Multigraded algebras and multigraded linear series

This paper is devoted to the study of multigraded algebras and multigraded linear series. For an Ns$\mathbb {N}^s$ ‐graded algebra A$A$ , we define and study its volume function FA:N+s→R$F_A:\mathbb {N}_+^s\rightarrow \mathbb {R}$ , which computes the asymptotics of the Hilbert function of A$A$ . We relate the volume function FA$F_A$ to the volume of the fibers of the global Newton–Okounkov body Δ(A)$\Delta (A)$ of A$A$ . Unlike the classical case of standard multigraded algebras, the volume function FA$F_A$ is not a polynomial in general. However, in the case when the algebra A$A$ has a decomposable grading, we show that the volume function FA$F_A$ is a polynomial with nonnegative coefficients. We then define mixed multiplicities in this case and provide a full characterization for their positivity. Furthermore, we apply our results on multigraded algebras to multigraded linear series. Our work recovers and unifies recent developments on mixed multiplicities. In particular, we recover results on the existence of mixed multiplicities for (not necessarily Noetherian) graded families of ideals and on the positivity of the multidegrees of multiprojective varieties.

1. Introduction 2. Notation and preliminaries 3. Singly graded algebras of almost integral type 3.1.The case where R is a domain 3.2.Valuations with bounded leaves 4. Multigraded algebras of almost integral type 4.1.Global Newton-Okounkov bodies 5. Multigraded algebras of almost integral type with decomposable grading 6. Application to graded families of ideals 6.1.Positivity for graded families of equally generated ideals 6.2.Positivity for mixed volumes of convex bodies 7. Multigraded linear series

INTRODUCTION
The main goal of this paper is to study and extend the current theory of graded algebras of almost integral type and graded linear series to multigraded algebras of almost integral type and multigraded linear series, respectively.The asymptotic behavior of graded algebras of almost integral type and graded linear series was extensively studied in the foundational papers of Kaveh and Khovanskii [21] and of Lazarsfeld and Mustat ¸ȃ [25].Following ideas from seminal works of Okounkov [29,30], the authors of [21] and [25] associated a convex body ∆(A) to a graded algebra A equipped with a valuation with one-dimensional leaves.The convex body ∆(A) is called the Newton-Okounkov body of the algebra A. One of the main results of [21,25] relates the asymptotic growth of the Hilbert function of a graded algebra of almost integral type with the corresponding Newton-Okounkov body.
More precisely, let k be an algebraically closed field and F be a field containing k. Recall that a graded ksubalgebra A ⊂ F[t] of the polynomial ring in one variable is of integral type if A is finitely generated over k and is a finitely generated module over the subalgebra generated by where B is a graded k-algebra of integral type.Then, one has the following theorem.
Theorem 1.1 ( [21,25]).Let A ⊂ F[t] be a graded k-algebra of almost integral type, and let q = dim R (∆(A)).Then the Hilbert function of A has a polynomial growth.Moreover, the main term of asymptotics has degree q and its coefficient is given by the volume of ∆(A).
We generalize Theorem 1.1 to a far-reaching general case of algebras of almost integral type.We consider an arbitrary field k (not necessarily algebraically closed) and we replace the field F with any reduced k-algebra R. We also relate the dimension of the Newton-Okounkov body of A ⊂ R[t] to the Krull dimension of A.
Theorem A (Theorem 3.6).Let k be a field and R be a reduced k-algebra.Let A ⊂ R[t] be a graded k-algebra of almost integral type.Suppose d = dim(A) > 0.Then, there exists an integer m > 0 such that the limit exists and it is a positive real number.
We prove this result by analyzing the graded algebras equipped with a valuation having leaves of bounded dimension.In particular, we show that any algebra A ⊂ R[t] of almost integral type has a valuation with bounded leaves (Proposition 3.13).Our treatment of singly graded algebras also profited from the works of Cutkosky [9,11].
We now focus on our developments of the multigraded case of algebras and linear series.
1.1.Multigraded algebras of almost integral type.Here we describe our main results regarding arbitrary multigraded algebras of almost integral type.
Let k be an arbitrary field and R be a k-domain.Our results in principle could cover the case when R is an arbitrary reduced ring (see the general reductions used in Section 3), however, in doing so the notation would be quite cumbersome in the notion of volume function that we define below.
First, we introduce our main object of study, which provides the multigraded extension of graded algebras of almost integral type (as introduced in [21]).Let t 1 , . . ., t s be new variables over R and consider R[t 1 , . . ., t s ] as a standard N s -graded polynomial ring where deg(t i ) = e i ∈ N s and e i denotes the i-th elementary vector (0, . . ., 1, . . ., 0).We have the following notions: (i) An N s -graded k-algebra A = n∈N s [A] n ⊂ R[t 1 , . . ., t s ] is said to be of integral type if A is finitely generated over k and is a finite module over the subalgebra generated by [A] e 1 , [A] e 2 , . . ., [A] e s .(ii) An N s -graded k-algebra A = n∈N s [A] n ⊂ R[t 1 , . . ., t s ] is said to be of almost integral type if A ⊂ B ⊂ R[t 1 , . . ., t s ], where B is an N s -graded algebra of integral type.
Let A ⊂ R[t 1 , . . ., t s ] be an N s -graded algebra of almost integral type.To simplify notation, we also need to assume that [A] e i = 0 for all 1 i s.One has that the Krull dimension of A is finite (see Theorem 2.5), i.e., dim(A) < ∞.Let d = dim(A) and q = d − s.Our main focus is on the volume function of A, which is defined as (1) As a direct consequence of Theorem 3.11, we obtain that the limit defining the volume function F A always exists.Note that, when A is a standard N s -graded algebra then the volume function F A encodes the mixed multiplicities of A (see Remark 4.1).
In a similar fashion to [25], we relate the volume function F A of A to a global Newton-Okounkov body that we define below.For that, we can safely assume that R is finitely generated over k (see Remark 3.4) and that there exists a valuation ν : Quot(R) → Z r with certain good properties (see Proposition 3.13).Of particular importance is the fact that the valuation ν has leaves of bounded dimension.Let Γ A be the valued semigroup of the N s -graded algebra A: We define ∆(A) = Con(Γ A ) ⊂ R r × R s 0 to be the closed convex cone generated by Γ A and we call it the global Newton-Okounkov body of A. We denote the index of A by ind(A) and the maximal dimension of leaves of A by ℓ A (these invariants refer to a uniform behavior of the Veronese subalgebras A (n) = ∞ n=0 [A] nn ; for more details, see Definition 4.10).One has the diagram below where π 1 : R r × R s 0 → R r and π 2 : R r × R s 0 → R s 0 denote the natural projections.We denote the fiber of the global Newton-Okounkov body ∆(A) over x ∈ R s 0 by ∆(A) x = ∆(A) ∩ π −1 2 (x).The following theorem contains our main results regarding the relations between the volume function F A and the global Newton-Okounkov body ∆(A).
Theorem B (Theorem 4.7, Corollary 4.11).Under the notations above the following results hold: (i) The fiber ∆(A) n of the global Newton-Okounkov body ∆(A) ⊂ R r × R s 0 coincides with the Newton-Okounkov body ∆(A (n) ) for each n ∈ Z s + , that is: (ii) There exists a unique continuous homogeneous function of degree q extending the volume function F A (n) defined in (1) to the positive orthant R s 0 .This function is given by Moreover, the function F A is log-concave: for all x, y ∈ R s 0 .
We illustrate the concrete computation of the volume functions in Example 4.4 and Example 4.15.In particular, we show that in general, part (ii) of Theorem B is the strongest result that can be obtained in this setting: for any non-negative, homogeneous of degree 1, concave function f : R s 0 → R 0 we construct an N s -graded algebra A whose volume function is f (see Example 4.12).
1.2.Algebras with decomposable grading.An important question is to determine the following: • When is the volume function F A a polynomial?By mimicking the classical case of standard multigraded algebras, the existence of such a polynomial yields a natural notion of mixed multiplicities for the algebra A (see Remark 4.1).It turns out that the volume function F A is not always a polynomial (see, e.g., Example 4.4, Example 4.12 and Example 4.15).However, we show that the volume function is a polynomial for the family of multigraded algebras with decomposable gradings.
An N s -graded algebra A is said to have a decomposable grading if the equality  The following theorem deals with the case of algebras with a decomposable grading.
Theorem C (Theorem 5.5).Assume the notations above with A having a decomposable grading.Then, there exists a homogeneous polynomial G A (n) ∈ R[n 1 , . . ., n s ] of degree q with non-negative real coefficients such that

