Simplicial generation of Chow rings of matroids

We introduce a presentation of the Chow ring of a matroid by a new set of generators, called"simplicial generators."These generators are analogous to nef divisors on projective toric varieties, and admit a combinatorial interpretation via the theory of matroid quotients. Using this combinatorial interpretation, we (i) produce a bijection between a monomial basis of the Chow ring and a relative generalization of Schubert matroids, (ii) recover the Poincar\'e duality property, (iii) give a formula for the volume polynomial, which we show is log-concave in the positive orthant, and (iv) recover the validity of Hodge-Riemann relations in degree 1, which is the part of the Hodge theory of matroids that currently accounts for all combinatorial applications of [AHK18]. Our work avoids the use of"flips,"the key technical tool employed in [AHK18].


INTRODUCTION
Chow rings of matroids were introduced in [FY04] as generalizations of cohomology rings of wonderful compactifications of hyperplane arrangement complements [DCP95]: Definition 1.0.1.Let M be a loopless matroid of rank r = d + 1 on a ground set E. The Chow ring of M is a graded ring The generators h F of A • ∇ (M ) are called the simplicial generators of the Chow ring of M .The simplicial presentation allows us to (i) interpret the simplicial generators and their monomials in A • ∇ (M ) combinatorially via the theory of matroid quotients; (ii) recover the Poincaré duality property of A • ∇ (M ); (iii) compute the volume polynomial of A • ∇ (M ), which we show to be Lorentzian in the sense of [BH20], and whose formula generalizes a main result of [Pos09]; and (iv) give a simplified proof of the Hodge-Riemann relations in degree 0 and 1 for matroids, which is the part of the Hodge theory of matroids in [AHK18] that accounts for all currently known combinatorial applications of [AHK18].
We achieve these by establishing combinatorial properties of the simplicial presentation that reflect the following two geometric observations: (A) Suppose base-point-free divisors D 1 , . . ., D s on a variety X generate the Chow ring A • (X), and Y ⊂ X is a subvariety.Then the pullback map A • (X) → A • (Y ) can be computed via the intersections of Y with general hyperplane pullbacks of X → P(H 0 (D i )).If furthermore the pullback map is surjective, then these intersections generate A • (Y ).(B) For base-point-free divisors D 1 , . . ., D s whose sum is an ample divisor on a projective variety Y , let vol(t 1 , . . ., t s ) be the volume polynomial Then vol(t), as a function R s → R, is positive and log-concave on the positive orthant R s >0 [Laz04, §1.6.A].When a matroid M is realizable, its Chow ring is isomorphic to the Chow ring of a projective variety Y R(M ) , which is a subvariety of a projective toric variety X An , and the pullback A • (X An ) → A • (Y R(M ) ) is surjective (see §2.3).In this case, unlike the classical presentation A • F Y (M ), the generators of our presentation A • ∇ (M ) represent base-point-free divisor classes on Y R(M ) , obtained as pullbacks of base-point-free divisor classes on X An that generate A • (X An ) (see §3.2).We demonstrate that even when M is not necessarily realizable, the generators of A • ∇ (M ) display behaviors analogous to those of base-point-free divisors on the projective varieties.
(i) Simplicial generators and their monomials as matroid quotients.The principal truncation of a matroid M by a flat F is a new matroid with bases {B \ f : B a basis of M and f ∈ B ∩ F = ∅}.When M is realizable, so that h F ∈ A1 ∇ (M ) represents a base-point-free divisor class on a projective variety Y R(M ) , the principal truncation by F corresponds to any general hyperplane pullback of the map defined by h F (Remarks 3.2.6 & 3.2.7).For an arbitrary matroid, the following analogue holds.
Theorem 3.2.3.Let M be a loopless matroid.The simplicial generator h F ∈ A • ∇ (M ) correspond, via the cap product (see §2.1 for a definition), to the principal truncation of M by the flat F .
We use this interpretation of the simplicial generators to study their monomials in A • ∇ (M ).A monomial basis for A • ∇ (M ) is obtained by carrying over a Gröbner basis computation for the classical presentation A • F Y (M ) in [FY04] (Proposition 3.3.2& Corollary 3.3.3).This monomial basis, which we call the nested basis, now has the following combinatorial interpretation.Definition 3.3.4.Let M ′ be a matroid quotient of M (that is, every flat of M ′ is a flat of M ), denoted f : M ′ և M .Write n f (S) := rk M (S) − rk M ′ (S) for a subset S of the ground set.An f -cyclic flat of f is a flat F of M ′ such that F is minimal (with respect to inclusion) among the flats G of M ′ with n f (G) = n f (F ).The matroid M ′ is a relative nested quotient of M if the f -cyclic flats form a chain.Theorem 3.3.8.There is a natural bijection, via the cap product, between the nested basis of A • ∇ (M ) and the set of relative nested quotients of M .
When the matroid M is a Boolean matroid, i.e. a matroid whose ground set is a basis, the relative nested quotients of M are known as "nested matroids," studied previously in the context of Chow rings of permutohedral varieties in [Ham17].
(ii) The Poincaré duality property.We build upon the bijection in Theorem 3.3.8 to establish the following theorem, which mirrors the fact that for certain classes of varieties, 1 Theorem 4.2.1.Let M be a loopless matroid on {0, 1, . . ., n}, and X An the permutohedral variety of dimension n.Let ∆ M be the Bergman class of the matroid M considered as an element of A • (X An ).Then A • (M ) ≃ A • (X An )/ ann(∆ M ), where ann(∆ The Poincaré duality property for A • ∇ (M ) was established in [AHK18,Theorem 6.19] by a double induction.In our case, the Poincaré duality property is a simple algebraic consequence of Theorem 4.2.1 (Corollary 4.2.2).
(iii) The volume polynomial and its log-concavity.The following formula computes the intersection numbers of the simplicial generators h F .Theorem 5.2.4.For M a loopless matroid of rank r = d + 1, let M : A d (M ) → R be the degree map.For a multiset of nonempty flats {F 1 , . . ., F d }, we have 1 if rk M ( j∈J F j ) ≥ |J| + 1 for every ∅ J ⊆ {1, . . ., d} 0 otherwise.
That these intersection numbers are either 1 or 0 stands in stark contrast to calculations in the classical presentation of the Chow ring of a matroid, which were carried out in [Eur20].
The intersection numbers collect together into the "volume polynomial" of A • ∇ (M ), defined in the following corollary.The corollary recovers [Pos09,Corollary 9.4] on volumes of generalized permutohedra when M is set to be the Boolean matroid.
The volume polynomial V P ∇ M , as a real-valued function, is therefore positive in its positive orthant, as are volume polynomials of base-point-free divisors on projective varieties.Moreover, it displays the logconcavity behavior described in (B).
Corollary 5.3.2.As a real-valued function, the volume polynomial V P ∇ M (t) is log-concave in its positive orthant.
Log-concavity is a consequence of the following statement.
Theorem 5.3.1.The volume polynomial V P ∇ M (t) of a loopless matroid M is Lorentzian in the sense of [BH20].
Lorentzian polynomials are multivariate polynomials characterized by two conditions: one on their supports and one on their partial derivatives (see §5.1).We show that V P ∇ M satisfies these two conditions by using Theorem 5.2.4 to understand the support of V P ∇ M (Proposition 5.3.3) and by using Theorem 3.2.3 to understand the partial derivatives of V P ∇ M as volume polynomials of principal truncations of M .
(iv) A simplified proof of the Hodge-Riemann relations in degree 1.We use log-concavity of V P ∇ M to give a simplified proof of the Hodge-Riemann relations in degree 1 for Chow rings of matroids: Theorem 6.2.1.Let M be a loopless matroid of rank r = d + 1 on a ground set E, and ℓ ∈ A 1 (M ) a combinatorially ample divisor class (see §2.2).Then the Chow ring A • (M ) with the degree map M satisfies the Kähler package in degree zero and one.That is, for i ≤ 1, (HL ≤1 ) (hard Lefschetz in degree ≤ 1) the multiplication by ℓ a is an isomorphism, and (HR ≤1 ) (Hodge-Riemann relations in degree ≤ 1) the symmetric form is non-degenerate on A i (M ) and positive-definite when restricted to the degree i primitive space The authors of [AHK18] establish the Hodge-Riemann relations in all degrees by a double-inductive argument that utilizes a generalization of Bergman fans of matroids.In our case, the log-concavity of V P ∇ M (Corollary 5.3.2) provides us a key step in the induction in degrees ≤ 1, and thus our proof of Theorem 6.2.1 is a single induction on ranks of matroids and involves only the classical Bergman fans of matroids.
The validity of the Hodge-Riemann relation in degree 1 accounts for all currently known combinatorial applications of [AHK18], including the Heron-Rota-Welsh conjecture.Huh has posed the discovery of combinatorial applications for the Hodge-Riemann relations in higher degrees as an open problem [Huh18].
Organization.In Section 2, we develop the necessary background on Chow rings of matroids.In Section 3, we introduce the simplicial presentation of the Chow ring, study its structural properties via principal truncations, and introduce the relative nested quotients.In Section 4, we establish the Poincaré duality property for matroids.In Section 5, we calculate intersection numbers with respect to the simplicial presentation, and prove that the volume polynomial of the Chow ring in the simplicial presentation is Lorentzian.Finally, in Section 6 we apply our results to give a simplified proof of the Hodge theory in degree 1 for matroids.
Acknowledgements.We are grateful to Vic Reiner and Federico Castillo for introducing the authors.The second author thanks June Huh for helpful conversations about Lorentzian polynomials.The first author was supported by a Zuckerman Postdoctoral Scholarship.We thank Alex Fink for suggesting some minor corrections, and we thank Matt Larson for pointing out a minor error in the proof of Proposition 3.3.10.

