When are multidegrees positive?

WHEN ARE MULTIDEGREES POSITIVE?

FEDERICO CASTILLO, YAIRON CID-RUIZ, BINGLIN LI, JONATHAN MONTA ˜NO^∗, AND NAIZHEN ZHANG

ABSTRACT. Letkbe an arbitrary field,P=P^m_k ¹×_k· · · ×_kP^m_k ^pbe a multiprojective space over k, andX⊆Pbe a closed subscheme ofP. We provide necessary and sufficient conditions for the positivity of the multidegrees ofX. As a consequence of our methods, we show that when Xis irreducible, the support of multidegrees forms a discrete algebraic polymatroid. In alge- braic terms, we characterize the positivity of the mixed multiplicities of a standard multigraded algebra over an Artinian local ring, and we apply this to the positivity of mixed multiplicities of ideals. Furthermore, we use our results to recover several results in the literature in the context of combinatorial algebraic geometry.

1. INTRODUCTION

Letkbe an arbitrary field,P=P^m_k¹×_k· · · ×_kP^m_k^pbe a multiprojective space overk, andX⊆ Pbe a closed subscheme ofP. Themultidegrees ofXare fundamental invariants that describe algebraic and geometric properties of X. For each n= (n₁, . . . ,n_p)∈N^p withn₁+· · ·+n_p= dim(X)one can define themultidegree ofXof typenwith respect toP, denoted by degⁿ_P(X), in different ways (seeDefinition 2.7,Remark 2.8andRemark 2.9). In classical geometrical terms, whenkis algebraically closed, degⁿ_P(X)equals the number of points (counting multiplicity) in the intersection ofXwith the productL₁×_k· · · ×_kL_p⊂P, whereL_i⊂P^m_kⁱ is a general linear subspace of dimensionm_i−n_ifor each 16i6p.

The study of multidegrees goes back to pioneering work by van der Waerden [60]. From a more algebraic point of view, multidegrees receive the name ofmixed multiplicities(seeDefini- tion 2.7). More recent papers where the notion of multidegree (or mixed multiplicity) is studied are, e.g., [1,9,11,23,33,36,41,42,57].

The main goal of this paper is to answer the following fundamental question considered by Trung [57] and by Huh [25] in the casep=2.

• Forn∈N^pwithn₁+· · ·+n_p=dim(X), when do we have thatdegⁿ_P(X)>0?

Our main result says that the positivity of degⁿ_P(X) is determined by the dimensions of the images of the natural projections from P restricted to the irreducible components ofX. First, we set a basic notation: for eachJ ={j₁, . . . ,j_k}⊆{1, . . . ,p}, letΠ_J be the natural projection

Π_J:P=P^m_k¹×_k· · · ×_kP^m_k^p → P^m_k^j1×_k· · · ×_kP^m_k^jk.

The following is the main theorem of this article. Here, we give necessary and sufficient condi- tions for the positivity of multidegrees.

Date: September 24, 2020.

2010Mathematics Subject Classification. Primary 14C17, 13H15; Secondary 52B40, 13A30.

Key words and phrases. positivity, multidegrees, mixed multiplicities, multiprojective scheme, projections, poly- matroids, Hilbert polynomial.

∗The fourth author is supported by NSF Grant DMS #2001645.

1

Theorem A(Theorem 3.12,Corollary 3.13). Letkbe an arbitrary field,P=P^m_k¹×_k· · · ×_kP^m_k^p be a multiprojective space overk, andX⊆Pbe a closed subscheme ofP. Letn= (n₁, . . . ,n_p)∈ N^pbe such thatn₁+· · ·+n_p=dim(X). Then,degⁿ_P(X)>0if and only if there is an irreducible componentY⊆XofXthat satisfies the following two conditions:

(a) dim(Y) =dim(X).

(b) For eachJ ={j₁, . . . ,j_k}⊆{1, . . . ,p}the inequality

n_j1+· · ·+n_jk6dim Π_J(Y) holds.

Whenkis the field of complex numbersTheorem Ais essentially covered by the geometric results in [34, Theorems 2.14, 2.19],¹however their methods do not extend to arbitrary fields.

Here we follow an algebraic approach that allows us to prove the result for all fields, and hence a general version for algebras over Artinian local rings (seeTheorem B). The main idea in the proof ofTheorem Ais the study of the dimensions of the images of the natural projections after cutting by a general hyperplane (seeTheorem 3.7).

We note that ifp=2 andXis arithmetically Cohen-Macaulay, the conclusion ofTheorem A in the irreducible case also holds forX(see [57, Corollary 2.8]). InExample 5.2we show that this is not necessarily true forp >2.

If X is irreducible, then the function r:2^{1,...,p} →Z defined by r(J) :=dim Π_J(Y) is a submodular function, i.e.,r(J₁∩J₂) +r(J₁∪J₂)6r(J₁) +r(J₂)for any two subsetsJ₁,J₂⊆ {1, . . . ,p}, as proved inProposition 5.1(see alsoDefinition 2.16). By the Submodular Theorem (see, e.g., [7, Theorem 3.11] or [44, Appendix B]) and the inequalities ofTheorem A, the points n∈N^pfor which degⁿ_P(X)>0 are the lattice points of ageneralized permutohedron. Defined by A. Postnikov in [49] generalized permutohedra are polytopes obtained by deforming usual permutohedra. In recent years this family of polytopes has been studied in relation to other fields such as probability, combinatorics, and representation theory (see [44,45,48]).

In a more algebraic flavor, we state the translation of Theorem Ato the mixed multiplicities of a standard multigraded algebra over an Artinian local ring (seeDefinition 2.13).