Additionally, we have
(n) p q for all n ∈ Z s + .
We illustrate Theorem C with an example of the Cox ring of a full flag variety (Example 5.6).
Moreover, we can write the polynomial G A of Theorem C as follows Then, for each d = (d 1 , . . ., d s ) ∈ N s with |d| = q, we define the non-negative real number e(d; A) 0 to be the mixed multiplicity of type d of A.
In Corollary 5.8, we provide an extension into a multigraded setting of the Fujita approximation theorem for graded algebras given in [21,Theorem 2.35].We show the following equalities Furthermore, in Theorem 5.9, we provide a full characterization for the positivity of the mixed multiplicities e(d; A) of A.
In our findings, we got that a number of interesting applications follow rather easily by studying certain multigraded algebras with decomposable grading.The list of applications includes the following: (1) In Section 6, we recover some results from [7,8,10] by showing the existence of mixed multiplicities for (not necessarily Noetherian) graded families of ideals.(2) In §6.1, we provide a full characterization for the positivity of the mixed multiplicities of graded families of equally generated ideals.(3) In §6.2, we recover a classical characterization for the positivity of the mixed volumes of convex bodies.
1.3.Multigraded linear series.Finally, we are interested in the notion of multigraded linear series.Let k be an arbitrary field and X be a proper variety over k.Let D 1 , . . ., D s be a sequence of Cartier divisors on X, and consider the corresponding section ring A multigraded linear series associated to the divisors D 1 , . . ., D s is an N s -graded k-subalgebra W of the section ring S(D 1 , . . ., D s ).In particular, W is of almost integral type (see Proposition 7.2).The Kodaira-Itaka dimension of W is given by κ where as before dim(W) denotes the Krull dimension of W. As simple consequence of our developments for multigraded algebras, we have the following theorem for multigraded linear series (which we enunciate below for the sake of completeness).
Theorem D (Theorem 7.4).Under the above notations, let W ⊂ S(D 1 , . . ., D s ) be a multigraded linear series and suppose that [W] e i = 0 for all 1 i s.Then, the following statements hold: (i) The volume function (ii) There exists a unique continuous function that is homogeneous of degree κ(W) and log-concave and that extends the volume function F W (n) to the positive orthant R s 0 .This function is given by (iii) If W has a decomposable grading, then there exists a homogeneous polynomial G W (n) ∈ R[n 1 , . . ., n s ] of degree κ(W) with non-negative real coefficients such that Finally, in Theorem 7.5, we express the mixed multiplicities of a multigraded linear series with decomposable grading in terms of the multidegrees of the multiprojective varieties obtained as the image of certain Kodaira rational maps.
1.4.Organization of the paper.The basic outline of this paper is as follows.In Section 2, we recall some important results and fix some notations.In Section 3, we deal with singly graded algebras and we prove Theorem A. In Section 4, we begin our study of multigraded algebras and we prove Theorem B. Our treatment of multigraded algebras with decomposable grading is made in Section 5, where we show Theorem C. In Section 6, we obtain some applications for the mixed multiplicities of graded families of ideals and for the mixed volumes of convex bodies.Finally, in Section 7, we apply our results on multigraded algebras to multigraded linear series and we prove Theorem D.