PRELIMINARIES
In this section, we review relevant background materials on Chow rings of matroids.Familiarity with toric and tropical geometry, from which the combinatorial notions here originate, can be helpful but not necessary.As references we point to [Ful93] and [CLS11] for toric geometry, and to [FS97], [MS15, Chapter 6], and [AHK18, §4- §5] for tropical geometry.
In §2.1, we describe Chow cohomology rings and Minkowski weights of fans, and in §2.2 we illustrate these notions in the setting of matroids.These first two subsections are purely combinatorial.In §2.3, we provide the underlying geometric picture that motivates many of the combinatorial constructions.
2.1.Chow cohomology rings and Minkowski weights.We give a brief account of Chow cohomology rings and Minkowski weights of smooth fans, which are combinatorial analogues of cohomology rings and homology classes of algebraic varieties 2 .
We set the following notation and definitions for rational fans over a lattice.
• Let N be a lattice of rank n, and N ∨ the dual lattice.We write N R := N ⊗ Z R.
• For Σ ⊂ N R a rational fan, let Σ(k) be the set of k-dimensional cones of Σ.
• A fan Σ is smooth if, for all cones σ of Σ, the set of primitive ray vectors of σ can be extended to a basis of N .A smooth fan is simplicial in that every k-dimensional cone is generated by k rays.
Convention.Throughout this section, we assume that Σ ⊂ N R is a smooth fan of dimension d, which is not necessarily complete.
Geometrically, the ring A • (Σ) is the Chow ring A • (X Σ ) of the toric variety X Σ associated to the fan Σ. See [Dan78, §10.1] for the case where Σ is complete, and [BDCP90] or [Bri96] for the general case.From this geometric description of A • (Σ), or directly from the algebraic definition above, one can check that A ℓ (Σ) = 0 unless 0 ≤ ℓ ≤ d.
We call a linear combination ρ c ρ x ρ ∈ R[x ρ : ρ ∈ Σ(1)] of the variables x ρ a divisor on Σ. Divisors of special interest in algebraic geometry are ample and nef divisors 3 .They have the following combinatorial description for a complete fan Σ (i.e. a complete toric variety X Σ ).
A divisor D = ρ∈Σ(1) c ρ x ρ on a complete fan Σ defines a piecewise-linear function ϕ D : N R → R, determined by being linear on each cone of Σ with ϕ D (u ρ ) = c ρ .We say that are strict whenever u and u ′ are not in a common cone of Σ, we say that D is ample.Nef (resp.ample) divisors on Σ correspond to polytopes in N ∨ R whose outer normal fans coarsen (resp.equal) Σ.
Theorem 2.1.2.[CLS11, Theorems 6.1.5-6.1.7]Let Σ be a smooth complete fan.A nef divisor D = ρ∈Σ(1) c ρ x ρ on Σ defines a polytope 2 We use real coefficients for Chow cohomology rings and Minkowski weights, although Chow rings of algebraic varieties initially take integral coefficients.The algebraic varieties that motivate the constructions here-smooth complete toric varieties and wonderful compactifications-share the feature that the Chow ring, the integral cohomology ring, and the ring of algebraic cycles modulo numerical equivalence all coincide [EH16, Appendix C.3.4].In particular, their Chow rings are torsion-free.In this paper, while most of our arguments work over Z, we will always work over R for convenience.
In agreement with the terminology of [AHK18], we will call certain divisors "combinatorially nef" (see §2.2) although they are furthermore base-point-free on wonderful compactifications of realizable matroids.
A nef divisor D is ample if the outer normal fan of P D is equal to Σ.
A divisor D defines an element [D] ∈ A 1 (Σ), which we call the divisor class (of D) on Σ.We say that a divisor class ζ ∈ A 1 (Σ) is nef (resp.ample) if any choice of a divisor D representing ζ is nef (resp.ample).This is well-defined because two divisors D and D ′ represent the same divisor class if and only if ϕ D − ϕ D ′ is a linear function on N R .In terms of polytopes, two nef divisors D and D ′ define the same divisor class in if and only if P D and P D ′ are parallel translates.
Remark 2.1.3.Any nef divisor class [D] ∈ A 1 (Σ) is effective; that is, it can be written as non-negative linear combination D = ρ∈Σ(1) c ρ x ρ (with c ρ ≥ 0 ∀ρ ∈ Σ(1)).This is an immediate consequence of Theorem 2.1.2:Given a nef divisor D, translating if necessary one can assume that the polytope P D contains the origin in its relative interior.
With A • (Σ) as an analogue of a cohomology ring, we describe analogues of homology groups.
The groups of Minkowski weights are analogues of homology groups because they are dual to the Chow cohomology ring in the following way.
Lemma 2.1.5.[MS15, Theorem 6.7.5]4For 0 ≤ ℓ ≤ d, we have an isomorphism This isomorphism is an analogue of the Kronecker duality map in algebraic topology.We use it to define combinatorial analogues of some standard operations in algebraic topology.We define the cap product by defined as its generator (unique up to scaling), and the cap product with the fundamental class defines the map In particular, noting that MW 0 (Σ) = R, the degree map is defined as If Σ is complete, one can check that MW n (Σ) ≃ R, where the fundamental class ∆ Σ is given by ∆ Σ (σ) = 1 for each cone σ ∈ Σ(n).In this case, we have the following analogue of the Poincaré duality theorem in algebraic topology.
is an isomorphism for each 0 ≤ k ≤ n.Equivalently (by Lemma 2.1.5),the pairing The isomorphisms in Theorem 2.1.6make n i=0 MW n−i (Σ) into a graded ring when Σ is complete.We write MW • (Σ) := MW n−• (Σ) for this graded ring.For complete fans, the resulting multiplication structure on the Minkowski weights is known as the stable intersection, denoted ∩ st , in tropical geometry.We will only need a special case of stable intersections5 , which we provide explicitly in the context of matroids in §3.1 (Proposition 3.1.8).
In the proof of Lemma 3.2.5, we will need the following explicit description of the map δ Σ : A 1 (Σ) Proposition 2.1.7.Let D be a nef divisor on Σ such that the corresponding polytope P D is a lattice polytope.Then the Minkowski weight given by Theorem 2.1.6 is defined by where P D (σ) is the edge of P D corresponding to the cone σ ∈ Σ Q (n − 1), and ℓ(P D (σ)) is its lattice length, i.e. the number of lattice points on P D (σ) minus one.
We end this subsection by noting the functoriality of the constructions here.An inclusion of fans ι : Σ ′ ֒→ Σ defines the pullback map ι * , which is a surjective map of graded rings Comparing the presentations of A • (Σ) and A • (Σ ′ ), one checks easily that this map coincides with the quotient of A • (Σ) by the ideal Dually, a Minkowski weight ∆ ′ on Σ ′ is naturally a Minkowski weight on Σ.In this case we often abuse the notation and write ∆ ′ for both Minkowski weights.
Remark 2.1.8.Unraveling the definitions, one checks that the cap product is functorial in the following sense: The pullback map ι * : A , where ∆ ′ on the right hand side is considered as a Minkowski weight on Σ.

Bergman classes and Chow rings of matroids.
We now specialize our discussion to matroids.We begin with the braid fan, on which matroids will arise as certain Minkowski weights.
First, we fix some notations.Let E := {0, 1, . . ., n}, and for a subset S ⊆ E write e S := i∈S e i , where e 0 , . . ., e n is the standard basis of Z E .Let N be the lattice N = Z E /Ze E , and write u S for the image of e S in N .The dual lattice of N is N ∨ = (Ze E ) ⊥ = {(y 0 , . . ., y n ) ∈ Z E | n i=0 y i = 0}.The braid fan (of dimension n), denoted Σ An , is the outer normal fan of the standard permutohedron (of dimension n), which is the polytope Concretely, the braid fan Σ An is a complete fan in N R whose cones are Cone(u S 1 , . . ., u S k ) ⊂ N R , one for each chain of nonempty proper subsets In particular, the primitive rays of Σ An are {u S | ∅ S E}.This fan is also known as the Coxeter complex of the type A root system, hence the notation Σ An .The associated toric variety of Σ An , denoted X An , is often called the permutohedral variety (of dimension n).
We assume familiarity with the basics of matroids, and refer to [Wel76; Oxl11] as general references.We fix the following notation for matroids: We write U r,E for the uniform matroid of rank r on E, and we set a matroid M to have ( The Chow ring of a matroid is defined as the Chow cohomology ring of its Bergman fan.Explicitly, we have the following.Definition 2.2.4.The Chow ring of a loopless matroid M is the graded ring . We call linear combinations of the variables x F divisors on M , and the elements of A 1 (M ) divisor classes on M .The divisor class of F ⊇a x F for any atom a ∈ A(M ) is called the hyperplane class of M , and is denoted α(M ) or just α if the matroid in question is clear.
Remark 2.2.5.The ring A • (M ) was first studied in [FY04] under a slightly different presentation, which for clarity is denoted A • F Y (M ) and is given as .
That is, we have x F = z F for every nonempty proper flat F ∈ L M , and z E = −α.As both presentataions A • (M ) and A • F Y (M ) are relevant for us, we will use the variable names x, z in a consistent manner; for example, in the summation Since MW(Σ M ) ≃ R by Proposition 2.2.1.(2),with the Bergman class as the fundamental class, the Chow ring of a matroid M has the degree map M : A d (Σ M ) → R, defined by the cap product ξ → ξ∩∆ M .Explicitly, it is determined by Note that the braid fan Σ An is the Bergman fan of the Boolean matroid U |E|,E , and its fundamental class ∆ Σ An is the Bergman class of U |E|,E .We will thus always identify We end this subsection with a discussion of nef and ample divisors on the braid fan, and the resulting analogous notions for Bergman fans.The following characterization of nef divisors on Σ An , which is a specialization of Theorem 2.1.2,was recognized in various works [Edm70; Mur03; Pos09; AA17]; for a modern treatment and generalization to arbitrary Coxeter root systems we point to [ACEP20].