Theorem B (Corollary 3.14). Let A be an Artinian local ring and R be a finitely generated standard N^p-gradedA-algebra. For each16j6p, letm_j⊂Rbe the ideal generated by the elements of degreee_j, wheree_j∈N^pdenotes thej-th elementary vector. LetN = m₁∩· · ·∩mp⊂ R. Let n= (n₁, . . . ,np) ∈N^p be such that n₁+· · ·+np =dim(R/(0:_RN^∞)) −p. Then, e(n;R)>0if and only if there is a minimal prime idealP∈Min(0:_RN^∞)of(0:_RN^∞)that satisfies the following two conditions:

(a) dim(R/P) =dim(R/(0:_RN^∞)).

(b) For eachJ ={j₁, . . . ,j_k}⊆{1, . . . ,p}the inequality n_j1+· · ·+n_jk 6dim R

P +P

j6∈Jm_j

!

−k

holds.

1InRemark 3.15 we briefly discuss how (over the complex numbers)Theorem Acan be obtained by using the results in [34, §2.2].

For a given finite set of ideals in a Noetherian local ring, such that one of them is zero- dimensional, we can define their mixed multiplicities by considering a certain associated stan- dard multigraded algebra (see [58] for more information). These multiplicities have a long history of interconnecting problems from commutative algebra, algebraic geometry, and com- binatorics, with applications to the topics of Milnor numbers, mixed volumes, and integral dependence (see, e.g., [25,27,55,58]). As a direct consequence ofTheorem Bwe are able to give a characterization for the positivity of mixed multiplicities of ideals (see Corollary 4.3).

In another related result, we focus on homogeneous ideals generated in one degree; this case is of particular importance due to its relation with rational maps between projective varieties. In this setting, we provide more explicit conditions for positivity in terms of the analytic spread of products of these ideals (seeTheorem 4.4).

Going back to the setting ofTheorem A, we switch our attention to the following discrete set MSupp_P(X) =

n∈N^p | degⁿ_P(X)>0 ,

which we call the support of X with respect to P. When X is irreducible, we show that MSupp_P(X) is a (discrete) polymatroid (see §2.3, Proposition 5.1). The latter result was in- cluded in an earlier version of this paper whenkis algebraically closed, and an alternative proof is given by Br¨and´en and Huh in [3, Corollary 4.7] using the theory of Lorentzian polynomials.

An advantage of our approach is that we can describe the corresponding rank submodular func- tions of the polymatroids, a fact that we exploit in the applications ofSection 6. Additionally, our results are valid when X is just irreducible and not necessarily geometrically irreducible overk(i.e., we do not need to assume thatX×_kkis irreducible for an algebraic closurekofk);

it should be noticed that this generality is not covered by the statements in [3] and [34].

Discrete polymatroids [24] have also been studied under the name of M-convex sets [46].

Polymatroids can also be described as the integer points in a generalized permutohedron [49], so they are closely related to submodular functions, which are well studied in optimization, see [38] and [52, Part IV] for comprehensive surveys on submodular functions, their applications, and their history. There are two distinguishable types of polymatroids, linear and algebraic polymatroids, whose main properties are inherited by their representation in terms of other algebraic structures. Theorem Aallows us to define another type of polymatroids, that we call Chow polymatroids, and which interestingly lies in between the other two. In the following theorem we summarize our main results in this direction.

Theorem C (Theorem 5.5). Over an arbitrary field k, we have the following inclusions of families of polymatroids

Linear polymatroids

⊆

Chow polymatroids

⊆

Algebraic polymatroids

. Moreover, whenkis a field of characteristic zero, the three families coincide.

Ifkhas positive characteristic, then these types of polymatroids do not agree. In fact, there exist examples of polymatroids which are algebraic over any field of positive characteristic but never linear (seeRemark 5.7).

Theorem A can be applied to particular examples of varieties coming from combinatorial algebraic geometry. In§6.1we do so to matrix Schubert varieties; in this case the multidegrees are the coefficients of Schubert polynomials, thus our results allow us to give an alternative proof to a recent conjecture regarding the support of these polynomials (seeTheorem 6.3). In§6.2and

§6.3we study certain embeddings of flag varieties and of the moduli spaceM_0,p+3, respectively

(seeProposition 6.7andProposition 6.8). In§6.4we recover a well-known characterization for the positivity of mixed volumes of convex bodies (seeTheorem 6.9).

We now outline the contents of the article. InSection 2we set up the notation used throughout the document. We also include key preliminary definitions and results, paying special attention to the connection between mixed multiplicities of standard multigraded graded algebras and multidegrees of their corresponding schemes. Section 3is devoted to the proof of Theorem A and Theorem B. Our results for mixed multiplicities of ideals are included in Section 4. In Section 5we relate our results to the theory of polymatroids. In particular, we show the proof of Theorem C. We finish the paper with Section 6 where the applications to combinatorial algebraic geometry are presented.

We conclude the Introduction with an illustrative example. The following example is con- structed following the same ideas inProposition 5.4.

Example 1.1. Consider the polynomial ring S=k[v₁,v₂,v₃][w₁,w₂,w₃]with the N³-grading deg(v_i) = (0, 0, 0), deg(w_i) =e_i for 16i 63. Let T be the N³-graded polynomial ringT = k[x₀, . . . ,x₃] [y₀, . . . ,y₃] [z₀, . . . ,z₃]where deg(x_i) =e₁, deg(y_i) =e₂and deg(z_i) =e₃. Consider theN³-gradedk-algebra homomorphism

ϕ=T→S,

x₀7→w₁, x₁7→v₁w₁, x₂7→v₁w₁, x₃7→v₁w₁, y₀7→w₂, y₁7→v₁w₂, y₂7→v₂w₂, y₃7→(v₁+v₂)w₂, z₀7→w₃, z₁7→v₁w₃, z₂7→v₂w₃, z₃7→v₃w₃.

Note that P =Ker(ϕ)⊂T is an N³-graded prime ideal. Let Y⊂P=P³_k×_kP³_k×_kP³_kbe the closed subscheme corresponding to P. In this case, one can easily compute the dimension of the projectionsΠ_J(Y)for eachJ⊆{1, 2, 3}, and soTheorem Aimplies that MSupp_P(Y)is given by alln= (n₁, . . . ,n₃)∈N³satisfying the following conditions:

n₁+n₂+n₃=3=dim(Y),

n₁+n₂62=dim Π_{1,2}(Y)

, n₁+n₃63=dim Π_{1,3}(Y)

, n₂+n₃63=dim Π_{2,3}(Y)

, n₁61=dim Π_{1}(Y)

, n₂62=dim Π_{2}(Y)

, n₃63=dim Π_{3}(Y) .