NOTATION AND PRELIMINARIES
In this preparatory section, we fix our notation and recall some important results to be used throughout the paper.We denote the set of non-negative integers by N = {0, 1, 2, . ..} and the set of positive integers by Z + = {1, 2, . ..}.Let d 1.For a vector n = (n 1 , . . ., n d ) ∈ N d we denote by |n| the sum of its entries.We also denote by e i ∈ N d the i-th elementary vector (0, . . ., 1, . . ., 0).For n = (n 1 , . . ., n d ) and m = (m 1 , . . ., m d ) in N d we write n m if n i m i for every 1 i d.The null vector (0, . . ., 0) ∈ N d is denoted by 0 ∈ N d .Moreover, we write n ≫ 0 if n i 0 for every 1 i d.We also use the abbreviations We first describe the notions and methods of Newton-Okounkov bodies and recall some important results from [21].Let d 1. Suppose that S ⊂ Z d+1 is a semigroup in Z d+1 .Let π : R d+1 → R be the projection into the last component.Let L = L(S) be the linear subspace of R d+1 which is generated by S. Let M = M(S) be the rational half-space M(S) and let ∂M Z = ∂M ∩ Z d+1 .Let Con(S) ⊂ L(S) be the closed convex cone which is the closure of the set of all linear combinations i λ i s i with s i ∈ S and λ i 0. Let G(S) ⊂ L(S) be the group generated by S.
We say that the semigroup S is non-negative if S ⊂ M; additionally, if Con(S) is strictly convex and intersects the space ∂M only at the origin, then S is strongly non-negative (see [21, Definition 1.9 and §1.4]).Following [21], when S is non-negative we fix the following notation: -ind(S) := [∂M Z : G(S) ∩ ∂M].
-Vol q (∆(S)) is the integral volume of ∆(M, S) (see [21,Definition 1.13]); this volume is computed using the translation of the integral measure on ∂M.
The following result is of remarkable importance for us.
Theorem 2.1 (Kaveh-Khovanskii, [21,Corollary 1.16]).Suppose that S is strongly non-negative.Let m = m(S) and q = dim R (∆(S)).Then For a subset U ⊂ S of S and n ∈ N, we define n ⋆ U := n i=1 u i | u 1 , . . ., u n ∈ U .For p ∈ N, we denote by S p the subsemigroup of S generated by [S] p , that is, The following approximation theorem relates the semigroup S with the semigroups S pm for p big enough.).Suppose that S is strongly non-negative.Let m = m(S) and q = dim(∆(S)).Let ε > 0 be a positive real number.Then, for p ≫ 0 the following statements hold: (ii) ind( S pm ) = ind(S).
(iii) We have the inequalities We now briefly recall Minkowski's theorem and the notion of mixed volume of convex bodies.Let K = (K 1 , . . ., K s ) be a sequence of convex bodies in R d .For any sequence λ = (λ 1 , . . ., λ s ) ∈ N s of nonnegative integers, we denote by λK the Minkowski sum λK : j=1 {K i } of λ i copies of K i for each 1 i s.Below is the classical Minkowski's theorem (see, e.g., [31,Theorem 5.1.7]).Next, we concentrate on describing the dimension of a subalgebra of an algebra finitely generated over a field.Following the terminology of [23], we say that these algebras are subfinite.Let k be a field and A be a k-algebra.We shall always denote by dim(A) the Krull dimension of A. We say that A is subfinite over k if there exists a finitely generated k-algebra B containing A. Denote by δ k (A) the maximal number of elements in A which are algebraically independent over k, that is, δ k (A) := sup d | there exists x 1 , . . ., x d ∈ A which are algebraically independent over k .
Equivalently, δ k (A) = d if d is the largest integer such that there is an inclusion k[x 1 , . . ., x d ] ֒→ A where k[x 1 , . . ., x d ] is a polynomial ring over k.
Remark 2.4.If A is finitely generated over k, then dim(A) = δ k (A).This is a classical result that follows for instance from the Noether normalization theorem (see, e.g., [12,Theorem 13.3]).
The following theorem extends the above remark to algebras that are subfinite over k (cf.[14]).

SINGLY GRADED ALGEBRAS OF ALMOST INTEGRAL TYPE
In this section, we extend the results of [21, Part II] regarding the asymptotic behavior of algebras of almost integral type.Our results are slightly more complete than the ones in [21, Part II] with respect to the following points: (1) We consider an arbitrary field k instead of assuming that k is algebraically closed.
(2) We substitute the field F by any reduced k-algebra R.
(3) We show that the asymptotic growth of an algebra of almost integral type is determined by its Krull dimension.
Similar results to the ones in this section were also obtained in [11] for graded linear series.
Let k be an arbitrary field and R be a reduced k-algebra.Let R[t] be a standard graded polynomial ring over R. Below we introduce the notion of almost integral type in our current setting.
k and is a finitely generated module over the subalgebra generated by By definition, one has that a graded k-algebra of almost integral type is a subfinite algebra over k.For any positively graded k-algebra A which is subfinite over k, we denote by G i (A) the finitely generated graded k-algebra that is generated by the graded components First, we discuss some properties of graded k-algebras that are subfinite over k.Lemma 3.2.Let A be a positively graded k-algebra which is subfinite over k.Then, the following statements hold: for all n > 0.
Proof.(i) This part follows from Theorem 2.5. (ii Since C is finitely generated over k, by [15, Lemma 13.10, Remark 13.11], one has a positive integer h > 0 such that the Veronese subalgebra Lemma 3.3.Let A, B and C be graded k-algebras which are subfinite over k.Suppose there exists n 0 0 such that for all n n 0 we have a short exact sequence 0 Proof.After choosing some h n 0 , we substitute A, B and C by the Veronese subalgebras A (h) , B (h) and C (h) , respectively.Thus we may assume that 0 By invoking Theorem 2.5, we choose i > 0 such that dim n for any graded k-algebra S. From the above short exact sequence, we obtain that the ideal G i (B) ∩ A + is finitely generated (as it is an ideal over G i (B)), which implies that G i (B) ∩ A is a finitely generated graded k-algebra (see, e.g., [3,Proposition 1.5.4]).We substitute A, B and C by G i (B) ∩ A, G i (B) and G i (C), respectively.So, we assume that A, B and C are finitely generated kalgebras.
The following easy remark shows that we can always substitute R by a reduced finitely generated kalgebra (cf.[21, Proposition 2.25]).Remark 3.4.Let A ⊂ R[t] be a graded k-algebra of almost integral type.By definition, let B ⊃ A be a graded k-algebra of integral type and suppose that f i 1 t i 1 , . . ., f i m t i m are generators of B as a k-algebra.Let R ′ ⊂ R be the reduced k-algebra generated by the elements Thus, as a consequence of the above remark, we now safely assume that R is a reduced finitely generated algebra.Hence, let p 1 , . . ., p ℓ ∈ Spec(R) be the minimal primes of R. Since R is assumed to be reduced, we have a canonical inclusion For each 1 i ℓ, let B i be the k-subalgebra defined by (2) where ).By construction, B i is a graded k-algebra of almost integral type and we have a canonical inclusion Lemma 3.5.Under the above notation, the following statements hold: Proof.For each 1 i ℓ we have the short exact sequence The following theorem contains the main result of this section.By using the general arguments above, the first idea in the proof is to reduce to the case when R is a domain.For organizational purposes, in the next subsection, we encapsulate a proof of Theorem 3.6 under the assumptions of R being a finitely generated k-domain (see Theorem 3.11).Theorem 3.6.Let k be a field and R be a reduced k-algebra.Let A ⊂ R[t] be a graded k-algebra of almost integral type.Suppose d = dim(A) > 0.Then, there exists an integer m > 0 such that the limit exists and it is a positive real number.
Proof.By Remark 3.4, we assume that R is finitely generated over k.Suppose that p 1 , . . ., p ℓ are the minimal primes of R, and consider the algebras B i constructed in (2).Since each B i ⊂ R/p i [t] is an algebra of almost integral type, Theorem 3.11 yields the existence of the following limits Therefore, after taking m = lcm(m 1 , . . ., m ℓ ), the result of the theorem follows from Lemma 3.5.