Proposition 2.2.6.The following are equivalent for a divisor (1) D is a nef divisor on Σ An , (2) the function c (•) : 2 E → R satisfies the submodular property (3) the normal fan of the polytope This polytope lives in the translate of N ∨ R where the points have coordinate sum c E .One translates P (c) to N ∨ R as follows.Fix an element i ∈ E. We have where α The notion of nef and ample divisors on a matroid is inherited from the braid fan.First, note that for a loopless matroid M , the inclusion of fans ι M : Σ M ֒→ Σ An induces the pullback map When we wish to clarify whether a variable x S is an element of A • (M ) or A • (Σ An ), we write The pullback map motivates the following notions regarding divisors on M .We say that a divisor (class) on M is combinatorially nef if it is a pullback of a nef divisor (class) on Σ An .A combinatorially ample divisor (class) is defined similarly.Explicitly, a divisor Say D is ample if the inequality is strict whenever A and B are incomparable.
Combinatorially nef (resp.ample) divisor classes on M are closed under nonnegative linear combinations, since nef (resp.ample) divisor classes on complete fans are in general.We thus let K M (resp.K M ) be the cone in A 1 (M ) of combinatorially nef (resp.ample) divisor classes on M , called the combinatorially nef 2.3.The geometry of matroids via wonderful compactifications.We provide the underlying algebraic geometry of the combinatorial constructions in the previous two subsections §2.1 and §2.2.
Let M be a loopless matroid on E = {0, 1, . . ., n} of rank r = d + 1 realizable over a field , which we may assume to be algebraically closed.A realization R(M ) of M consists of any of the following equivalent pieces of data: • a list of vectors E = {v 0 , . . ., v n } spanning a -vector space V ≃ r , or • a surjection n+1 ։ V where e i → v i , or • an injection PV * ֒→ P n , dualizing the surjection n+1 ։ V .
For a realization R(M ) of M with PV * ֒→ P n , the coordinate hyperplanes of P n intersect with PV * to give the associated hyperplane arrangement A R(M ) on PV * , which is encoded by the flats of M in the following way.For each nonempty flat F of M , let L F be a linear subspace of V * defined by and let PL F be the linear subvariety of PV * .The hyperplanes of A R(M ) are {PL a } a∈A(M ) corresponding to the atoms, and more generally, a flat F of rank c corresponds to the c-codimensional linear subvariety PL F .
We denote by YR(M) the hyperplane arrangement complement PV * \ A R(M ) .It is a linear subvariety of an algebraic torus in the following way: The algebraic torus T N = ( * ) n+1 / * of the lattice N = Z n+1 /Z(1, 1, . . ., 1) is the complement of the union of coordinate hyperplanes in P n , and hence YR(M) is the intersection of PV * with T N .The linear subvariety YR(M) ⊂ T N is related to the Bergman class of M by tropicalization in the following way (see [MS15,§3] for tropicalizations of subvarieties of an algebraic torus).
Theorem 2.3.1.Let R(M ) be a realization of a loopless matroid M , and let YR(M) ⊂ T N be the associated hyperplane arrangement complement.Recall that the permutohedral variety X An is a toric variety with the torus T N .
(1) [MS15, Theorem 4.1.11]The support of the tropicalization of YR(M) , denoted trop( YR(M) ) equals the support of the Bergman fan ∆ M .(2) [MS15, Proposition 6.4.17 & Theorem 6.7.7]It follows that the closure Y R(M ) of YR(M) in the permutohedral variety X An satisfies where (1) The variety Y R(M ) is obtained as a series of blow-ups on PV * by the following process: First blowup the points {PL F } rk(F )=rk(E)−1 , then blow-up the strict transforms of the lines {PL F } rk(F )=rk(E)−2 , and continue until having blown-up strict transforms of {PL F } rk(F )=1 .We denote by π R(M ) : Y R(M ) → PV * be the blow-down map.
(2) For each nonempty flat F of M , the projection away from the linear subvariety PL F ⊂ PV * is a rational map PV * P(V * /L F ).The variety Y R(M ) then is the (closure of) the graph of the rational map When U n+1,n+1 is realized as the standard basis of n+1 , the associated wonderful compactification is the toric variety X An of the braid fan.It is obtained from P n by blowing up the coordinate points, then the (strict transforms of) coordinate lines, and so forth.Let us write π An : X Σ An → P n for the blow-down map.Then for a realization PV * ֒→ P n of a loopless matroid M , Remark 2.3.2.(1) above expresses the wonderful compactification Y R(M ) as the strict transform of PV * ⊂ P n under the sequence of blow-ups π An .In other words, we have a diagram The boundary of Y R(M ) \ YR(M) consists of the exceptional divisors E F obtained by blowing up (strict transforms of) PL F .These divisors have simple-normal-crossings [DCP95], and consequently the intersection theory of the boundary divisors of Y R(M ) is encoded in the matroid.More precisely, the Chow ring Remark 2.3.3.We note the following geometric observations about the presentation .
(1) The variables x F correspond to the exceptional divisors E F obtained by blowing up (strict transforms of) PL F .(2) The quadric relations x F x F ′ = 0 reflect that two exceptional divisors from blowing up two nonintersecting linear subspaces do not intersect.
(3) The linear relations defining A • (M ) reflect that for any atom a ∈ A(M ), we have Remark 2.3.4 (Relation to (A)).The geometry and the combinatorics of the permutohedral variety X An have been widely studied in various contexts including moduli spaces [LM00; BB11], convex optimization [Edm70; Mur03], Hopf monoids [DF10; AA17], and lattice polyhedra [PRW08; Pos09].In our case, the variety X An plays the role of ambient variety X in (A), and the wonderful compactification Y R(M ) the subvariety Y ⊂ X.In §3.2, we introduce base-point-free divisor classes on X An that generate A • (X An ).

THE SIMPLICIAL PRESENTATION AND ITS MONOMIALS
In this section, we introduce the paper's main object of study: a new presentation of the Chow ring of a matroid which we call the simplicial presentation A • ∇ (M ) of A • (M ).After a combinatorial preparation in §3.1, we introduce the simplicial presentation in §3.2 and show that multiplying by its generators corresponds to a matroid operation called principal truncations.In §3.3, we extend this correspondence to establish a combinatorial interpretation of a monomial basis of the Chow ring of a matroid.
Let M and M ′ be matroids on a common ground set E = {0, 1, . . ., n}.Definition 3.1.1.The matroid M ′ is a (matroid) quotient of M , written f : M ′ և M , if any every flat of M ′ is also a flat of M .In particular, if M and M ′ are loopless, then f : Example 3.1.2.Any matroid on ground set E is a quotient of the Boolean matroid U |E|,E .Any Bergman fan of a loopless matroid is a subfan of the braid fan.
Example 3.1.3(Realizable matroid quotients).Matroid quotients model linear surjections (dually, linear injections) in the following way.Let M and M ′ have realizations by E ։ V and E ։ V ′ (respectively).If there exists a commuting diagram of linear maps or dually then M ′ is a matroid quotient of M .Matroid quotients M ′ և M arising in this way are called realizable matroid quotients.We caution that a matroid quotient M ′ և M with both M ′ and M realizable over the same field need not be realizable (for an example, see [BGW03, §1.7.5]).
For a matroid quotient f : We say that M ′ is an elementary (matroid A modular cut K of M defines an elementary quotient M ′ և M by Conversely, given an elementary quotient f : M ′ և M , one recovers the modular cut K of M defining the elementary quotient by We write M ′ K և M to denote an elementary quotient of M given by a modular cut K. Example 3.1.4.Let M have a realization E ։ V .For K a modular cut of M , let v K be a nonzero vector contained in F ∈K span (F ) and not contained in any span of a flat that is not in K 7 .Dually, with 7 For such vK to exist, the field must be large enough, and the elementary matroid quotient defined by the modular cut K must be realizable.
the notation as in §2.3, we have a general hyperplane or dually, The map E ։ V / span (v K ) is a realization of the matroid M ′ of the elementary quotient M ′ K և M defined by K. Dually, with the notation as in §2.3, the associated hyperplane arrangement A R(M ′ ) is the intersection of PH K with the coordinate hyperplanes in P E .Equivalently, the hyperplane arrangement A R(M ′ ) is the intersection of PH K with the hyperplanes in the hyperplane arrangement A R(M ) under the inclusion PH K ⊂ PV * .
Of particular interest in our case is when K is the interval [F, E] ⊂ L M , since an interval in L M is always a modular cut.We call the resulting elementary quotient, denoted T F (M ), the principal truncation of M associated to the flat F .An explicit description of principal truncations follows.
Proposition 3.1.5.[Oxl11, Exercise 7.2.4.]The principal truncation T F (M ) of a matroid M associated to a flat F ∈ L M of rank at least 1 has bases and the flats of T F (M ) partition into two sets L T F (M ) = K ⊔ L according to their f -nullities by Remark 3.1.6.In Example 3.1.4,if K = [F, E] for some flat F , then we can set v K = v F , a general vector in span (F ), and dually, we can set H K = H F , a general hyperplane in V * containing L F .
We end our combinatorial preparation by connecting matroid quotients to Minkowski weights on braid fans via the notion of matroid intersections.Recall that a spanning set of a matroid is a subset that contains a basis of the matroid.