Hence MSupp_P(Y) ={(0, 0, 3),(0, 1, 2),(0, 2, 1),(1, 0, 2),(1, 1, 1)}⊂N³. This set can also be represented graphically as follows:

(0, 0, 3) (0, 3, 0) (3, 0, 0)

Additionally, by usingMacaulay2[21] we can compute that itsmultidegree polynomial(see Definition 2.10) is equal to:

deg_P(Y;t₁,t₂,t₃) = t³₁t³₂+t³₁t²₂t₃+t³₁t₂t²₃+t²₁t³₂t₃+t²₁t²₂t²₃.

We note that here we are following the convention that MSupp_P(Y)is given by the complemen- tary degrees of the polynomial deg_P(Y;t₁,t₂,t₃); for instance, the termt³₁t³₂corresponds to the point(3, 3, 3) − (3, 3, 0) = (0, 0, 3)∈MSupp_P(Y).

2. NOTATION ANDPRELIMINARIES

In this section, we set up the notation that is used throughout the paper. We also present some preliminary results needed in the proofs of our main theorems.

Let p>1 be a positive integer. If n= (n₁, . . . ,n_p),m= (m₁, . . . ,m_p)∈Z^p are two multi- indexes, we writen>mwhenevern_i>m_ifor all 16i6p, andn>mwhenevern_j> m_j for all 16j6p. For each 16i6p, lete_i∈N^pbe thei-th elementary vectore_i= (0, . . . , 1, . . . , 0).

Let 0∈N^p and1∈N^p be the vectors0= (0, . . . , 0) and1= (1, . . . , 1)ofpcopies of 0 and 1, respectively. For anyn= (n₁, . . . ,n_p)∈Z^p, we define its weight as|n|:=n₁+· · ·+n_p. Let[p]

denote the set[p] :={1, . . . ,p}.

For clarity of exposition we first introduce the main concepts in the theory of multidegrees over an arbitrary field. Later, we also work over Artinian local rings; we highlight important details in this more general setting in§2.2.

2.1. The case over a field. We begin by introducing a general setup for Theorem A and its preparatory results.

Setup 2.1. Letkbe an arbitrary field. LetRbe a finitely generated standardN^p-graded algebra overk, that is,[R]₀=kandRis finitely generated overkby elements of degreee_iwith 16i6p.

For each subset J ={j₁, . . . ,j_k}⊆[p] ={1, . . . ,p} denote by R_(J) the standard N^k-graded k- algebra given by

R_(J):= M

i1>0,...,ip>0 ij=0 ifj6∈J

[R]_(i

1,...,ip);

for instance, for each 16j6p,R_(j)denotes the standardN-gradedk-algebraR_(j):=L

k>0[R]_k·e

j. For each 16j6p, letm_j⊂Rbe the idealm_j:= [R]_ej

. LetN⊂Rbe the multigraded irrele- vant idealN := m₁∩ · · · ∩m_p. For eachJ⊆[p], letN_J⊂R_(J)be the corresponding multigraded irrelevant idealN_J:=

T

j∈Jm_j

∩R_(J). LetXbe the multiprojective schemeX:=MultiProj(R) (seeDefinition 2.2below) andX_J be the multiprojective schemeX_J:=MultiProj(R_(J))for each J⊆[p]. To avoid trivial situations, we always assume thatX6=∅.

Definition 2.2. The multiprojective scheme MultiProj(R) is given by MultiProj(R) :=

P∈ Spec(R)|PisN^p-graded andP6⊇N , and its scheme structure is obtained by using multi- homogeneous localizations (see, e.g., [28, §1]).

The inclusionR_(J),→Rinduces the natural projection

Π_J :X→X_J, P∈X7→P∩R_(J)∈X_J.

We embedXas a closed subscheme of a multiprojective spaceP:=P^m_k¹×_k· · · ×_kP^m_k^p. Then, for each J ={j₁, . . . ,j_k}⊆[p], Π_J:X→X_J corresponds with the restriction to Xand toX_J of the natural projection

Π_J:P → P^m_k^j1×_k· · · ×_kP^m_k^jk, andX_Jbecomes a closed subscheme ofP^m_k^j1×_k· · · ×_kP^m_k^jk.

For any multi-homogeneous elementx∈R, the closed subscheme MultiProj(R/xR)⊆X is denoted byX∩V(x).

Notation 2.3. From now on,J ={j₁, . . . ,j_k}denotes a subset of[p]. Setr:=dim(X)andr(J) :=

dim(Π_J(X))for eachJ⊆[p]. For a singleton set{i}⊆[p], r({i})andΠ_{i} are simply denoted byr(i)andΠ_i, respectively.

Note that the image ofΠ_J:X→X_J can be described by the following isomorphism

(1) Π_J(X) =∼ MultiProj R_(J)

R_(J)∩(0:_RN^∞)

! . Remark 2.4. SinceR=R_(J)⊕P

j6∈Jm_j

, we obtain a natural isomorphismR_(J)=∼ ^{P R}

j6∈Jm_j of N^k-gradedk-algebras wherek=|J|.

We now provide some preparatory results.

Lemma 2.5. UnderSetup 2.1, the following statements hold:

(i) r=dim(X) =dim(R/(0:_RN^∞)) −p.

(ii) There is an isomorphism

(2) Π_J(X) =∼ MultiProj



R/

(0:_RN^∞) +X

j6∈J

m_j



. (iii) If(0:_RN^∞) =0, thenΠ_J(X)=∼MultiProj(R_(J)) =X_J.

(iv) If(0:_RN^∞) =0, then

0:_R(J) N^∞_J

=0.