3.1.
The case where R is a domain.The setup below is used throughout this subsection.
be the multiplicity of the standard graded k-algebra A p .
Remark 3.8.We fix an order on Z r which is compatible with an addition, i.e., for any n, m, p ∈ Z r with n < m, we have n + p < m + p.One way to do this is to fix a linear function l on Z r with rationally independent coefficients and set n < m ⇐⇒ l(n) < l(m) for n, m ∈ Z r .
We assume that Quot(R) admits a valuation ν : Quot(R) → Z r such that ν(α) = 0 for all α ∈ k ⊂ Quot(R).We further suppose that ν : Quot(R) → Z r is faithful and has leaves of bounded dimension.By Proposition 3.13 we can always construct such a valuation.The faithfulness of ν means that ν(Quot(R)) = Z r .For any n = (n 1 , . . ., n r ) ∈ Z r , we define and we say that the leaf with value n is given by We say that ν has leaves with bounded dimension if sup n∈Z r (dim k (L n )) < ∞.Let ℓ be the maximal dimension of the leaves of ν: For any n 0 and n ∈ Z r one has the inequalities We consider the integer (4) Then, we have the following equalities for all n 0. We have the following general lemma.
In the next proposition, we study some properties of Γ (t) A .For the special case t = 1, we simply write A ⊆ {0}).Then, the following statements hold: A is a semigroup ideal of Γ A .In particular, one has that Thus, there exist a 1 . . ., a t ∈ k such that ν(a A .Finally, Lemma 3.9 implies that ∆(Γ The following theorem deals with the asymptotic behavior of the growth of A. The semigroup Γ A is given by n and we call it the valued semigroup of the graded k-algebra A. To simplify notation, we denote ∆(Γ A ) and ind(Γ A ) by ∆(A) and ind(A), respectively.Theorem 3.11.Assume Setup 3.7, set m = m(A) and let d = dim(A) and ℓ A be as in (4).Then and we have the equality Proof.For any p > 0 and t > 0 we consider the semigroup these semigroups play the same role for A pm than the semigroups in (3) for A. By using the inclusions and Theorem 2.2, we can choose From Theorem 2.5, by possibly making p larger, we can assume that dim( A pm ) = d.After regrading A pm and considering the standard graded k-algebra A pm , we obtain the existence of a polynomial From Proposition 3.10, we also have ∆(Γ A ) = m(A) = m for all 1 t ℓ A .Therefore, by using (5), Proposition 3.10 and Theorem 2.1, we obtain the equality So, the first statement of the theorem holds.From (6) and Theorem 2.2, for any ε > 0, we can choose p ≫ 0 such that for all t.Finally, by (7) we obtain which gives the second claimed equality.

3.2.
Valuations with bounded leaves.In this subsection, we show that the valuations used in §3.1 can always be constructed.Here we continue using Setup 3.7.We start with the following lemma that will allow us construct a valuation on Quot(R).
Lemma 3.12.There exists a regular local ring (S, m, k ′ ) such that R ⊂ S, Quot(S) = Quot(R) and the residue field k ′ = S/m is finite over k.
Proof.From [32, Proposition 07PJ] and [32, Definition 07P7], we have is an open subset of Spec(R).As R is a domain, Reg(R) is non-empty.Then we can choose 0 = g ∈ R such that Spec(R g ) ⊂ Reg(R).Let m ⊂ Spec(R g ) be a maximal ideal of R g .Therefore, by setting S = (R g ) m the result follows.
From Lemma 3.12, fix a regular local ring (S, m, k ′ ) such that R ⊂ S, Quot(S) = Quot(R) and the residue field k ′ = S/m is finite over k.Let r = dim(S) = trdeg k (Quot(R)), and choose y 1 , . . ., y r ⊂ m a regular system of parameters for S. Let b 1 , . . ., b r be rationally independent real numbers with b i 1 for every 1 i r.Since S is a regular local ring and Quot(S) = Quot(R), we can construct a valuation ω on Quot(R) with values in R by setting Proposition 3.13.There is a valuation ν : Quot(R) → Z r that satisfies the following conditions: (ii) ν is faithful.
(iii) ν has leaves of bounded dimension.

MULTIGRADED ALGEBRAS OF ALMOST INTEGRAL TYPE
Here we define and study a volume function for multigraded algebras of almost integral type (see ( 9)), and we relate this function to certain global Newton-Okounkov bodies.In [25], similar volume functions have been considered for the case of multigraded linear series in an irreducible projective variety over an algebraically closed field.First, for the sake of completeness, we recall the notion of mixed multiplicities for the well-studied case of standard multigraded algebras, and we explain how this volume function encodes the mixed multiplicities.
Remark 4.1.Let k be a field and A be a standard N s -graded k-algebra (i.e., it is finitely generated over k by elements of degree e i for 1 i s).Suppose A is a domain and set d = dim(A).Let P A (n) = P A (n 1 , . . ., n s ) be the multigraded Hilbert polynomial of A (see, e.g., [18,Theorem 4.1], [6,Theorem 3.4]).Then, the degree of P A is equal to q = d − s and P A (n) = dim k ([A] n ) for all n ∈ N s such that n ≫ 0. Furthermore, if we write is the mixed multiplicity of A of type d.We define a function For all n ∈ Z s + , we have So, the function F A (n) encodes the mixed multiplicities of A.
Let k be a field and R be a k-domain.We introduce new variables t 1 , . . ., t s over R and consider R[t 1 , . . ., t s ] as a standard N s -graded polynomial ring where deg(t i ) = e i ∈ N s .We have the following definition.
is finitely generated over k and is a finite module over the subalgebra generated by where B is an N s -graded algebra of integral type.
The following setup is used throughout the rest of this section.
Setup 4.3.Let k be a field and R be a k-domain.Let A ⊂ R[t 1 , . . ., t s ] be an N s -graded k-algebra of almost integral type.Set d = dim(A) and q = d − s.We assume that [A] e i = 0 for all 1 i s.
Let n = (n 1 , . . ., n s ) ∈ Z s + .We consider the graded k-algebra By definition, A (n) is a graded k-algebra of almost integral type with a canonical inclusion into a standard graded polynomial ring R[t].Denote by A (0) ⊂ Quot(A) the subfield of fractions of multihomogeneous elements with the same degree, that is, Let 0 = f i ∈ [A] e i for each 1 i s.Since Quot(A) = A (0) (f 1 , . . ., f s ) with f 1 , . . ., f s transcendental elements over A (0) , Theorem 2.5 yields that dim(A) = s + trdeg k A (0) .In the same way, Quot(A , and Theorem 2.5 yields that dim(A Therefore, for n ∈ Z s + , Theorem 3.11 applied to the algebra A (n) gives a well-defined function with q = d − s.We say that F A (n) is the volume function of the N s -graded k-algebra A.
The following simple example shows that, for an arbitrary algebra of almost integral type A, F A (n) may not coincide with a polynomial (also, see [10, §7]).So, in general we do not have a suitable extension of Remark 4.1.
, where ⌈β⌉ denotes the ceiling function of a real number β ∈ R. We define the family of vector spaces However, in this case the corresponding function Therefore, in this case F A (n 1 , n 2 ) is not a polynomial function.