3.2.The variables of the simplicial presentation.We now define a new presentation A • ∇ (M ) of the Chow ring of a matroid M , and discuss its first properties.The key result here is that the variables of A • ∇ (M ) correspond to principal truncations of M .
We prepare by noting a distinguished set of nef divisor classes on Σ An and their polytopes considered in [Pos09].For a nonempty subset S of E, let denote the negative standard simplex of S. As the edges of ∇ S are parallel translates of e i −e j for i = j ∈ S, Proposition 2.2.6 (in the form of Remark 2.2.7) implies that ∇ S is a polytope with the corresponding nef divisor class These divisors were considered in [Pos09] 9 and implicitly in [Ham17].We now consider the presentation of A • (M ) given by pullbacks of these nef divisor classes corresponding to (negative) standard simplices.For M a loopless matroid on E, and is the smallest flat containing S, note that we have By construction, the elements h F (M ) ∈ A 1 (M ) are (combinatorially) nef divisor classes on M .We will simply write h F for h F (M ) when there is no confusion.Definition 3.2.1.For M a loopless matroid on E, the simplicial presentation A • ∇ (M ) of the Chow ring of M is the presentation of A • (M ) whose generators are {h F } F ∈L M \{∅} where We call {h F } F ∈L M \{∅} the simplicial generators.
The variable h here stands for "hyperplane"; for the geometric origin of the simplicial presentation see Remarks 3.2.6 and 3.2.7.The linear change of variables from {z F } F ∈L M \{∅} to {h F } F ∈L M \{∅} is evidently invertible, given by an upper triangular matrix.Explicitly, by Möbius inversion we have where µ is the Möbius function on the lattice L M .Thus, the explicit presentation of

Denote by L ≥2
M the set of flats of M of rank at least 2. Noting that h a = 0 ∈ A • ∇ (M ) for any atom a ∈ A(M ), we define {h F | F ∈ L ≥2 M } to be the nontrivial simplicial generators of the Chow ring of M .They form a basis of A 1 ∇ (M ).
9 In [Pos09] the author uses yS to denote the nef divisor of the standard simplex of S instead of the negative standard simplex.
This difference is due to using inner normal fans instead of outer normal fans of polytopes.
Remark 3.2.2.When the matroid M is the cyclic matroid of the complete graph K n−1 on n − 1 vertices, the Chow ring of M is the cohomology ring of the Deligne-Knudson-Mumford space M 0,n of rational curves with n marked points [DCP95, §4.3], [MS15, Theorem 6.4.12].In this case, by using the minimal building set instead of the maximal building set, the simplicial presentation recovers the Etingof-Henriques-Kamnitzer-Rains-Singh presentation of the cohomology ring of M 0,n [EHKR10;Sin04].In this presentation, the author of [Dot19] showed that the cohomology ring of M 0,n is Koszul because it has a quadratic Gröbner basis.In the classical presentation, the Chow ring of any matroid with rank > 3 has no quadratic Gröbner basis.
The following theorem, which relates the variables of the simplicial presentation to principal truncations, is the key property of the simplicial presentation that we use throughout this paper.Let us first fix a notation: Notation.For a nonempty subset S ⊆ E, we denote by H S the matroid with bases Theorem 3.2.3.Let M be a loopless matroid on E, and S a nonempty subset of E. Write F for the smallest flat of M containing S. Then H S ∧ M = T F (M ), and the nef divisor class The theorem will follow mostly from the following lemma.
Lemma 3.2.4.Let M and H S be as in the statement of Theorem 3.2.3, and let F be the smallest flat of M containing S. Then we have (2) and consequently, (3) Proof.We first show that (3) follows from (2).A matroid is loopless if and only if the empty set ∅ is a flat.By Proposition 3.1.5,the matroid T F (M ) is thus loopless if and only if ∅ is not covered by an element in the interval [F, E].As claimed, this happens if and only if rk M (F ) > 1.
We now prove (2).By definition of H S ∧ M , the minimal elements in the set of spanning sets S(H S ∧ M ) are B \ i where B ∈ B(M ) and i ∈ B ∩ S = ∅.Since minimal spanning sets are bases, we have When S = F , this is the description of the bases of T F (M ) in Proposition 3.1.5,so it remains to show be extended to a basis by an element in S. That is, there exists an element s ∈ S such that (B \ f ) ∪ s is a basis of M , as desired.
Proof of Theorem 3.2.3.Let δ Σ An : A • (Σ An ) ∼ → MW n−• (Σ An ) be the isomorphism map in Theorem 2.1.6.We claim that δ Σ An (h S ) = ∆ H S , which is proved in Lemma 3.2.5 below.Our desired statement then follows immediately from Lemma 3.2.4,since h S ∩ ∆ M = (δ Σ An (h S )) ∩ st ∆ M by the definition of stable intersection ∩ st .Lemma 3.2.5.Let δ Σ An : A • (Σ An ) ∼ → MW n−• (Σ An ) be the isomorphism map in Theorem 2.1.6.Then we have Proof.We claim that the support |∆ H S | of ∆ H S is equal to the support of the (n − 1)-skeleton of the outer normal fan of negative standard simplex ∇ S .If this is the case, then Proposition 2.1.7implies δ Σ An (h S ) = ∆ H S because all the edges of the negative standard simplex ∇ S have lattice length 1.Now, for the claim, note first that the translate Theorem 3.2.3encodes the combinatorics of the following geometric motivation for the simplicial presentation.
Remark 3.2.6.We recall the following standard fact in algebraic geometry.Let L ⊂ V * be an inclusion of vector spaces, so that PL is a linear subvariety of PV * .Let X = Bl PL PV * be the blow-up of PV * along PL, which is also the graph of the rational map PV * P(V * /L), and let π : X → PV * be the blow-down map.Let h = c 1 (O PV * (1)) denote the hyperplane class, and E denote the exceptional divisor of the blow-up.Then the map Bl PL PV * → P(V * /L) corresponds to the linear series |O X (π * h − E)|, so the sections of the line bundle O X (π * h − E) correspond to the hyperplanes in PV * that contain PL.Now, suppose M has a realization R(M ) over an algebraically closed field as PV * ֒→ P n , and let notations be as in §2.3.The geometry of A and hence h F represents the divisor class of the strict transform of a general hyperplane in PV * containing the linear subvariety PL F .Thus multiplying by h F corresponds to intersecting by a general hyperplane in PV * containing PL F , which corresponds to the principal truncation T F (M ) by Example 3.1.4(in the form of Remark 3.1.6).More precisely, we have 3 is the combinatorial mirror of this geometric observation.
Remark 3.2.7 (Relation to (A)).Suppose M has a realization R(M ) by PV * ֒→ P n .By the second description in Remark 2.3.2.(2), the wonderful compactification Y R(M ) is embedded in the product of projective spaces F ∈L M \{∅} P(V * /L F ).We described h F as a divisor class represented by the strict transform of a general hyperplane in PV * containing PL F in the previous Remark 3.2.6.Alternatively, the variable h F thus represents the base-point-free divisor obtained as the hyperplane class pullback of the map Y R(M ) → P(V * /L F ).In other words, the divisor classes h S ∈ A • ∇ (M ) play the role of "base-point-free divisor classes," and we have interpreted the hyperplane class pullbacks of the maps they define as principal truncations.
Remark 3.2.8.In the classical presentation A • (Σ An ), the cap product x S ∩ ∆ M is almost never a Bergman class of a matroid-it is a Minkowski weight which may have negative weights on some cones.This reflects the geometry that the divisor x S is effective but usually not nef.

3.3.
A monomial basis of the simplicial presentation and relative nested quotients.We introduce the notion of relative nested quotients, which are relative generalizations of (loopless) Schubert matroids in matroid theory, and we show that they are in in bijection with elements of a monomial basis of A • ∇ (M ).We start by producing a monomial basis of A • ∇ (M ) via the Gröbner basis computation in [FY04].Pick a total order > on elements of L M such that F > G if rk M (F ) ≤ rk M (G), and take the induced lex monomial order on A • F Y (M ).A Gröbner basis for A • F Y (M ) was given as follows.Theorem 3.3.1.[FY04, Theorem 1] The following form a Gröbner basis for the ideal of F and G are incomparable nonempty flats G a nonempty flat .
In [FY04], the authors associate a ring to an atomic lattice with a chosen "building set," and [FY04, Theorem 1] provides a Gröbner basis for the defining ideal of these more general rings.In our setting, the atomic lattice is the lattice of flats of a matroid, and the building set is the collection of all nonempty flats.We now note that the Gröbner basis in [FY04] carries over to the simplicial presentation as follows.Again, pick a total ordering > of L M such that if rk M (F ) ≤ rk M (G), then F > G.
Proposition 3.3.2.The following is a Gröbner basis for the defining ideal of A • ∇ (M ) with respect to the lex monomial ordering induced by >: and define ϕ : S F Y → S ∇ to be the substitution Observe that ϕ is lower triangular with −1's on the diagonal when the variables z F and h F are written in descending order with respect to >.Hence, if f ∈ S with initial monomial The proposition now follows from the fact that the elements of the Gröbner basis above are the images under ϕ of the elements of the Gröbner basis given in Theorem 3.3.1.

As a result, we obtain a monomial basis of A
We call this basis of A • ∇ (M ) the nested basis of the Chow ring of M .Proof.If B is a Gröbner basis for an ideal I of a polynomial ring R over a field k, then the monomials not divisible by the initial term of any element of B are a k-basis for R/I [Pee11, Theorem 39.6].Applying this fact to the Gröbner basis of Proposition 3.3.2yields the result.