Proof. (i) This formula follows from [28, Lemma 1.2] (also, see [10, Corollary 3.5]).

(ii) From the natural mapsR_(J),→RR/(0:_RN^∞), we obtain a natural isomorphism R_(J)/ R_(J)∩(0:_RN^∞) =^∼

−→ R/(0:_RN^∞)

(J). By using Remark 2.4 it follows that R/(0:_RN^∞)

(J)=∼ R/

(0:_RN^∞) +P

j6∈Jm_j

. There- fore, the claimed isomorphism is obtained from(1).

(iii) It follows directly from part (ii) andRemark 2.4.

(iv) This part is clear.

Let P_R(t) =P_R(t₁, . . . ,tp)∈Q[t] =Q[t₁, . . . ,tp] be the Hilbert polynomial of R (see, e.g., [23, Theorem 4.1], [9, Theorem 3.4]). Then, the degree ofP_R is equal torand

P_R(ν) =dim_k([R]_ν) for allν∈N^p such thatν0. Furthermore, if we write

(3) P_R(t) = X

n1,...,np>0

e(n₁, . . . ,np)

t₁+n₁ n₁

· · ·

t_p+n_p n_p

, then 06e(n₁, . . . ,nr)∈Zfor alln₁+· · ·+np=r.

Remark 2.6. The following are basic properties of Hilbert polynomials.

(i) SinceN^k(0:_RN^∞) =0 fork0 we have dim_k([R]_ν) =dim_k([R/(0:_RN^∞)]_ν)forν0.

Thus,

P_R(t) =P_R/(0:RN∞)(t).

(ii) Let Lbe a field extension of k. Then, R⊗_kLis a finitely generated standardN^p-graded L-algebra and dim_L([R⊗_kL]_ν) =dim_k([R]_ν)for allν∈N^p. Thus,

P_R⊗kL(t) =P_R(t).

In particular, one can always assumekis an infinite field (for instance, we can substitute kby a purely transcendental field extensionk(ξ)).

Under the notation of(3)we define the following invariants.

Definition 2.7. Letn= (n₁, . . . ,n_p)∈N^pwith|n|=r. Then:

(i) e(n,R) :=e(n₁, . . . ,n_p)is themixed multiplicity ofRof typen.

(ii) degⁿ_P(X) :=e(n₁, . . . ,n_p)is themultidegree ofX=MultiProj(R)of typenwith respect to P.

As stated in the Introduction, in classical geometrical terms, whenkis algebraically closed, degⁿ_P(X) is also equal to the number of points (counting multiplicity) in the intersection of X with the productL₁×_k· · · ×_kL_p⊂P, whereL_i⊆P^m_kⁱis a general linear subspace of dimension m_i−n_ifor each 16i6p(see [60], [9, Theorem 4.7]).

The multidegrees of Xcan be defined easily in terms of Chow rings and in terms of Hilbert series.

Remark 2.8. The Chow ring ofP=P^m_k¹×_k· · · ×_kP^m_k^pis given by A^∗(P) = Z[H₁, . . . ,Hp]

H^m₁¹⁺¹, . . . ,H^m_p^p+1

where H_i represents the class of the inverse image of a hyperplane of P^m_kⁱ under the natural projectionΠ_i:P→P^m_kⁱ. Then, the class of the cycle associated toXcoincides with

[X] = X

06n_i6m_i

|n|=r

degⁿ_P(X)H^m₁ ¹⁻ⁿ¹· · ·H^m_p^p−np ∈A^∗(P).

Remark 2.9. By considering the Hilbert series Hilb_R(t₁, . . . ,t_p) :=P

ν∈N^pdim_k([R]_ν)t^ν₁¹· · ·t^ν_p^p of R, one can analogously define the notions of mixed multiplicities and multidegrees (see [43, §8.5], [9, Theorem A]). Here we quickly derive this analogous definition because we shall use it in§6.1. LetS=k[x_1,0,x_1,1, . . . ,x_1,m1]⊗_k· · · ⊗_kk[x_p,0,x_p,1, . . . ,xp,m_p]be the multigraded polynomial ring corresponding with P =P^m_k¹×_k· · · ×_kP^m_k^p, that is P =MultiProj(S). By considering anS-free resolution ofR, we can write

HilbR(t) =K(R;t)/(1−t)^m+1=K(R;t₁, . . . ,t_p)/

Yp i=1

(1−t_i)^mi+1,

whereK(R;t)is called theK-polynomial ofR(see [43, Definition 8.21]). LetC(R;t)∈Z[t₁, . . . ,t_p] be the sum of all the terms inK(R;1−t)of total degree equal to dim(S) −dim(R)(see [43, Def- inition 8.45]). Then, if(0:_RN^∞) =0, we obtain the equality

C(R;t) = X

06n_i6m_i

|n|=r

degⁿ_P(X)t^m₁¹⁻ⁿ¹· · ·t^m_p^p−np.

Proof. From [9, Theorem A(I)] we have Hilb_R(t) =P

|k|=dim(R)Q_k(t)/(1−t)^k whereQ_k(t)∈ Z[t]. The assumption(0:_RN^∞) =0 gives that dim(R) =r+p(seeLemma 2.5(i)). Hence, by using [9, Theorem A(II,III)] we obtain that degⁿ_P(X) =e(n;R) =Q_n+1(1)for all|n|=r. Also, the assumption(0:_RN^∞) =0 and [9, Theorem 2.8(ii)] imply thatQ_k(1) =0 when|k|=dim(R) andk_i=0 for some 16i6p.

After writing Hilb_R(t) =P

|k|=dim(R)Q_k(t)(1−t)^m+1−k/(1−t)^m+1, we obtain the equality

K(R;t) = X

|k|=dim(R)

Q_k(t)(1−t)^m+1−k.

Making the substitutiont_i7→(1−t_i)and choosing the terms of total degree dim(S) −dim(R) = Pp

i=1m_i−r, it follows thatC(R;t) =P

|k|=dim(R)Q_k(1)t^m+1−k=P

|n|=rQ_n+1(1)t^m−n. So, the

result is clear.