Global Newton-Okounkov bodies.
Here we study the function F A (n) for an arbitrary algebra of almost integral type and we relate it to the construction of certain "global Newton-Okounkov bodies".We continue using Setup 4.3.Since we may assume that R is finitely generated over k (see Remark 3.4), we fix a valuation ν : Quot(R) → Z r that satisfies the conditions of Proposition 3.13; in particular, ν has leaves of bounded dimension.By using the fixed valuation ν : Quot(R) → Z r , we make the following construction.
Definition 4.5.Let Γ A be the valued semigroup of the N s -graded algebra A: We define ∆(A) = Con(Γ A ) ⊂ R r × R s 0 to be the closed convex cone generated by Γ A and we call it the global Newton-Okounkov body of A.
We consider the following diagram where π 1 : R r × R s 0 → R r and π 2 : R r × R s 0 → R s 0 denote the natural projections.We denote the fiber of the global Newton-Okounkov body ∆(A) over x ∈ R s 0 by ∆(A) x := ∆(A) ∩ π −1 2 (x).Since A is a domain and [A] e i = 0 for all 1 i s, it follows that m(A (n) ) = 1 and ∆(A (n) ) ⊂ R r × {1} ⊂ R r × R 0 .The following theorem describes the Newton-Okounkov bodies of the graded k-algebra A (n) (see (8)).+ , that is: For the proof of Theorem 4.7 it is convenient to use the following well-known result.
Lemma 4.8.Let S ⊂ Z r × N be a strongly non-negative semigroup.The Newton-Okounkov body ∆(S) can be computed as where m = m(S).
Proof of Theorem 4.7.Fix a vector n ∈ Z s + of positive integers.The statement follows from the fact that the valued semigroup of A (n) is given by Hence, by Lemma 4.8 we have that which completes the proof.
The following lemma provides a required uniformity result for the algebras A (n) .Lemma 4.9.Let n ∈ Z s + and m ∈ Z s + .Then, one has ind(A (n) ) = ind(A (m) ) and ℓ Proof.For each n 0 we have a canonical multiplication map Then, by proceeding similarly to Proposition 3.10, we obtain that ind(A (n) ) ind(A (n+m) ) and ℓ A (n) ℓ A (n+m) .Take k > 0 big enough such that (k − 1) • n > m.As above, by considering the multiplication maps Symmetrically, we also have that ind(A (m) ) = ind(A (n+m) ) and ℓ A (m) = ℓ A (n+m) .
Definition 4.10.By using Lemma 4.9, we set: (i) ind(A) be the index of A which is the constant value of ind(A (n) ) for all n ∈ Z s + .(ii) ℓ A be the maximal dimension of leaves of A which is the constant value of ℓ A (n) for all n ∈ Z s + .
The corollary below gives an important characterization of the volume function of A in terms of the global Newton-Okounkov body ∆(A).
Corollary 4.11.Assume Setup 4.3.There exists a unique continuous homogeneous function of degree q extending the volume function F A (n) defined in (9) to the positive orthant R s 0 .This function is given by Moreover, the function F A is log-concave: for all x, y ∈ R s 0 .
Proof.By combining Theorem 3.11, Theorem 4.7 and Lemma 4.9, it follows that extends the function F A (n) defined in ( 9) for all n ∈ Z s + .Now, the homogeneity follows from the fact that ∆(A) λ•x = λ • ∆(A) x .Since ∆(A) is convex we have ∆(A) x + ∆(A) y ⊂ ∆(A) x+y and hence the log-concavity follows from Brunn-Minkowski inequalities for the volume of convex bodies.
As our next example shows, in general, Corollary 4.11 is the most general statement about volume function F A for some multigraded algebra A.
Example 4.12 (Every concave function is a volume function).Let R = k[u] be a polynomial ring and consider the polynomial ring R[t 1 , . . ., t s ].Let further f : R s 0 → R 0 be any non-negative, homogeneous of degree 1, concave function.We define a family of vector spaces indexed by n ∈ N s : is naturally an N s -graded algebra.Since f is concave and homogeneous, the global Newton-Okounkov body ∆(A f ) of A f is a cone in R s+1 given by: In particular, we get However in some cases one can say more about function F A (x).Let C ⊂ R s be a convex cone and consider a family {∆ x } x∈C of convex bodies of dimension q parametrised by C. We say {∆ x } x∈C is linear if ∆ λ 1 x+λ 2 y = λ 1 ∆ x + λ 2 ∆ y for any x, y ∈ C and λ 1 , λ 2 > 0. According to Minkowski's theorem (see Theorem 2.3), the volume of a linear family of convex bodies is a homogeneous polynomial.Thus we obtain the following proposition.Proof.Since the statement of the proposition follows from Minkowski's theorem and the fact that Another important case is when the global Newton-Okounkov body is a polyhedral cone.In this case, the multiplicity function is a piecewise polynomial with respect to some fan supported on the positive orthant.Proof.Indeed, by [22,Proposition 4.1], for any polyhedral cone C ⊂ R q × R s with π 2 (C) = R s 0 , the family {C ∩ π −1 2 (x)} x∈C of polytopes is piecewise linear with respect to some fan Σ supported on R s 0 .Therefore, the proposition follows from Minkowski's theorem.0 → R be a function defined by f(x 1 , x 2 ) = min(x 1 , x 2 ).We define a family of vector spaces: The algebra is naturally an N 2 -graded algebra.The global Newton-Okounkov body of A is a cone C in R 3 generated by vectors (1, 0, 0), (0, 1, 0), (1, 1, 1) and, as in Example 4.12,