While the nested basis of A • ∇ (M ) looks identical to the one given for A • F Y (M ) in [FY04, Corollary 1], we show here that with the simplicial presentation the monomials in the basis now allow for a combinatorial interpretation as a distinguished set of matroid quotients of M .
Let f : M ′ և M be a matroid quotient on a ground set E.
Relative nested quotients are relative generalizations of (loopless) Schubert matroids: , which occurs if and only if A\B is a set of coloops in the restriction M ′ | A .In other words, the f -cyclic flats of M ′ are precisely the cyclic flats of M ′ , which are flats F of M ′ such that M ′ | F has no coloops.Moreover, the relative nested quotients of U |E|,E are called nested matroids, which in the literature also go by (loopless) Schubert matroids because they are realized by general points in the appropriate Schubert subvarieties of Grassmannians.See [Ham17, §2.2] for more on cyclic flats and nested matroids.
The data of cyclic flats of a matroid and their ranks determine the matroid [Bry75, Proposition 2.1].We generalize the statement to f -cyclic flats of a matroid quotient.We first need the following fact about obtaining any matroid quotient as a sequence of elementary quotients.Lemma 3.3.6.
(1) [Hig68], [Bry86, Exercise 7.20] Any matroid quotient f : M ′ և M can be obtained as a sequence of elementary quotients in a canonical way called the Higgs factorization of f .The Higgs factorization of a quotient f : where the bases of M i for i = 1, . . ., c are defined as (2) [KK78, Theorem 3.4] The modular cuts K i of the Higgs factorization are Proposition 3.3.7.The data of the f -cyclic flats, their f -nullities, and the matroid M determine the quotient f : M ′ և M .More precisely, writing n f (E) = c, the data recovers the Higgs factorization f by specifying the modular cuts K i to be Proof.For each i = 1, . . ., c, the modular cut (2).This can equivalently be written as Let us prepare with a lemma.For a flat F of a matroid M , and an integer a ≥ 0, denote by T F a (M ) the matroid obtained from M by performing a many times the principal truncation associated to F .Such iteration of T F is well-defined because principal truncation associated to F preserves F as a flat.
Lemma 3.3.9.Let f : M ′ և M be a matroid quotient formed by two loopless matroids, and let F be a nonempty flat of M ′ .Then, the following hold: (1) Suppose G is another flat of M ′ such that G F , and a < rk M ′ (F )− rk M ′ (G).Then, the subset G is a flat of T F a (M ′ ).In particular, setting G = ∅, we have that (2) Suppose F ⊆ S for every nonempty f -cyclic flat S of M ′ .Denote by g the matroid quotient g : T F (M ′ ) և M .Then, the set cyc(g) of g-cyclic flats equals {F } ∪ cyc(f ).
Proof.For statement (1), we first recall the description of the flats in a principal truncation (Proposition 3.1.5 Moreover, the rank of F decreases by 1 each time one performs the principal truncation T F .Hence, the condition a < rk M ′ (F ) − rk M ′ (G) ensures that G is not covered by F in the matroid T F a−1 (M ′ ), and so G is a flat of T F a (M ′ ).For statement (2), again by Proposition 3.1.5,we have that the set and the other flats of T F (M ′ ) do not change in rank when considered as flats of M ′ .Thus, since F is contained in every nonempty f -cyclic flat of M ′ , the subset F is a g-cyclic flat with n g (F ) = n f (F ) + 1, and if S is a nonempty f -cyclic flat with n f (S) = k, then S is a g-cyclic flat with n g (S) = k + 1.
F k be an element of the monomial basis given in Corollary 3.3.3.By repeated application of Theorem 3.2.3combined with Lemma 3.3.9.(1), we have where M ′ is the loopless matroid obtained from M by a sequence of principal truncations, first by F k repeated a k times, then by F k−1 repeated a k−1 times, and so forth.Moreover, Lemma 3.3.9.(2) implies that f : M ′ և M is a matroid quotient with cyc(f ) = {∅, F 1 , . . ., F k } and n f (F j ) = j i=1 a i .We have thus shown that an element of the nested basis defines a loopless relative nested quotient by the cap product.
Conversely, let f : M ′ և M be a loopless relative nested quotient with cyc(f i=1 a i for all j = 1, . . ., k. Proposition 3.3.7 implies that the f -cyclic flats and their f -nullities of a relative nested quotient f : M ′ և M recovers the Higgs factorization of f by specifying the modular cuts to be Thus, since by assumption the f -cyclic flats form a chain, the descriptions of the modular cuts K i imply that the matroid M ′ is obtained by a sequence of principal truncations, first by F k repeated a k times, then by F k−1 repeated a k−1 times, and so forth, where a j = n f (F j ) − n f (F j−1 ) for j > 1, and a 1 = n f (F 1 ).
Moreover, the bijection given in the previous theorem respects linear independence.
Proposition 3.3.10.The elements The proof given below is a modification of the one given for nested matroids in [Ham17, Proposition 3.2].is an isomorphism for all 0 ≤ k ≤ d, or equivalently, the pairing We write (A • , ) for a Poincaré duality algebra with a chosen degree map .In Section 6, we will often drop the degree symbol when the context is clear.In particular, for ζ ∈ A 1 we will often write (1) The tensor product (2) A surjection A • ։ B • of Poincaré duality algebras of the same dimension is an isomorphism.
We will use the following construction to establish that Chow rings of matroids are Poincaré duality algebras.Proof.This is a straightforward check; see [MS05, Corollary I.2.3.] for example.
It will sometimes be convenient to identify elements of the ring A • / ann(f ) to elements of the principal ideal f ⊂ A • , with multiplication is by af • bf = (ab) • f .The construction in Proposition 4.1.3will arise in next subsection §4.2 with f being the Bergman class of a matroid.
The rest of this subsection on Poincaré duality algebras will not be needed until Section 6.
We describe another way the construction in Proposition 4.1.3arises in the context of Chow cohomology rings of fans.Let Σ be a d-dimensional smooth rational fan in N R for a lattice N , and let ρ ∈ Σ(1) be ray.Denote by u the image of u ∈ N R under the projection By definition of the Chow cohomology ring, one can check that there is a surjection A • (Σ) ։ A • (star(ρ, Σ)) determined by Since x ρ ′ | ρ ′ and ρ do not form a cone in Σ ⊂ ann A • (Σ) (x ρ ), thus we get an induced map In Section 6, we will use the following criterion for when the map π ρ is an isomorphism.4.2.Poincaré duality for matroids.We show that the Chow ring A • (M ) of a loopless matroid M is a Poincaré duality algebra with M as the degree map.While this was proved in [AHK18, Theorem 6.19], we give a non-inductive proof by using the simplicial presentation.
Our main theorem of the section is the following.
is an isomorphism of R-vector spaces.Thus, the Bergman classes of relative nested quotients form a basis of MW • (Σ M ).
Proof.The first statement follows from Corollary 4.2.2 and Hom(A c (M ), R) ≃ MW d−c (Σ M ) (Lemma 2.1.5).Theorem 3.3.8then implies the second statement.
Remark 4.2.5 (cf.(A)).Let Y R(M ) be the wonderful compactification of a realization of M .Since the pullback map Proof of Theorem 4.2.1.Recall that the isomorphism A • (Σ An ) ≃ MW n−• (Σ An ) makes the set of Minkowski weights into a graded ring, denoted MW • (Σ An ).Let ι * M be the pullback map of the inclusion ι M : Σ M ֒→ Σ An .A formal property of cap products, given below in Lemma 4.2.6, gives us a commuting diagram of surjections Here, we have identified the elements of MW • (Σ An )/ ann(∆ M ) with the elements of the principal ideal ∆ M ⊂ MW • (Σ An ), which is generated by Minkowski weights of the form ξ ∩ ∆ M where ξ ∈ A • (M ).Proposition 3.3.10then states that the bottom horizontal map preserves linear independence, and hence is injective as well.
Lemma 4.2.6.Let ι * be the pullback map of an inclusion of fans ι : Σ ′ ֒→ Σ where Σ is complete, and let Proof.We need show that the kernel we may consider ∆ as a Minkowski weight on Σ ′ , and thus by functoriality of the cap product (Remark 2.1.8),we have

LOG-CONCAVITY OF THE VOLUME POLYNOMIAL
A presentation of a graded Poincaré duality algebra A • can be encoded via the Macaulay inverse system into a single polynomial V P A , called the volume polynomial of A • [Eis95, §21.2].In geometric contexts, the volume polynomial takes on an additional meaning: • If A • is the ring of algebraic cycles modulo numerical equivalence on a smooth projective variety, then V P A measures degrees of ample divisors (see [ELMNP05]).• If A • is the Chow cohomology ring of a complete smooth fan Σ, then V P A measures the volumes of polytopes whose normal fans coarsen Σ (see [CLS11,§13]).
In both geometric contexts, the volume polynomial of A • is positive and log-concave on the ample cone when considered as a function A 1 → R.
In this section, we give a combinatorial formula for the volume polynomial V P ∇ M of the Chow ring A • ∇ (M ) of a loopless matroid M .We show that, as in the geometric cases, the volume polynomial V P ∇ M when regarded as a function A 1 ∇ (M ) → R is both positive and log-concave on a subcone K ∇ M of the ample cone K M generated by the nontrivial simplicial generators.While the results of [AHK18] imply that the volume polynomial of a matroid satisfies such properties, we give here an independent proof by establishing that V P ∇ M is a Lorentzian polynomial as defined in [BH20].In the next section §6, we build upon the results of this section to conclude that V P ∇ M is both positive and log-concave on the entire ample cone K M .5.1.Volume polynomials and Lorentzian polynomials.Here we review the notion of volume polynomials and how they generalize to Lorentzian polynomials.