Although in the proofs of Theorem A and Theorem B we do not exploit the fact that mul- tidegrees can be defined as inRemark 2.8, we do encode the multidegrees in a homogeneous polynomial that mimics the cycle associated toXin the Chow ringA^∗(P). The following ob- jects are the main focus of this paper.

Definition 2.10. LetX⊆P=P^m_k¹×_k· · · ×_kP^m_k^p be a closed subscheme withr=dim(X). We denote themultidegree polynomial ofXwith respect toPas the homogeneous polynomial

deg_P(X;t₁, . . . ,tp) := X

06ni6mi

|n|=r

degⁿ_P(X)t^m₁¹⁻ⁿ¹· · ·t^mp^p−np ∈ N[t₁, . . . ,tp]

of degreem₁+· · ·+m_p−r. We say that thesupport ofXwith respect toPis given by MSupp_P(X) :=

n∈N^p | degⁿ_P(X)>0 .

Remark 2.11. Note that under the assumption(0:_RN^∞) =0 we obtain the equality deg_P(X;t) = C(R;t).

2.2. The case over an Artinian local ring. In this subsection, we show how the mixed multi- plicities are defined for a standard multigraded algebra over an Artinian local ring.

Setup 2.12. Keep the notations and assumptions introduced inSetup 2.1and now substitute the fieldkby an Artinian local ringA.

In this setting, the notion of mixed multiplicities is defined essentially in the same way as in Definition 2.7.

Definition 2.13. LetP_R(t) =P_R(t₁, . . . ,t_p)∈Q[t] =Q[t₁, . . . ,t_p]be the Hilbert polynomialof R(see, e.g., [23, Theorem 4.1], [9, Theorem 3.4]). Then, as before, the degree ofP_R is equal to dim(R/(0:_RN^∞)) −pand

P_R(ν) =length_A([R]_ν) for allν∈N^psuch thatν0. If we writeP_R(t) =P

n₁,...,np>0e(n₁, . . . ,n_p) ^t1_n⁺ⁿ¹

1

· · · ^tp_n^+np

p

, then 0 6e(n₁, . . . ,n_r) ∈ Z for all n₁+· · ·+n_p =dim(R/(0:_RN^∞)) −p. For each n = (n₁, . . . ,n_p)∈N^p with |n|=dim(R/(0:_RN^∞)) −p, we set that e(n,R) :=e(n₁, . . . ,n_p) is themixed multiplicity ofRof typen.

2.3. Polymatroids. In this subsection we include some relevant information about polyma- troids.

Definition 2.14. LetEbe a finite set andra functionr:2^E→Z_>0satisfying the following two properties: (i) it isnon-decreasing, i.e.,r(T₁)6r(T₂)ifT₁⊆T₂⊆E, and (ii) it issubmodular, i.e., r(T₁∩T₂) +r(T₁∪T₂)6r(T₁) +r(T₂) if T₁,T₂⊆E. The function r is called a rank function onE. We usually letE= [p].

A(discrete)polymatroidPon[p]with rank functionris a collection of points inN^p of the following form

P=



x= (x₁, . . . ,x_p)∈N^p | X

j∈J

x_j6r(J), ∀J([p], X

i∈[p]

x_i=r([p])



.

By definition, a polymatroid consists of the integer points of a polytope (the convex hull ofP), we call that polytope a base polymatroid polytope. We note that a polymatroid is completely determined by its rank function.

Remark 2.15. If the rank function of P satisfiesr({i})61 for everyi∈[p], then Pis called a matroid. In other words, matroids are discrete polymatroids where every integer point is an element of{0, 1}^p. A general reference for matroids is [47].

In the following definition we consider the standard notions of linear and algebraic matroids (see [47, Chapter 6]) and adapt them to the polymatroid case.

Definition 2.16. LetPbe a polymatroid.

• We say P is linear over a field k if there exists a k-vector space V and subspaces V_i,i ∈[p] such that for every J⊆[p] we have r(J) =dim_kP

j∈JV_j

[47, Proposi- tion 1.1.1]. The vector space V together with the subspaces V_i for 16i6p, are a linear representationofP.

• We sayPisalgebraicover a fieldkif there exists a field extensionk,→Land intermedi- ate field extensionsL_i,i∈[p]such that for everyJ⊆[p]we haver(J) =trdeg_k

V

j∈JL_j , whereV

j∈JL_j is the compositumof the subfields, i.e., the smallest subfield inL con- taining all of them [47, Theorem 6.7.1]. The fieldLtogether with the subfields L_i for 16i6p, are analgebraic representationofP.

3. ACHARACTERIZATION FOR THE POSITIVITY OF MULTIDEGREES

In this section, we focus on characterizing the positivity of multidegrees and our main goal is to prove Theorem A and Theorem B. Throughout this section we continue using the same notations and assumptions ofSection 2.

We begin with the following result that relates the Hilbert polynomialP_R(t)∈Q[t]ofRwith the dimensions r(J) =dim(Π_J(X))of the schemes Π_J(X). It extends [57, Theorem 1.7] to a multigraded setting.

Proposition 3.1. AssumeSetup 2.1. For eachJ ={j₁, . . . ,j_k}⊆[p], letdeg(P_R;J)be the degree of the Hilbert polynomialP_Rin the variablest_j1, . . . ,t_jk. Then, for every suchJ ={j₁, . . . ,j_k}we have that

deg(P_R;J) =r(J).

Proof. We may assume that (0:_R N^∞) =0 andkis an infinite field by Remark 2.6. Fix J = {j₁, . . . ,j_k}⊆[p]and letw∈Nbe such that dim_k([R]_n) =P_R(n)for everyn= (n₁, . . . ,n_p)>w1.