MULTIGRADED ALGEBRAS OF ALMOST INTEGRAL TYPE WITH DECOMPOSABLE GRADING
In this section, we introduce and study the notion of mixed multiplicities for certain multigraded algebras of almost integral type.We treat a family of algebras that we call algebras with decomposable grading.Our approach is inspired by the methods used in [7,8,10].Definition 5.1.An N s -graded algebra A is said to have a decomposable grading if we have the equality We now proceed to define the mixed multiplicities of a multigraded algebra A of almost integral type with decomposable grading.Here we extend Remark 4.1: our approach relies on showing that the corresponding function F A (n) coincides with a polynomial G A (n) when n ∈ Z s + .We use this polynomial to define the mixed multiplicities of A. For the rest of this section we use the following setting.Setup 5.2.Let k be a field and R be a k-domain.Let A ⊂ R[t 1 , . . ., t s ] be an N s -graded k-algebra of almost integral type with decomposable grading.Set d = dim(A) and q = d − s.We assume that [A] e i = 0 for all 1 i s. [A] je i ⊂ A of A which is the subalgebra generated by the graded components [A] je i with 1 i s, 1 j a.The next proposition says that the a-truncations can be used to approximate F A (n).
Proposition 5.3.Assume Setup 5.2.For each n = (n 1 , . . ., n s ) ∈ Z s + , we have the equality Proof.Fix n = (n 1 , . . ., n s ) ∈ Z s + .To simplify notation, set C = A (n) and D a = G [a] (A) (n) .Note that . By following the same steps as in §3.1, we can define strongly non-negative semigroups and . Consider the integer a ′ = ⌊a/ max{n 1 , . . ., n s }⌋ (where ⌊β⌋ denotes the floor function of β ∈ R) and the inclusions Then, Theorem 2.2 implies that lim and so the result follows.
The following proposition deals with the case when A is also a finitely generated k-algebra.
Proposition 5.4.Assume Setup 5.2 with A being finitely generated over k.For each n = (n 1 , . . ., n s ) ∈ Z s + , we have the equality Proof.Fix n = (n 1 , . . ., n s ) ∈ Z s + .We define the graded k-algebra For each p 1, let C p be the graded k-algebra Once again, we can define strongly non-negative semigroups . By Theorem 2.2, it follows that As a consequence, we get lim p→∞ . To finish the proof, it remains to show the equality . As A has a decomposable grading, the algebras A (e 1 ) , . . ., A (e s ) are also finitely generated over k.Hence, by [15,Lemma 13.10] we can choose h > 0 such that for all n 0. We then obtain So, the proof of the proposition is complete.
The next theorem contains the main result of this section.It shows that F A (n) is a polynomial like function when A has a decomposable grading.
Theorem 5.5.Assume Setup 5.2.Then, there exists a homogeneous polynomial G A (n) ∈ R[n 1 , . . ., n s ] of degree q with non-negative real coefficients such that Additionally, we have p q for all n ∈ Z s + .
Proof.Fix n = (n 1 , . . ., n s ) ∈ Z s + .For any a 1 and p 1, we have the following inequalities where B a = G [a] (A).Therefore, by combining Proposition 5.3 and Proposition 5.4, we obtain the equalities From Remark 4.1, the function (see, e.g., [10,Lemma 3.2]).So, the result of the theorem follows.
Below we have an example of a family of algebras with decomposable grading.
Example 5.6.Let G be a complex semisimple group and let B ⊂ G be a Borel subgroup with a character lattice M. Denote by A the Cox ring of G/B, i.e.
with a product induced by the natural maps: The Cox ring A is naturally an algebra graded by the Picard group Pic(G/B) of the flag variety G/B.By Borel's theorem, Pic(G/B) is isomorphic to M, for a character λ ∈ M, we will denote by L λ ∈ Pic(G/B) the corresponding line bundle.Moreover, by Borel-Weil theorem one has Therefore, the Cox ring A of G/B is given by where Λ + is the positive Weyl chamber, i.e., the direct sum is over dominant weights.The product maps λ+µ can be described in the following way.The tensor product λ+µ is the projection on V λ+µ in the above decomposition.In particular, we have where (n 1 , . . ., n s ) ∈ N s and ω 1 , . . ., ω s are fundamental weights of G. Therefore, the Cox ring A has decomposable grading and by Theorem 5.5 the volume function F A is a polynomial.
Finally the global Newton-Okounkov body of A has a nice description.By [20] there exists a valuation ν on A such that for any dominant weight λ, the Newton-Okounkov body ∆(A (λ) ) is the string polytope St λ .Therefore, the global Newton-Okounkov body ∆(A) is the weighted string cone, which is in particular a polyhedral cone ( [1,26]).
Note that, in general, string polytopes St λ provide only piecewise linear family of polytopes on the positive Weyl chamber.So a priory by Proposition 4.14, the function F A is only piecewise polynomial with respect to some fan decomposition of the positive Weyl chamber.However, using virtual polytopes, one can construct a linear family of virtual string polytopes which makes the polynomiality of F A evident.See [19,Section 10] for more details.With Theorem 5.5 in hand we are now able to define the mixed multiplicities of A. Definition 5.7.Assume Setup 5.2 and let G A (n) be as in Theorem 5.5.Write For each d = (d 1 , . . ., d s ) ∈ N s with |d| = q, we define the non-negative real number e(d; A) 0 to be the mixed multiplicity of type d of A.
The next straightforward corollary shows that the mixed multiplicities e(d; A) of A can be expressed as a limit that depends on the multiplicities e d; A Proof.It follows directly from (11).
Finally, we provide a complete characterization for the positivity of the mixed multiplicities of a multigraded algebra of almost integral type with decomposable grading.This result is a direct consequence of Corollary 5.8 and the general criterion of [5].
Following the notation of [5], for an N s -graded algebra T and for each subset J = {j 1 , . . ., j k } ⊆ {1, . . ., s} denote by T (J) the N k -graded k-algebra given by We obtain a full characterization for the positivity of the mixed multiplicities e(d, A) of A in terms of the dimensions dim A (J) .Theorem 5.9.Assume Setup 5.2.Let d = (d 1 , . . ., d s ) ∈ N s such that |d| = q.Then, e(d, A) > 0 if and only if for each J = {j 1 , . . ., j k } ⊆ {1, . . ., s} the inequality Proof.For each p 1, [5, Theorem B] characterizes the positivity of the mixed multiplicities of the standard N s -graded algebra A [p] , namely: Since A is an algebra with decomposable grading, by Theorem 2.5, we can choose p big enough such that dim for all J = {j 1 , . . ., j k } ⊆ {1, . . ., s}.Therefore, the result follows from Corollary 5.8.

