}}(N_{σ} (D)) + Δ_{Y_{•}}(P_{σ} (D)) ⊆ Δ_{Y_{•}}(D) via Proposition 1.4. For an effective ℝ-divisor D′∼_{ℝ} D, [N, III.1.14] gives that the divisor D_{σ} = D′− N_{σ} (D) ∼_{ℝ} P_{σ} (D) is effective. Thus ν_{Y_{•}}(D′) = ν_{Y_{•}}(D_{σ}) + ν_{Y_{•}}(N_{σ} (D)), which completes the proof. □

Next, we study the variation of Zariski decomposition after Nakayama when varying the divisors inside the pseudo-effective cone. We start with the following lemma. Although it follows from [N, III.1.9], we present a proof to illustrate the use of Newton– Okounkov bodies techniques.

Lemma 4.3. Suppose D is a big ℝ-divisor on X and E-prime effective divisor. If σ_{E}(D) = 0, then σ_{E}(D − tE) = 0 for all t ⩾ 0.

Proof. The condition σ_{E}(D) = 0 implies E ⊈ B_{−} (D), thus, by Theorem 2.1, for a flag Y_{•} : X ⊇ E ⊇ ... ⊇ {x}, with x ∈ E very general point, we have that 0 ∈ Δ_{Y_{•}}(D). Again, by the very general choice of x ∈ E, Theorem 4.2 says that σ_{E}(D − tE) · e_{1} ∈ Δ_{Y_{•}}(D − tE). On the other hand, by Proposition 1.6 we know that Δ_{Y_{•}}(D)_{ν_{1}⩾t} = Δ_{Y_{•}}(D − tE) + te_{1}, therefore (σ_{E}(D − tE) + t)e_{1} ∈ Δ_{Y_{•}}(D). By convexity, this implies t · e_{1} ∈ Δ_{Y_{•}}(D), again by Proposition 1.6 we have 0 ∈ Δ_{Y_{•}}(D − tE), hence σ_{E}(D − tE) = 0 by the choice of x ∈ E and Theorem 4.2. □

The next proposition shows how the negative part of the Zariski decomposition varies inside the big cone.

Proposition 4.4. Suppose D is a big ℝ-divisor on X and E a prime effective divisor. Then

(1) If σ_{E}(D) > 0, then N_{σ} (D − tE) = N_{σ} (D) − tE, for any t ∈ [0,σ_{E}(D)].

(2) If σ_{E}(D) = 0, then the function t → N_{σ} (D − tE) is an increasing function, i.e. for any t_{1} ⩾ t_{2} the divisor N_{σ} (D − t_{1}) − N_{σ} (D − t_{2}E) is effective.

Proof. (1) This statement is proved in Lemma 1.8 from [N].

(2) Since σ_{E}(D) = 0, then Lemma 4.3 implies that σ_{E}(D − tE) = 0 for any t ⩾ 0 and in particular E ⊈ Supp(N_{σ} (D − tE)) for any t ⩾ 0. So, take Γ ⊆ Supp(N_{σ} (D − t_{2}E)) a prime divisor. The goal is to prove that σ_{Γ}(D−t_{1}E) ⩾ σ_{Γ}(D−t_{2}E). Without loss of generality, we assume that t_{2} = 0 and t_{1} = t ⩾ 0 and we need to show that σ_{Γ}(D−tE) ⩾ σ_{Γ}(D). Now take a flag Y_{•} : X ⊇ Γ ⊇ ... ⊇ {x}, where x ∈ Γ is a very general point and x ∉ E. Then by Theorem 4.2 we have

σ_{Γ}(D) · e_{1} ∈ Δ_{Y_{•}}(D) ⊆ σ_{Γ}(D) · e_{1}+ ℝ_{+}^{n}

and

σ_{Γ}(D − tE) · e_{1} ∈ Δ_{Y_{•}}(D − tE) ⊆ σ_{Γ}(D − tE) · e_{1}+ ℝ_{+}^{n} .

On the other hand, it is not hard to see that Δ_{Y_{•}}(D − tE) ⊆ Δ_{Y_{•}}(D). For any D′∼_{ℝ} D − tE effective ℝ-divisor, the ℝ-divisor D′+ tE ∼_{ℝ} D is also effective. Since x ∉ E, then ν_{Y_{•}}(D′) = ν_{Y_{•}}(D′+ tE) and the inclusion follows naturally. Combining this and the above information we obtain that σ_{Γ}(D−tE)·e_{1} ∈ Δ_{Y_{•}}(D) and thus σ_{Γ}(D−tE) ⩾ σ_{Γ}(D). □

References

[AKL] Dave Anderson, Alex Küronya, and Victor Lozovanu, Okounkov bodies of finitely generated divisors, International Mathematics Research Notices 132 (2013), no. 5, 1205–1221. ↑(document), 1.3

[BKS] Thomas Bauer, Alex Küronya, and Tomasz Szemberg, Zariski decompositions, volumes, and stable base loci, Journal für die reine und angewandte Mathematik 576 (2004), 209–233. ↑(document)

[B1] Sébastien Boucksom, Corps D’Okounkov, Séminaire Bourbaki 65 (Unknown Month 2012), no. 1059, 1–38. ↑(document), 1.3

[B2] Sébastien Boucksom, Divisorial Zariski decompositions on compact complex manifolds, Ann. Sci. École Norm. Sup. (4) 37 (2004), no. 1, 45–76, DOI 10.1016/j.ansens.2003.04.002. ↑(document), 4

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[ELMNP1] Lawrence Ein, Robert Lazarsfeld, Mircea Mustaţă, Michael Nakamaye, and Mihnea Popa, Asymptotic invariants of base loci, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 6, 1701–1734. ↑1.2, 1.3, 1.2, 2, 2.3, 2, 2, 3, 3, 3, 3

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[LM] Robert Lazarsfeld and Mircea Mustaţă, Convex bodies associated to linear series, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), no. 5, 783–835. ↑(document), 1.1, 1.3, 1.3, 2.7, 2

[L] John Lesieutre, The diminished base locus is not always closed, Compos. Math. 150 (2014), no. 10, 1729–1741, DOI 10.1112/S0010437X14007544. ↑1.2

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[PAG1] Robert Lazarsfeld, Positivity in algebraic geometry. I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 48, Springer-Verlag, Berlin, 2004. ↑2

[PAG2] , Positivity in algebraic geometry. II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 49, Springer-Verlag, Berlin, 2004. ↑1.2

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Alex Küronya Johann-Wolfgang-Goethe-Universität Institut für Mathematik Robert-Mayer-Straße 6-10. D-60325 Frankfurt am Main Germany

and

Budapest University of Technology and Economics Department of Algebra Egry József u. 1. H-1111 Budapest Hungary kuronya@math.uni-frankfurt.de Victor Lozovanu Universitá degli Studi di Milano–Bicocca Dipartimento di Matematica e Applicazioni, Via Cozzi 53 I-20125 Milano, Italy

and

Leibniz-Universität Hannover Institut für Algebraische Geometrie Welfengarten 1 D-30167 Hannover Germany lozovanu@math.uni-hannover.de