One can encode a graded Poincaré duality algebra into a single polynomial called the volume polynomial as follows.
Definition 5.1.1.Let (A • , ) be a graded Poincaré duality algebra of dimension d that is generated in degree 1, with a chosen presentation A • = [x 1 , . . ., x s ]/I and a degree map : A d → .Then its volume polynomial V P A is a multivariate polynomial in [t 1 , . . ., t s ] defined by where we extend the degree map to A[t 1 , . . ., t s ] → [t 1 , . . ., t s ].
If (A • , ) is a Poincaré duality algebra with a chosen presentation A • = [x 1 , . . ., x s ]/I, then the defining ideal I can be recovered from the volume polynomial V P A as follows [CLS11, Lemma 13.4.7] In [BH20], the authors define Lorentzian polynomials as a generalization of volume polynomials in algebraic geometry and stable polynomials in optimization.Here we briefly summarize the relevant results.
Definition 5.1.2.A homogeneous polynomial f ∈ R[x 1 , . . ., x n ] of degree d is strictly Lorentzian if its support consists of all monomials in x of degree d, all of its coefficients are positive, and any of its Lorentzian polynomials are polynomials that can be obtained as a limit of strictly Lorentzian polynomials.
To characterize Lorentzian polynomials, we need a combinatorial notion which generalizes the exchange axiom for matroids: a collection of points J ⊂ Z n ≥0 is M-convex if for any α, β ∈ J and i ∈ [n] with α i > β i there exists j ∈ [n] such that α j < β j and α − e i + e j ∈ J.When the elements of J all have the same coordinate sum, this is equivalent to stating that the convex hull of J is a generalized permutohedra [Mur03, Theorem 4.15].
The following characterization can be seen as a linear algebraic abstraction of the proof of Teissier-Khovanskii inequalities via the Hodge index theorem for algebraic surfaces.The Lorentzian property is preserved under nonnegative linear change of variables: for any n × m matrix A with non-negative entries.
Applications to log-concavity phenomena in combinatorics arise from the following properties of Lorentzian polynomials.
Theorem 5.1.5.Let f ∈ R[x 1 , . . ., x n ] be a homogeneous polynomial with nonnegative coefficients.The Lorentzian property of f can be characterized via log-concavity properties as follows.
(1) [BH20, Theorem 5.3] A homogeneous polynomial f is Lorentzian if and only if f is strongly logconcave, in the sense that if g is any partial derivative of f of any order, then either g is identically zero or log g is concave on the positive orthant 2 , then f is Lorentzian if and only if (a 0 , a 1 , . . ., a d ) has no internal zeroes and is ultra log-concave, that is, We remark that (strictly) Lorentzian polynomials arise in classical algebraic geometry whenever one has a set of nef (ample) divisors on a smooth projective variety.
Remark 5.1.6.Let {D 1 , . . ., D s } be nef (ample) divisors on a smooth projective -variety X of dimension d, and A(X) its Chow ring.Let X : A d (X) → R be the degree map obtained as the pushforward map along the structure map X → Spec .Then 5.2.The Dragon Hall-Rado formula.We prepare our formula for the volume polynomial of A • ∇ (M ) by describing the combinatorial notions in [Pos09] that we generalize to arbitrary matroids.
We first recall Hall's marriage theorem and Rado's generalization; for proofs we point to The well-known Hall's marriage theorem [Hal35] states that a transversal of {A 0 , . . ., A n } exists if and only if j∈J A i ≥ |J| for all J ⊆ [n].The following theorem of Rado gives a matroid generalization of the condition given in Hall's theorem.
The dragon marriage theorem above follows easily from the original Hall's marriage theorem, and conversely, one can obtain Hall's marriage theorem from the dragon marriage theorem as follows: given A 0 , . . ., A n ⊆ E as in Hall's theorem, set E ′ = E ⊔ { * } and A ′ i := A 0 ⊔ { * } for each 0 ≤ i ≤ n and apply Postnikov's theorem to {A ′ 0 , . . ., A ′ n }.We now consider a variant of Rado's theorem in the same spirit.Proof.This follows from Theorem 5.2.1 and the observation that independent transversals I ⊆ E \ {e} of {A 1 , . . ., A m } are the same as independent transversals of {A 1 \ {e}, . . ., A m \ {e}}.
We can obtain Rado's theorem from the dragon Hall-Rado theorem by an argument analogous to how Hall's marriage theorem is obtained from dragon marriage theorem.In summary, the combinatorics introduced in this subsection thus far can be schematically laid out as follows with the indicated logical implications: Hall's marriage theorem Rado's theorem

Dragon marriage theorem Dragon Hall-Rado theorem
We are now ready to compute the intersection numbers of the variables h F in the simplicial presentation of the Chow ring of a matroid M .Theorem 5.2.4.Let A 1 , . . ., A d be a collection of subsets of E, and M a loopless matroid on E of rank d + 1.Let H A 1 , . . ., H A d be matroids as defined in Proposition 3.2.4.Then

Thus, we have
Proof.For the first assertion, we begin by making two easy observations.
Then, for , so that M j∈J H A j already has a loop.For the ⇐= direction, we induct on d.The base case d = 1 is easily verified.Now, we claim that if {A 1 , . . ., A d } satisfies the dragon Hall-Rado condition for M , then so does {A 1 , . . ., A d−1 } for M := M ∧ H A d .For the sake of contradiction, suppose {A 1 , . . ., A d−1 } fails DHR( M ).That is, without loss of generality suppose that rk For the second assertion, we first note that is the unique loopless matroid of rank 1 on E, which defines the Bergman class ∆ U 1,E by ∆ U 1,E (0) = 1 where 0 is the zero-dimensional cone of Σ An , we have that satisfies DHR(M ) and equals 0 otherwise.We obtain as an immediate corollary the promised generalization of [Pos09,Corollary 9.4].Recall that L ≥2 M denotes the flats of M of rank at least two.
Corollary 5.2.5.Let M be a loopless matroid on E of rank d + 1.The volume polynomial where the sum is over ordered collections of nonempty flats F 1 , . . ., F d of M satisfying DHR(M ).Alternatively, we have where the sum is over size d multisets {F d 1 1 , . . ., F d k k } of flats of M satisfying DHR(M ).One recovers the following central result of [Pos09] by setting M = U |E|,E .
The volume polynomial V P M of the more classical presentation A • (M ) of the Chow ring of a matroid M by generators {x F | F ∈ L M \ {∅, E}} was computed in [Eur20].5.3.Volume polynomial of a matroid is Lorentzian.Motivated by Remark 5.1.6and the fact that the simplicial generators are combinatorially nef, we prove here that the volume polynomial V P ∇ M of the simplicial presentation A • ∇ (M ) is Lorentzian.
Theorem 5.3.1.The volume polynomial M ] of a loopless matroid M is Lorentzian.As an immediate corollary, by applying Theorem 5.1.5 to Theorem 5.3.1 we obtain: , is strongly logconcave in the positive orthant R L ≥2 M >0 .In other words, as a function A 1 (M ) → R, the polynomial V P ∇ M is strongly log-concave in the interior of the cone K ∇ M generated by the nontrivial simplicial generators.We will show that the volume polynomial V P ∇ M of a loopless matroid M satisfies the two conditions listed in Theorem 5.1.3.First, we see that the dragon Hall-Rado condition description of the support of V P ∇ M implies that V P ∇ M has M-convex support.
Proof.First note that{F 1 , . . ., F d } and {G 1 , . . ., G d } satisfy the Dragon-Hall-Rado condition.In this argument, we borrow standard language from (poly)matroid theory [HH02].Let us call a multiset of flats {A 1 , . . .A k } dependent if rk M ( k j=1 A j ) ≤ k, and independent otherwise.We will also call a minimally dependent set of flats a circuit.We claim that the multiset of flats {F 1 , . . ., F d , G d } contains a unique circuit X.The proposition will follow from this claim because we can take F m to be any flat in X which appears more times in {F 1 , . . ., F d } than it does in {G 1 , . . ., G d }, and {F 1 , . . ., F d , G d } \ {F m } will satisfy the Dragon-Hall-Rado condition.
Remark 5.3.4.Suppose M has a realization R(M ).By Remark 2.3.2.(2), the wonderful compactification Y R(M ) is embedded in the product F ∈L M \{∅} P(V * /L F ).Our simplicial generators are pullbacks of the hyperplane classes of the projective spaces P(V * /L F ) (see Remark 3.2.7).Thus, in this case, that the support of V P ∇ M is M-convex follows from the result of [CCRLMZ20] that the multidegree of an irreducible variety in a product of projective spaces has M-convex support.
Proof of Theorem 5.3.1.Let M be a loopless matroid of rank r = d + 1.There is nothing to prove if d = 1, so we assume d ≥ 2. The support of V P ∇ M is M-convex by the previous proposition.We now show that the quadrics obtained as the (d − 2)-th partial derivatives have the desired signature.Observe that for a flat G of rank ≥ 2, we have is a loopless matroid of rank 3.By Proposition 5.1.4,it suffices to check that V P ∇ M ′ is Lorentzian.For any loopless matroid M ′ of rank 3, the degree 1 part A 1 ∇ (M ) of its Chow ring has the simplicial basis {h and 0 otherwise, the Hessian matrix of the quadratic form V P ∇ M ′ is 2 times the matrix via the symmetric Gaussian elimination: Explicitly, let E i be the elementary matrix such that multiplying E i to a matrix A results in adding −1 times the first row of A to the i-th row of A, and let E T i be its transpose.Then, the claimed congruence is given by

THE HODGE THEORY OF MATROIDS IN DEGREES AT MOST 1
The reduced characteristic polynomial of M is defined as where µ(−, −) is the Möbius function of the lattice L M and µ i (M ) is the absolute value of the ith coefficient of χ M (t).The Heron-Rota-Welsh conjecture [Rot71; Her72; Wel76] stated that To resolve the conjecture, the authors of [AHK18] showed that the Chow ring of a matroid satisfies properties enjoyed by the cohomology ring of a smooth projective complex variety.Namely, these are the Poincaré duality property, the hard Lefschetz property, and the Hodge-Riemann relations, which together form the "Kähler package."In particular, the Hodge-Riemann relation in degree 1 implies the Heron-Rota-Welsh conjecture [AHK18, §9].