APPLICATION TO GRADED FAMILIES OF IDEALS
In this section, we apply the results of Section 5 to the case of graded families of ideals and we recover some results from [7,8,10].We also obtain a characterization for the positivity of mixed multiplicities of certain graded families of ideals.First, we recall the notion of mixed multiplicities introduced by Bhattacharya in [2] for the case of ideals, and extended for (not necessarily Noetherian) graded families of ideals in [8].
Let k be a field and R be a finitely generated positively graded k-domain.We denote the graded irrelevant ideal of R by m Remark 6.1.Let I, J 1 , . . ., J s be non-zero homogeneous ideals in R such that I is m-primary.Then, for n 0 ≫ 0 and n = (n 1 , . . ., n s ) ≫ 0 the Bhattacharya function dim k (I coincides with a polynomial of total degree d − 1 whose homogeneous term in degree d − 1 can be written as Using standard techniques (see [7, proof of Lemma 4.2]), one may show that for each n 0 ∈ N and n = (n 1 , . . ., n s ) ∈ N s the limit n d exists and coincides with the following polynomial G (I;J 1 ,...,J s ) (n 0 , n 1 , . . ., n s ) := The numbers e (d 0 ,d) (I | J 1 , . . ., J s ) are non-negative integers called the mixed multiplicities of J 1 , . . ., J s with respect to I.
A sequence of ideals I = {I n } n∈N is a graded family if I 0 = R and I i I j ⊆ I i+j for every i, j ∈ N. The graded family is Noetherian if the corresponding Rees algebra R The graded family is m-primary when each I n is m-primary, and it is a filtration when I n+1 ⊆ I n for every n ∈ N.For a homogeneous ideal J ⊂ R we denote maxdeg(J) := max{j | [J ⊗ R R/m] j = 0}, that is, the maximum degree of a minimal set of homogeneous generators of J.
Throughout this section, we assume the following setup.
Setup 6.2.Let k be a field and R be a finitely generated positively graded k-domain.Let I = {I n } n∈N be a (not necessarily Noetherian) graded family of m-primary homogeneous ideals in R. Let J(1) = {J(1) n } n∈N , . .., J(s) = {J(s) n } n∈N be (not necessarily Noetherian) graded families of non-zero homogeneous ideals in R. We assume that there exists β ∈ N satisfying (12) maxdeg(J(i) n ) βn for all 1 i s and n ∈ N.
Remark 6.3.The condition ( 12) is automatically satisfied in the case of m-primary graded families of ideals.Similar assumptions to the one in (12) have been considered in previous works regarding limits of graded families of ideals [11,Theorem 6.1], [7], [8].
Notice that both A and B are of almost integral type and have decomposable gradings.Since dim(A) = dim(B) = d + s + 1, we can rewrite (14) in terms of the volume functions of A and B (see ( 9)), that is (15) Therefore, as a simple consequence of Theorem 5.5 we obtain the following result, which extends Remark 6.1 and allows us to define mixed multiplicities for graded families of ideals.
Proof.By using the equality (15), the result follows by applying Theorem 5.5 to the N s+1 -graded algebras of almost integral type A and B that have decomposable grading.
As a consequence of Theorem 6.4 we have the following definition.
The following result recovers the "Volume = Multiplicity formula" for the mixed multiplicities of graded families of ideals (see [7,8]).Proof.It follows from Theorem 6.4 and Remark 6.1.
6.1.Positivity for graded families of equally generated ideals.Here, we restrict to the following graded families of ideals and we provide a criterion for the positivity of their mixed multiplicities.
Setup 6.7.Assume Setup 6.2.Let M be the filtration M := {m n } n∈N .For each 1 i s, assume that there is an integer h i 1 such that the ideal J(i) n is generated in degree nh i for all n ∈ N, that is, Our focus is on characterizing the positivity of the mixed multiplicities e (d 0 ,d) (M | J(1), . . ., J(s)).We define the following N s+1 -graded k-algebra (16) T := Since each ideal J(i) n i is generated in degree n i h i , by Nakayama's lemma we get the isomorphism , . . ., J(s)).
As a consequence we obtain that T is a k-domain of almost integral type with decomposable grading, and so by Theorem 5. Proof.For each p 1, we define the following standard N s+1 -graded algebra Hence e (d 0 , d; C p ) = e (d 0 ,d) (m | J(1) p , . . ., J(s) p ).
Since the following polynomials are equal if and only if for all p ≫ 0 and J = {j 1 , . . ., j k } ⊆ {1, . . ., s} the inequality Proof.By using Proposition 6.For any p > 0, [5,Theorem 4.4] implies that the inequality e (d 0 ,d) m | J(j 1 ) p , . . ., J(j k ) p > 0 is equivalent to the condition of having for all J = {j 1 , . . ., j k } ⊆ {1, . . ., s}.So, the result follows.6.2.Positivity for mixed volumes of convex bodies.In this subsection, by exploiting the known close relation between mixed multiplicities and mixed volumes (see [7,33]), we provide a positivity criteria for mixed volumes.Following the notation of [7], we use the setup below.
0 .The corresponding homogenization of K i (with respect to h i ) is defined as the convex body We consider the (not necessarily Noetherian) graded family of monomial ideals Let M be the filtration M := {m n } n∈N .
As a direct consequence of our previous developments, we recover a classical criterion for the positivity of mixed volumes (see [31,Theorem 5.1.8]).

MULTIGRADED LINEAR SERIES
In this section, we apply our main results from the previous sections to the case of multigraded linear series.Here, we extend the results of [25] on the multigraded linear series.Our main improvement is the fact that we deal with valuations with leaves of bounded dimension instead of restricting to valuations with only one-dimensional leaves.Furthermore, with the family of multigraded linear series with decomposable grading, we provide an interesting family for which the volume function is a polynomial.
Following the notation of [17], we say that X is a variety over a field k if X is a reduced and irreducible separated scheme of finite type over k.
Throughout this section the following setup is used.Proof.First, by Chow's lemma (see, e.g.[15,Theorem 13.100] or [17,Exercise II.4.10]), there exists a proper birational morphism π : X ′ → X where X ′ is a normal projective variety over k.Since for all n ∈ N s we have H 0 X, O(nD) ֒→ H 0 X ′ , π * O(nD) , it suffices to assume that X is a normal projective variety over k, and we do so.
We can find a very ample divisor H on X such that D i H, and so O(nD) ⊆ O(|n|H) for all n ∈ N s (see, e.g., [24,Example 1.2.10] and [21,Theorem 3.9]).From the fact that the section ring S(H) = ∞ n=0 H 0 (X, O(nH)) is of integral type (see [17, Exercise II.5.14]), we obtain that S(D 1 , . . ., D s ) is an N s -graded algebra of almost integral type.
Here, we study multigraded linear series as defined below.Definition 7.3.A multigraded linear series associated to the divisors D 1 , . . ., D s is an N s -graded ksubalgebra W of the section ring S(D 1 , . . ., D s ).
Let W ⊆ S(D 1 , . . ., D s ) be a multigraded linear series and suppose that [W] e i = 0 for all 1 i s.By Proposition 7.2, it follows that W is an N s -graded algebra of almost integral type.The Kodaira-Itaka dimension of W is denoted and given by κ(W) := dim(W) − s, where as before dim(W) denotes the Krull dimension of the N s -graded algebra W of almost integral type.This value was the correct asymptotic for the volume function of an N s -graded algebra of almost integral type (see (9)).Additionally, notice that this agrees with the definition used in [11,Section 7] for the case of singly graded linear series.Following Definition 4.5, let ∆(W) be the global Newton-Okounkov body of W. As in Definition 4.10, consider the integers ind(W) and ℓ W .
Our main result regarding multigraded linear series is the theorem below, and it follows rather easily from our previous developments.(iii) If W has a decomposable grading, then there exists a homogeneous polynomial G W (n) ∈ R[n 1 , . . ., n s ] of degree κ(W) with non-negative real coefficients such that for all n ∈ Z s + .
Proof.(i) The function is obtained as in ( 9).The following theorem is an extension of [21,Theorem 3.3] to a multigraded setting.It expresses the mixed multiplicities of a multigraded linear series with decomposable grading in terms of the multidegrees of the image of the corresponding Kodaira rational maps.Proof.The result follows by applying Corollary 5.8 to the multigraded linear series W that has a decomposable grading.
Finally, below we have an example where the section ring has a decomposable grading.
Example 7.6.Let C be a smooth projective algebraic curve of genus g over an algebraically closed field k.
Let D 1 , . . ., D s be divisors on C with deg(D i ) 2g + 1.Then by [28,Theorem 6], the tensor product map is surjective.Hence, the multigraded linear series W = S(D 1 , . . ., D s ) has a decomposable grading and by part (iii) of Theorem 7.4 the function F W is a homogeneous polynomial of degree κ(W).
Finally, we would like to remark that the conditions on C and D 1 , . . ., D s can be relaxed slightly by considering generalizations of Mumford's theorem, see [4,13,16] for details.