Let(d₁, . . . .d_k)be such thatδ:=deg(P_R;J) =d₁+· · ·+d_k andt^d_j¹

1 · · ·t^d_j^k

k divides a term ofP_R. Let q be a polynomial in the variables{t_i|i6∈J}such thatP_R−q·t^d_j¹

1 · · ·t^d_j^k

k has no term divisible byt^d_j¹

1 · · ·t^d_j^k

k . Lets= (s_i|i6∈J)∈N^p−|J|be a vector of integers such thats>w1and q(s)6=0. Thus, if one evaluatest_i=s_iinP_Rfor everyi6∈Jone obtains a polynomialQon the variablest_j1,· · ·,t_jk of degreeδ. On the other hand, by [9, Theorem 3.4], forn_j1,· · ·,n_jk >w this polynomialQcoincides with the Hilbert polynomial of theR_(J)-module generated by[R]_s⁰, wheres_i⁰=s_iifi6∈Jands_i⁰=0 otherwise. Call this moduleM.

Since(0:_RN^∞) =0, for every 16i6pwe have grade(m_i)>1, and then there exist ele- mentsy_i∈[R]_ei which are non-zero-divisors (see, e.g., [6, Lemma 1.5.12]). From the fact that y^s

0 1

1 · · ·y^s

p0

p ∈M, it follows that Ann_R(J)(M) =0. Therefore,δ=dim Supp(M)∩X_(J)

=r(J),

by [9, Theorem 3.4], finishing the proof.

In the following remark we gather some basic relations for the radicals of certain ideals.

Remark 3.2. (i) Let I,J,K⊂R be ideals. If J⊂√

K, then I+J⊂√

I+K. In particular, if

√J=√

K, then√

I+J=√ I+K.

(ii) For any elementx∈m₁, since (x:_Rm^∞₁ )m^k₁ ⊂(x)for somek >0, it follows thatp (x) = q

(x:_Rm^∞₁ )m^k₁=p

(x:_Rm^∞₁ )∩m₁.

If k is an infinite field, then for each 16i6p we say that a property P is satisfied by a general element in thek-vector space [R]_ei, if there exists a dense open subsetUof[R]_e

i with the Zariski topology such that every element inUsatisfies the propertyP.

The following three technical lemmas are important steps for the proof ofTheorem 3.7.

Lemma 3.3. AssumeSetup 2.1withkbeing an infinite field. Suppose thatRis a domain. Let x∈[R]_e1 be a general element. Then, we have the equalityp

(x:_RN^∞) = q

x:_Rm^∞₁ . Proof. Since(0:_RN^∞) =0, we have that ht(m_j)>1 for every 16j6p. Consider the following finite set of prime ideals

S =

P∈Spec(R)|P∈Min(m_j)for some 26j6pandP6⊇m₁ .

By using the Prime Avoidance Lemma and the fact that k is infinite, for a general element x∈[R]e₁ we have thatx6∈S

P∈SP. IfP∈Min x:_Rm^∞₁

, then ht(P)61 by Krull’s Principal Ideal Theorem, and so we would have thatP∈SwheneverP6⊇m₁andP⊇m_jfor some 26 j6p. Therefore, for anyP∈Spec(R)and a general elementx∈[R]e₁, ifP∈Min x:_Rm^∞₁ we getP⊇(x:_RN^∞) = (x:_R(m₁∩m₂∩ · · · ∩m_p)^∞); so,p

(x:_RN^∞) =p

(x:_Rm^∞₁ ).

The lemma below is necessary for some reduction arguments inTheorem 3.7.

Lemma 3.4. Assume Setup 2.1 with k being an infinite field. Suppose that R is a domain.

Let x∈[R]_e1 be a general element and set Z=X∩V(x) =MultiProj(R/xR). Then, for each J ={1,j₂, . . . ,j_k}⊆[p], the following statements hold:

(i) dim(Π_J(Z)) =dim(X_J∩V(x)), whereX_J∩V(x) =MultiProj R_(J)/xR_(J) . (ii) dim(Π_L(Z)) =dim Π_L⁰(X_J∩V(x))

, whereL = J\ {1}andΠ_L⁰ denotes the natural pro- jectionΠ_L⁰ :P^m_k¹×_kP^m_k^j2×_k· · · ×_kP^m_k^jk →P^m_k^j2×_k· · · ×_kP^m_k^jk.

Proof. For notational purposes, letb_J:= m₁∩R_(J).

(i) From(2)we have thatΠ_J(Z)=∼ MultiProj R/ (x:_RN^∞) +P

j6∈Jm_j

. Since we are as- suming 1∈J, fromRemark 2.4we obtain the natural isomorphism

R_(J)/

x:_R(J) b^∞_{J =∼}

−→ R/

(x:_Rm^∞₁ ) +X

j6∈J

m_j

;

indeed, for`>0 andy∈R<sub>(J)</sub>, one notices thatb<sup>`</sup>_J·y∈xR_(J)if and only ifm^`₁·y∈xR.

ByLemma 3.3andRemark 3.2(i) we haveq

(x:_Rm^∞₁ ) +P

j6∈Jm_j=q

(x:_RN^∞) +P

j6∈Jm_j, and by applyingLemma 3.3to the ringR_(J)we obtainq

(x:_R(J) b^∞_J ) =q

(x:_R(J) N^∞_J ). It fol- lows that

(4) R_(J)q

(x:_R(J) N^∞_J ) =∼ R s

(x:_RN^∞) +X

j6∈J

m_j, which gives the result.

(ii) By using(2)we obtain thatΠ_L(Z)=∼MultiProj R/ (x:_RN^∞) +P

j6∈Jm_j+ m₁

and that Π_L⁰ (X_J∩V(x))=∼MultiProj

R_(J)/

(x:_R(J) N^∞_J ) + b_J

. Since the isomorphism in(4)can be extended to

R_(J). q

(x:_R(J)N^∞_J ) + b_J

=∼ R.



 s

(x:_RN^∞) +X

j6∈J

m_j+ m₁



,

the result follows fromRemark 3.2(i).