To prove that the Chow ring of a matroid satisfies the Hodge-Riemann relations, the authors of [AHK18] adapt a line of argument that originally appeared in McMullen's work on simple polytopes [McM93].Their method employs a double induction on the rank of the matroid and the size of an order filter on the matroid's lattice of flats: the outer induction on rank shows that the Hodge-Riemann relations hold for all ample classes if they hold for a single ample class, and the inner induction on the size of an order filter is then used to construct an ample class for which the Hodge-Riemann relations can be verified.
In this section, we independently establish the Hodge-Riemann relations in degree 1 using a similar argument.As we have established in the previous section that the volume polynomial V P ∇ M of a matroid M is strongly log-concave in the subcone K ∇ M of the ample cone K M , we are able to avoid working with generalized Bergman fans associated to order filters and the flipping operation which interpolates between them.Thus our proof involves only classical Bergman fans associated to matroids and takes the form of a single induction on rank alone.6.1.The Kähler package in degree 1 and log-concavity.We begin by discussing here the statements of the Kähler package, and how in degree 1 they relate to log-concavity.We then provide some generalities on the inductive paradigm for proving Kähler package for Chow cohomology rings of fans, similar to one in [AHK18] adapted from the earlier work [McM93].Definition 6.1.1.Let (A • , ) be a Poincaré duality -algebra of dimension d with degree map .For ℓ ∈ A 1 and 0 ≤ i ≤ ⌊ d 2 ⌋, we define L k ℓ to be the Lefschetz operator L i ℓ : A i → A d−i , a → ℓ d−2i a, and define Q i ℓ to be Hodge-Riemann symmetric bilinear form We define the set of degree i primitive classes of ℓ to be P i ℓ := {x ∈ A k : xℓ d−2i+1 = 0}.The Poincaré duality property (PD) of (A • , ) implies that the form Q i ℓ is non-degenerate if and only if HL i ℓ holds.The properties (PD), (HL), and (HR) together are called the Kähler package for a graded ring A • .We will write HL ≤i and HR ≤i to mean hard Lefschetz property and Hodge-Riemann in degrees at most i, respectively.The relation between log-concavity and the Kähler package in degree ≤ 1 was realized in various contexts; for a survey we point to [Huh18].We will only need the following, adapted from [BH20, Proposition 5.6].It also appeared in [AOGV18, §2.3], and is a consequence of the Cauchy interlacing theorem.Proposition 6.1.3.Let A • be a Poincaré duality R-algebra of dimension d with degree map , and K a convex cone in A 1 .Suppose (A • , , K ) satisfy HL 0 K and HR 0 K .Then the following are equivalent: (1) The volume function vol : A 1 → R, ℓ → ℓ d is log-concave on K , and (2) for any ℓ ∈ K , the symmetric form Q 1 ℓ has exactly one positive eigenvalue.In particular, if the volume polynomial V P A of A • = R[x 1 , . . ., x s ]/I is Lorentzian, then (A • , , K ) satisfies HR ≤1 K where K is the interior of Cone(x 1 , . . ., x s ), provided that A • satisfies HL ≤1 K .
We now turn to an inductive paradigm for establishing (HL) and (HR).We assume all Poincaré duality algebras to be over R. We begin by noting an easy linear algebraic observation also made in [AHK18, Proposition 7.16].Proposition 6.1.4.Let (A • , , K ) be a Poincaré duality algebra which satisfies HL i K for K in a convex cone in A 1 .Suppose that (A • , ) satisfies HR i ℓ for some ℓ ∈ K .Then A • satisfies HR i K .
Proof.Let ℓ ′ ∈ K , and let l(t) = tℓ + (1 − t)ℓ ′ for t ∈ [0, 1] be a line segment connecting ℓ and ℓ ′ .By convexity of K , we know that every point on l is in K .If the signature of the bilinear pairing Q i l(t) changes along l(t) starting at ℓ, then it must degenerate at some point l(t 0 ) for t 0 ∈ [0, 1], but this violates HL i K .We now note how properties (HL) and (HR) behave under tensor products and quotients by annihilators of an element.While these are adapted from [AHK18,§7] where they are phrased in terms of Chow cohomology rings of fans, because we restrict ourselves Kähler package up to degree 1, we can provide here easier and more direct proofs for general Poincaré duality algebras.Before giving the proof, we remark that if d A = 0 then (A ⊗ B) • ≃ B • (likewise if d B = 0) so that the statement in the proposition is trivially satisfied after suitable modifications.If d A = 1 (or d B = 1), so that HR 1 ℓ A (or HR 1 ℓ B ) is vacuous, our proof below implies that (A • ⊗ B • ) satisfies HR 1 ℓ A ⊗1+1⊗ℓ B if HR 0 ℓ A and HR 0 ℓ B are satisfied.
The symmetric matrix for Q 1 ℓ with respect to the above basis is given by One computes that det(M ) < 0, and hence M has signature (+, −).We conclude that Q 1 ℓ (a, b) is nondegenerate and has exactly one positive eigenvalue completing the proof.Proposition 6.1.6.Let (A • = R[x 1 , . . ., x s ]/I, ) be a Poincaré duality algebra of dimension d, and let ℓ ∈ A 1 be an effective divisor-that is, a non-negative linear combination of {x 1 , . . ., x s }.Denote by ℓ k the image of ℓ in A • / ann(x k ).For 0 ≤ i ≤ ⌊ d−1 2 ⌋, if (A • / ann(x k ), x k ) satisfies HR i ℓ k for every k = 1, . . ., s, then (A • , ) satisfies HL i ℓ .
Proof.Let ℓ = s k=1 c k x k with c k ∈ R ≥0 , and suppose ℓ d−2i f = 0 for some f ∈ A i .We will show that f = 0 necessarily.Let f i the image of f in A • / ann(x i ).As 0 = ℓ d−2i f , we have 0 = ℓ d−2i k f k , and because A • / ann(x k ) is a Poincaré duality algebra of dimension d − 1, we conclude that f k belongs to the primitive space P i ℓ k .Now, we note that for all k, k ≤ 0 where the last inequalities follow from HR i ℓ k .Moreover, as Q 1 ℓ k is negative-definite on P i ℓ k , we conclude each f k to be 0, that is, x k f = 0 for all k = 1, . . ., s.Since {x 1 , . . ., x s } generate A • , the Poincaré duality property of A • implies that if f = 0 then there exists a polynomial g(x) of degree d − i such that g(x)f = 0, and hence we conclude that f = 0.
6.2.Kähler package in degree at most one for matroids.We now specialize our discussion to the setting of matroids, and establish Kähler package in degree at most one for Chow rings of matroids.As a consequence, we recover the proof of Heron-Rota-Welsh conjecture as in [AHK18].
Theorem 6.2.1.The Chow ring of a matroid (A • (M ), M , K M ) satisfies HL ≤1 K M and HR ≤1 K M .We will prove the theorem by induction on the rank of the matroid.The key combinatorial observation that allows one to reduce the rank is the following, adapted from [AHK18, Proposition 3.5].It underlies the well-known Hopf algebraic structure for the lattice of flats of a matroid; see [KRS99; AA17] for a detailed discussion of Hopf algebraic structures for matroids.Lemma 6.2.2.Let M be a loopless matroid, and F a nonempty proper flat of M .Let ρ F be the ray corresponding to F in the Bergman fan Σ M of M .We have The remaining key part of the induction in the proof of Theorem 6.2.1, in light of Proposition 6.1.4,is to establish HR 1 ℓ for some divisor ℓ ∈ K M .In [AHK18] the authors employ the method of order filters and flips for this purpose; in our case, the Lorentzian property of the volume polynomial provides the desired key step.Lemma 6.2.3.Let M be a loopless matroid of rank r = d + 1 ≥ 2, and recall that K ∇ M is the interior of the cone generated by the nontrivial simplicial generators of A • ∇ (M ).(It is a subcone of K M .)For any ℓ ∈ K ∇ M , we have M ℓ d > 0. If further r = d + 1 ≥ 3, then the form Q 1 ℓ has exactly one positive eigenvalue.In particular, HR 1 K M is satisfied for M with rank 3.
Proof.The statement M ℓ d > 0 follows from our dragon Hall-Rado formula (Corollary 5.2.5).When the rank of M is at least 3, that the form Q 1 ℓ has exactly one positive eigenvalue follows from combining Theorem 5.3.1 and Proposition 6.1.3.Lastly, when rank of M equals 3, the bilinear form Q 1 ℓ is the Poincaré an atom in the lattice of flats of M .We introduce a presentation of the Chow ring by a new set of generators: Definition 3.2.1.Let M be a loopless matroid on E. The simplicial presentation A • ∇ (M ) of the Chow ring of M is the quotient of a polynomial ring R[h F | F ⊆ E nonempty flat of M ] by the kernel of the surjective map Theorem 2.1.6.[FS97, Theorem 3.1, Proposition 4.1.(b),Theorem 4.2] For Σ a smooth complete fan, the cap product with the fundamental class ∆ Σ δ
the set of bases of M , • rk M the rank function of M , or simply rk when the matroid in question is clear, • L M the lattice of flats of M , which we also use to denote the set of flats, • A(M ) the set of atoms of L M , which are flats of rank 1, • S(M ) the set of spanning sets of M , which are subsets of E that contain a basis of M , • cl M (S) the closure of a subset S ⊆ E, which is the smallest flat of M containing S, and • Q(M ) the base polytope of M , which is the polytope Conv(e B | B ∈ B(M )) ⊂ R E .Matroids define Minkowski weights on Σ An in the following way.Proposition 2.2.1.Let M be a loopless matroid M of rank r = d + 1.