Setup 3 . 7 .
Let k be a field, R be a finitely generated k-domain and r = trdeg k (Quot(R)).Let A ⊂ R[t] be a graded k-algebra of almost integral type.Let d = dim(A).Let m(A) be the index m(A) := [Z : G] of the subgroup G of Z generated by {n ∈ N | [A] n = 0}.For any p > 0, we denote by A p the graded k-subalgebra A p := k [A] p ⊂ A generated by the graded part [A] p .By regrading the algebra A p , we obtain the standard graded k-algebra A p defined by [ A p ] n := [ A p ] np for all n 0. As customary, let e( A p ) := every (n 1 , . . ., n d ) ∈ N r , and ω(γ) = 0 if γ ∈ S has non-zero residue in k ′ = S/m.Let (V ω , m ω , k ω ) be the valuation ring of ω in Quot(R).Notice that V ω dominates S and we have the equality k ω = k ′ .Since b 1 , . . ., b r are rationally independent, we can define a function ϕ : Quot(R) → Z r determined by ϕ(f) = (n 1 , . . ., n r ) if f ∈ Quot(R) and ω(f) = n 1 b 1 + • • • + n r b r .As in Remark 3.8, we now fix the order on Z r that is determined by the linear function l : Z r → R, (n 1 , . . ., n r ) → n 1 b 1 + • • • + n r b r .Therefore, we obtain the valuation ν : Quot(R) → Z r , f → ϕ(f) for which the proposition below is valid.

Theorem 4 . 7 .
Assume Setup 4.3.The fiber ∆(A) n of the global Newton-Okounkov body ∆(A) ⊂ R r × R s 0 coincides with the Newton-Okounkov body ∆(A (n) ) for each n ∈ Z s

Proposition 4 . 13 .
Assume that the fibers of the global Newton-Okounkov body form a linear family of convex bodies.Then the function F A (x) is a homogeneous polynomial of degree q.

Proposition 4 . 14 .
Assume that the global Newton-Okounkov body ∆(A) is a polyhedral cone.Then there exists a fan Σ supported on R s 0 such that the function F A (x) is polynomial at each cone of Σ.

Example 4 . 15 .
Let R = k[u] be a polynomial ring and consider the polynomial ring R[t 1 , t 2 ].Let f : R 2

For each p 1
, let A [p] := k [[A] pe 1 , . . ., [A] pe s ] ⊂ A be the N s -graded algebra generated by [A] pe 1 , . . ., [A] pe s , and denote by A [p] := n∈N s A [p] pn the standard N s -graded algebra obtained by regrading A [p] .For p ≫ 0, Theorem 2.5 and the fact that A has a decomposable grading imply that dim A [p] = d.Then, by Remark 4.1 the function F A [p] A [p] n d for all n ∈ Z s + , where q = d − s and e d; A [p] denotes the mixed multiplicity of A [p] of type d ∈ N s .For each a 1, we consider the a-truncation G [a] (A) := k ∪ s i=1 ∪ a j=1 [p] of the standard multigraded algebras A [p] .It can be seen as an extension into a multigraded setting of the Fujita approximation theorem for graded algebras given in [21, Theorem 2.35].Corollary 5.8.Assume Setup 5.2.Then, the following equalities hold e(d; A) = lim p→∞ e d; A [p] p q = sup p∈Z + e d; A [p] p q .

Setup 7 . 1 .
Let k be a field and X be a proper variety over k.Let D 1 , . . ., D s be a sequence of Cartier divisors on X.We consider the section ring of the divisors D 1 , . . ., D s , which given byS(D 1 , . . ., D s ) := (n 1 ,...,n s )∈N s H 0 X, O(n 1 D 1 + • • • + n s D s ) .Notice that S(D 1 , . . ., D s ) is by construction an N s -graded k-algebra.To simplify notation, for any n = (n 1 , . . ., n s ) ∈ N s , we denote the divisorn 1 D 1 + • • • + n s D s by nD := n 1 D 1 + • • • + n s D s .The following basic result shows that the section ring of D 1 , . . ., D s is an N s -graded algebra of almost integral type (in the sense of Definition 4.2).For the single graded case, see[21, Theorem 3.7].Proposition 7.2.S(D 1 , . . ., D s ) is an N s -graded algebra of almost integral type.

Theorem 7 . 4 .
Assume Setup 7.1.Let W ⊂ S(D 1 , . . ., D s ) be a multigraded linear series and suppose that [W] e i = 0 for all 1 i s.Then, the following statements hold:(i) The volume functionF W (n) := lim n→∞ dim k [W] nn n κ(W)of W is well-defined for all n ∈ Z s + .(ii) There exists a unique continuous function that is homogeneous of degree κ(W) and log-concave and that extends the volume function F W (n) of part (i) to the positive orthant R s 0 .This function is given byF W : R s 0 → R, x → ℓ W • Vol κ(W) ∆(W) x ind(W) .
(ii) It follows from Corollary 4.11.(iii) It follows from Theorem 5.5.For a closed subscheme Y ⊂ P m 1 k × k • • • × k P m s k of a multiprojective space over k, we can consider the multidegrees of Y.These fundamental invariants go back to the work of van der Waerden [34].If S is a standard N s -graded algebra that coincides with the multihomogeneous coordinate ring of Y, then for each d ∈ N s with |d| = dim(Y), one way of defining the multidegree of Y of type d is by setting deg(d; Y) := e(d; S).

(
Cid-Ruiz) DEPARTMENT OF MATHEMATICS, NORTH CAROLINA STATE UNIVERSITY, RALEIGH, NC 27695, USA Email address: ycidrui@ncsu.edu(Mohammadi) DEPARTMENT OF MATHEMATICS, KU LEUVEN, CELESTIJNENLAAN 200B, LEUVEN, BELGIUM AND DE- 2 pe 1 , . . ., [A] pe s ] ⊂ A be the N s -graded algebra generated by [A] pe 1 , . . ., [A] pe s , and denote by A [p] = n∈N s A [p] pn the standard N • • • • • [A] n s e s holds for each (n 1 , n 2 , . . ., n s ) ∈ N s .Now, we additionally assume that the N s -graded algebra A ⊂ R[t 1 , . . ., t s ] of almost integral type has a decomposable grading.For each p 1, let A [p] = k [[A] s -graded algebra obtained by regrading A [p] .