We continue with the next auxiliary lemma that allows us to simplify the proof of Theo- rem 3.7.

Lemma 3.5. AssumeSetup 2.1 withkbeing an infinite field. Suppose thatRis a domain and r(1)>1. Let x∈[R]_e1 be a general element and setZ=X∩V(x) =MultiProj(R/xR). Then, the following statements hold:

(i) If1∈J⊆[p], thendim(Π_J(Z)) =r(J) −1; in particular,dim(Z) =r−1.

(ii) If16∈Jandr(K)> r(J), whereK ={1}∪J⊆[p], thendim(Π_J(Z)) =r(J).

Proof. (i) First, fromLemma 3.4(i) it suffices to compute dim(X_J∩V(x)), whereX_J∩V(x) = MultiProj R_(J)/xR_(J)

. For J ={1,j₂, . . . ,j_k}⊆[p], note that Π₁(X)=∼ Π₁⁰(X_J), whereΠ₁⁰ de- notes the natural projection Π₁⁰ :P^m_k¹×_kP^m_k^j2×_k· · · ×_kP^m_k^jk →P^m_k¹. Therefore, neither the assumption nor the conclusion changes if we substituteRbyR_(J)andXbyX_J, and we do so.

From the short exact sequence

0→R(−e₁)−→^x R→R/xR→0,

we obtain P_R/xR(t) =P_R(t) −P_R(t−e₁). By using Proposition 3.1, deg(P_R;t₁) =r(1) >1 and so P_R is non-constant as a univariate polynomial in the variable t₁. Thus, P_R/xR(t)6=0 which implies that(x:_RN^∞)is a proper ideal. So, Krull’s Principal Ideal Theorem yields that ht(x:_RN^∞) =1 and that

dim(Z) =dim(R/(x:_RN^∞)) −p=dim(R) −1−p= (r+p) −1−p=r−1.

(ii) By usingLemma 3.4(ii), we can substituteRbyR_(K) andXbyX_K, and we do so. So, we may assume thatK = [p]andJ ={2, . . . ,p}. From(2)we get the isomorphism

(5) Π_J(Z)=∼MultiProj

R/((x:_RN^∞) + m₁) . The equality

(6) p

(x:_RN^∞) + m₁= q

(x:_Rm^∞₁ ) + m₁ follows fromLemma 3.3andRemark 3.2(i). The assumption yields that

ht(m₁) =dim(R) −dim R_(J)

= (r+p) − (r(J) +p−1)>2, then as a consequence Krull’s Principal Ideal Theorem it follows that p

(x) =p

(x:_Rm^∞₁ );

therefore,Remark 3.2(i) implies thatp

(x:_Rm^∞₁ ) + m₁=√

m₁. By summing up, we obtain the equalities dim R/((x:_RN^∞) + m₁)

=dim(R/m₁) =r(J) +p−1, and so the result follows.

The next important theorem computes the dimension of the image of the projectionsΠ_J after cutting with a general hyperplane under certain conditions. For the proof of this result, we need the following version of Grothendieck’s Connectedness Theorem. For that, we recall the definitions

c(R) :=min

dim(R/a)|a⊂Ris an ideal and Spec(R)\V(a)is disconnected , sdim(R) :=min

dim(R/P)|P∈Min(R) and ara(a) :=min{n|p

(a₁, . . . ,a_n) =√

aanda_i∈R} for any ideala⊂R.

Lemma 3.6([5, Proposition 2.1], [57, Lemma 2.6]). For two proper homogeneous idealsa,b⊂ R, ifmin{dim(R/a), dim(R/b)}>dim(R/(a + b)), then

dim(R/(a + b))>min{c(R), sdim(R) −1}−ara(a∩b).

We are now ready to present the following theorem.

Theorem 3.7. AssumeSetup 2.1withkbeing an infinite field. Suppose thatRis a domain and r(1)>1. Let x∈[R]_e1 be a general element and setZ=X∩V(x) =MultiProj(R/xR). Then, for eachJ⊆[p]we have that

dim(Π_J(Z)) =min

r(J), r J∪{1}

−1

.

Proof. For each J ={j₁, . . . ,j_k}⊆K = {h₁, . . . ,h<sub>`</sub>}⊆[p] we have that Π<sub>J</sub>(Z) = Π<sup>0</sup><sub>J</sub>(Π<sub>K</sub>(Z)) whereΠ<sup>0</sup><sub>J</sub>denotes the natural projectionΠ<sup>0</sup><sub>J</sub>:P<sup>m</sup><sub>k</sub><sup>h1</sup>×<sub>k</sub>· · · ×<sub>k</sub>P<sup>m</sup><sub>k</sub><sup>h`</sup> →P^m_k^j1×_k· · · ×_kP^m_k^jk. So, fromLemma 3.5(i) it follows that the inequality “6” holds in the desired equality.

Due to Lemma 3.5, in order to show the reversed inequality “>”, it suffices to show that dim(Π_J(Z)) >r(J) −1 when 16∈J and r(K) =r(J), where K ={1}∪J⊆ [p]. By using Lemma 3.4(ii), we assume may thatK = [p] and J = {2, . . . ,p}. From (5) and (6), the proof would be complete if we prove the inequality dim R/ (x:_Rm^∞₁ ) + m₁ >(r_J−1) + (p−1) = r+p−2.

By usingLemma 3.3andLemma 3.5(i) we obtain that

dim(R/(x:_Rm^∞₁ )) =dim(R/(x:_RN^∞)) = (r−1) +p=r+p−1, and sincer(J) =r, we have

dim(R/m₁) =dim(R_(J)) =r(J) + (p−1) =r+ (p−1) =r+p−1.

Moreover,(6)andLemma 3.5(i) yield that dim R/((x:_Rm^∞₁ ) + m₁)

=dim R/((x:_RN^∞) + m₁)

6(r−1) + (p−1) =r+p−2.