Theorem 2.1.6.In other words, the Bergman class ∆ M corresponds to the Chow homology class of the closure Y R(M ) of YR(M) in the permutohedral variety X An .The variety Y R(M ) is called the wonderful compactification of the hyperplane arrangement complement YR(M) .Remark 2.3.2.The wonderful compactification Y R(M ) can be described in two equivalent ways [DCP95, §3.2].
is the pullback map of Chow cohomology rings of Σ M and Σ An induced by the inclusion of fans Σ M ֒→ Σ An .In particular, the pullback map is surjective.(5)A divisor class D ∈ A 1 (M ) is an combinatorially ample (nef) if and only if there exists an ample (nef) divisor class L on X An such that ι * M L = D. Combinatorially ample (nef) divisors are ample (nef) on the variety Y R(M ) .
Definition 3.1.7.The matroid intersection 8 of two matroids M and N on a common ground set E is a new matroid M ∧ N on E whose spanning sets S(M ∧ N ) are {S ∩ S ′ | S ∈ S(M ), S ′ ∈ S(N )}.The matroid M ∧ N is a matroid quotient of both M and N .Matroid intersection behaves well in relation to Minkowski weights in the following way.Recall that the isomorphism A • (Σ An ) ≃ MW n−• (Σ An ) of Theorem 2.1.6makes MW n−• (Σ An ) into a graded ring, with multiplication called the stable intersection ∩ st .The following proposition states that stable intersections of Bergman classes are Bergman classes of matroid intersections.Proposition 3.1.8.[Spe08, Proposition 4.4], [Ham17, Remark 2.31] Let M and N be two matroids on a common ground set E, and let ∆ M and ∆ N be their Bergman classes, which are Minkowski weights on Σ An .Then we have which is equal to the base polytope Q(H S ) of H S .Since every face of Q(H S ), except for the vertices, are base polytopes of loopless matroids, by Lemma 2.2.3 the support of |∆ H S | equals the support of the (n − 1)dimensional skeleton of the outer normal fan of ∇ S .
by the definition of f -cyclic flats.We now show that the nested basis of A • ∇ (M ) given in Corollary 3.3.3 is in bijection with the set of relative nested quotients of M .Theorem 3.3.8.Let M be a loopless matroid of rank r = d + 1.For each 0 ≤ c ≤ d, the cap product map A c ∇ (M ) → MW d−c (Σ M ), ξ → ξ ∩ ∆ M induces a bijection between the monomial basis for A c ∇ (M ) given in Corollary 3.3.3and the set of Bergman classes ∆ M ′ of loopless relative nested quotients M ′ և M with rk(M ′ ) = rk(M ) − c.
ζ d to mean ζ d .Two useful facts about Poincaré duality algebras follow.Both are straightforward to check.Proposition 4.1.2.Let (A • , A ) and (B • , B ) be Poincaré duality algebras of dimension d A and d B over a common field .
Proposition 4.1.3.If (A • , ) is a Poincaré duality algebra of dimension d, and f ∈ A • a homogeneous element of degree k.Then the -algebra A • / ann(f ), where ann(f ) = {a ∈ A • | af = 0} is a Poincaré duality algebra of dimension d − k with the induced degree map f defined by f (a + ann(f )) := af for a ∈ A d−k .
Corollary 4.2.2.The Chow ring A • (M ) is a graded Poincaré duality algebra of dimension rk(M ) − 1 with M as the degree map.Remark 4.2.3.Because Σ An is a smooth projective fan, there exists a purely combinatorial proof of the Poincaré duality for its Chow cohomology ring via the line shelling of the fan; see [McM93; FK10].While Bergman fans of matroids are also shellable [Bjo92], they are not complete, and the arguments of [McM93; FK10] do not readily modify to give Poincaré duality for Chow rings of matroids.Rephrasing Corollary 4.2.2 yields the following generalization of [Ham17, Corollary 3.13].Corollary 4.2.4.For each 0 ≤ c ≤ d, the cap product map Theorem 5.1.3.[BH20, Theorem 5.1] A homogeneous polynomial f ∈ R[x 1 , . . ., x n ] of degree d with nonnegative coefficients is Lorentzian if and only if the following two conditions are satisfied:(1) The support of f is M-convex, and (2) The Hessian matrix of∂ i 1 • • • ∂ i d−2 fhas at most one positive eigenvalue for any choice of (d − 2)nd order partial differentiation.
Hall's condition can be recovered from Rado's by setting M = U |E|,E and m = n.See [Oxl11, Theorem 11.2.2] for more information and a proof of Rado's theorem.The following variant of Hall's marriage theorem was investigated by Postnikov as a combinatorial interpretation of a formula for volumes of generalized permutohedra [Pos09, §5, §9].
Proposition 5.2.3 (Dragon Hall-Rado condition).Let M be a matroid on E, and {A 1 , . . ., A m } a collection of subsets of E. There is a transversal I ⊆ E \ {e} of {A 1 , . . ., A m } for every e ∈ E if and only ifrk M j∈J A j ≥ |J| + 1, ∀∅ J ⊆ {1, . . ., m}and when this condition is satisfied, we say that {A 1 , . . ., A m } satisfy the dragon Hall-Rado condition of M , or DHR(M ) for short.
[Oxl11, Exercise 7.3.10]for the elementary quotient f : M ։ M ∧ H S we have by Proposition 3.1.5that {T ⊆ E | n f (T ) = 1} = {T ⊆ E | cl M (T ) ⊇ S}.In particular, the rank of T ⊆ E decreases by 1 in the elementary quotient if T ⊇ S.

( 1 )
star(Σ M , ρ F ) ≃ Σ M |F × Σ M/F , and consequently, (2) an isomorphism of Poincaré duality algebrasA • (M )/ ann(x F ) ≃ (A(M |F ) ⊗ A(M/F )) • such that if ℓ ∈ K M then its image in A • (M )/ ann(x F ) is in (K M |F ⊗ 1) ⊕ (1 ⊗ K M/F ).Proof.A face of Σ M is in star(Σ M , F ) if and only if it corresponds to a flag of flats which contains F .Any such flag naturally factors as the concatenation of two flags, one with maximal element strictly contained in F , and the other with minimal element F .This geometrically corresponds to the factorization of fans in the statement (1).For the second statement (2), first note that M |F and M/F are loopless since F is a flat.Then, combine Proposition 4.1.2and Proposition 4.1.4with the easily verifiable fact that A• (Σ × Σ ′ ) ≃ (A(Σ) ⊗ A(Σ ′ ))• for rational fans Σ and Σ ′ .This proves the isomorphism A • (M )/ ann(x F ) ≃ (A(M |F ) ⊗ A(M/F )) • .Lastly, the statement about ℓ ∈ K M follows from the fact that restriction of submodular functions on lattices remain submodular under restriction to intervals in the lattice.
M be the smooth fan structure on the support |∆ M | inherited from Σ An .That is, Σ M is a subfan of Σ An whose cones are Cone(u F 1 , ..., u F k ) ⊂ N R , one for each chain of nonempty proper flats ∅ F 1 • • • F k E of M .Then the Bergman class ∆ M is the unique d-dimensional Minkowski weight on Σ M (up to scaling).With notations as in Proposition 2.2.1 above, we call the Minkowski weight ∆ M the Bergman class of M , and we call the fan Σ M the Bergman fan 6 of M .thatcoarsensΣMcorresponds to a choice of a building set on the lattice of flats LM [FS05; AK06].Here we will always take the smooth fan structure for ΣM as a subfan of ΣA n .We will need the following description of supports of Bergman classes in Section 3.Lemma 2.2.3.[MS15, Corollary 4.2.11]Let M be a loopless matroid, and Q(M ) its base polytope.The support |∆ M | of its Bergman class is equal to the union of cones σ in the outer normal fan of Q(M ) satisfying the following condition: The corresponding face Q(M )(σ) of σ is a base polytope of a loopless matroid.
Proposition 5.3.3.Let {F 1 , ..., F d } and {G 1 , ..., G d } be two multisets of flats of M such that botht F 1 • • • t F d and t G 1 • • • t G d are in the support of V P ∇ M .Without loss of generality, suppose G d is a flat which appears more times in {G 1 , . . ., G d } than it does in {F 1 , . . ., F d }.Then, there exists another flat F m which appears more times in {F 1 , . . ., F d } than it does in {G 1 Definition 6.1.2.Let (A • , ) be a Poincaré duality R-algebra of dimension d, and let ℓ ∈ A 1 .For0 ≤ i ≤ ⌊ d 2 ⌋, we say that (A • , ) satisfies • HL i ℓ if L iℓ induces an isomorphism between A i and A d−i , and • HR i ℓ if the symmetric form (−1) i Q i ℓ is positive-definite when restricted to P i ℓ .Moreover, for K a convex cone in A 1 , we say that (A • , , K ) satisfies the hard Lefschetz property (HL i K ), resp.the Hodge-Riemann relation (HR i K ), in degree i if A • satisfies HL i ℓ , resp.HR i ℓ , for all ℓ ∈ K .