Sincex∈m₁,Remark 3.2(ii) gives that ara (x:_Rm^∞₁ )∩m₁

=ara((x)) =1. AsRis a domain, c(R) =sdim(R) =r+p. Therefore, fromLemma 3.6we obtain that

dim R/((x:_Rm^∞₁ ) + m₁)

>min{r+p,(r+p) −1}−1=r+p−2.

So, the proof is complete.

Notation 3.8. Let {x₀, . . . ,x_s} be a basis of the k-vector space [R]_e1. Consider a purely tran- scendental field extension L:=k(z₀, . . . ,z_s) of k, and set R_L :=R⊗_kL and X_L :=X⊗_kL= MultiProj(R_L)⊆P⊗_kL=P^m_L¹×_L· · · ×_LP^m_L^p. We say thatz:=z₀x₀+· · ·+z_sx_s∈[R_L]_e

1 is thegeneric elementof[R_L]_e

1.

In the following remark we explain that field extensions as in Notation 3.8preserve the do- main assumption.

Remark 3.9. Suppose thatRis a domain and consider a purely transcendental field extension k(ξ). Then, R⊗_kk(ξ) is also a domain; indeed, one can see that R⊗_kk(ξ) is a subring of the field of fractions Quot(R[ξ])of the polynomial ringR[ξ]. So, whenRis a domain one can extendkto an infinite field without loosing the assumption ofRbeing a domain.

The lemma below shows that the Hilbert function modulo a generic element coincides with the one module a general element.

Lemma 3.10. Assume Notation 3.8 with kbeing an infinite field. Let x∈[R]_e

1 be a general element, then

dim_k [R/xR]_ν

=dimL [R_L/zR_L]_ν for allν∈N^p.

Proof. Let T be the polynomial ring T =k[z₀, . . . ,z_s] and consider the finitely generated T- algebra given by S= (R⊗_kT)/w(R⊗_kT)where w=z₀x₀+· · ·+z_sx_s ∈R⊗_kT. From the Grothendieck’s Generic Freeness Lemma (see, e.g., [39, Theorem 24.1], [13, Theorem 14.4]) there exists an element 06=a∈T such thatS_a is a freeT_a-module. Hence, for anyp∈Spec(T)

inside the dense open subsetD(a)⊂Spec(T), ifk(p)denotes the residue fieldk(p) =T_p/pT_p ofT_p, one has that

dim_k(p) [S_a⊗_Tak(p)]_ν

=dim_Quot(T) [S_a⊗_TaQuot(T)]_ν

=dim_L [R_L/zR_L]_ν for allν∈N^p. Note that for anyβ= (β₀, . . . ,β_s)∈k^s+1withp_β= (z₀−β₀, . . . ,z_s−β_s)∈D(a) one has the isomorphisms

S_a⊗_Tak(p_β) =∼ R⊗_kT

(z₀x₀+· · ·+zsxs,z₀−β₀, . . . ,zs−βs) =∼ R/(β₀x₀+· · ·+β_sx_s)R.

So, the result follows.

We now obtainTheorem AwhenXis an irreducible scheme.

Remark 3.11. We first provide a couple of general words regarding the proof ofTheorem 3.12 below and where the irreducibility assumption comes into play. The proof is achieved by itera- tively cutting with generic hyperplanes (followingNotation 3.8) to arrive to a zero-dimensional situation, and the main constraint is to control the dimension of the image of all the possible pro- jections after cutting with a general hyperplane (see(7)). Our main tool to control those dimen- sions isTheorem 3.7, where it is needed to assume thatRis a domain. WhenXis irreducible, by just taking the reduced scheme structure X_red=MultiProj(R/√

0)we can easily reduce to the case whereRis a domain. To maintain the irreducibility assumption during the inductive pro- cess, we use a “generic” version of Bertini’s Theorem as presented in [16, Proposition 1.5.10].

It should be noted that the usual versions of Bertini’s Theorem for irreducibility require X to be geometrically irreducible and that the dimension of the image of certain morphism is bigger or equal than two (see [32, Theoreme 6.10, Corollaire 6.11]). Finally,Lemma 3.10is used to relate the process of cutting with a generichyperplane with the one of cutting with ageneral hyperplane.

Theorem 3.12. AssumeSetup 2.1. Suppose that Xis irreducible. Let n= (n₁, . . . ,n_p)∈N^p such that|n|=r. Then,degⁿ_P(X)>0if and only if for eachJ ={j₁, . . . ,j_k}⊆[p]the inequality n_j1+· · ·+n_jk6r(J)holds.

Proof. FromProposition 3.1it is clear that the inequalitiesn_j1+· · ·+n_jk6r(J)are a necessary condition for degⁿ_P(X) =e(n;R)>0. Therefore, it suffices to show that they are also sufficient.

Assume that n_j1+· · ·+n_jk 6r(J) for every J ={j₁, . . . ,j_k}⊆[p]. We may also assume that(0:_RN^∞) =0 byRemark 2.6(i). Hence, the condition of Xbeing irreducible implies that

√0⊂R is a prime ideal. Since the associativity formula for mixed multiplicities (see, e.g., [9, Lemma 2.7]) yields that

e(n;R) =length_R√ 0

R^√₀

·e

n;R/√ 0

,

we can assume thatRis a domain, and we do so. In addition, byRemark 2.6(ii),Proposition 3.1, andRemark 3.9we may also assume thatkis an infinite field.

We proceed by induction onr. Ifr=0, then [9, Theorem 3.10] impliese(0;R)>0.

Suppose now thatr>1. Without any loss of generality, perhaps after changing the grading, we can assume that n₁> 1. Let L, R_L, X_L and z be defined as in Notation 3.8. Let x ∈ [R]_e1 be a general element. Set S=R/xR, Z=X∩V(x) =MultiProj(S), T =R_L/zR_L, W= X_L∩V(z) =MultiProj(T) and n⁰ =n−e₁. Then, [9, Lemma 3.9] and Lemma 3.10 yield that e(n;R) =e(n⁰;S) =e(n⁰;T). From [16, Proposition 1.5.10] we obtain thatW is also an