On the rank of an A-hypergeometric D-module versus the normalized volume of A

ON THE RANK OF AN A-HYPERGEOMETRIC D-MODULE VERSUS THE NORMALIZED VOLUME OF A CHRISTINE BERKESCH AND MAR´ IA-CRUZ FERN ´ ANDEZ-FERN ´ ANDEZ ABSTRACT. The rank of an A-hypergeometric D-module MA(β), associated with a full rank (d× n)-matrix Aand a vector of parameters β∈Cd, is known to be the normalized volume of A, denoted vol(A), when βlies outside the exceptional arrangement E(A), an affine subspace arrangement of codimension at least two. If β∈ E(A)is simple, we prove that d−1is a tight upper bound for the ratio rank(MA(β))/vol(A)for any d≥3. We also prove that the set of parameters βsuch that this ratio is at least 2is an affine subspace arrangement of codimension at least 3. INTRODUCTION The systematic study of A-hypergeometric D-modules, also known as GKZ-systems, was initiated by Gelfand, Graev, Kapranov, and Zelevinski [GGZ87], [GKZ89]. These are systems of linear partial differential equations in several complex variables that generalize classical hypergeometric equations. They are determined by a matrix A= (a1···an) = (ai,j)with columns ak∈Zdand a parameter vector β∈Cd. Let x1, x2, . . . , xnbe coordinates on Cn, with corresponding partial derivatives ∂1, ∂2, . . . , ∂n, so that the Weyl algebra Don Cnis generated by x1, . . . , xn, ∂1, . . . , ∂n. Let IA. .=h∂u−∂v|u, v ∈Nn, Au =Avi ⊆ C[∂1, . . . , ∂n] denote the toric ideal of A. Denote by Ei. .=Pn j=1 ai,jxj∂jthe ith Euler operator of A. The A-hypergeometric D-module with parameter β∈Cdis the left D-module MA(β). .=D/D ·hIA, E1−β1, . . . , Ed−βdi. For any choice of Aand β, the module MA(β)is holonomic [GGZ87, Ado94]. When β∈Cdis generic, the dimension of the space of germs of holomorphic solutions of MA(β)at a nonsingular point, also known as its (holonomic) rank, is equal to the normalized volume vol(A)of the matrix A, see (1.1) [GKZ89, Ado94]. In general, this is only a lower bound; see [SST00] for the case when IAis homogeneous and [MMW05] for the general case. The set E(A). .={β∈Cd|rank(MA(β)) >vol(A)} is called the exceptional arrangement of A, which is an affine subspace arrangement of codimension at least two that is closely related to the local cohomology modules of the toric ring C[∂]/IA [MMW05]. A parameter β∈ E(A)is called a rank jumping parameter. There is a combinatorial formula to compute the rank of MA(β)in terms of the ranking lattices Eβ of Aat β[Ber11], with previous results in [CDD99] with d= 2 and in [Oku06] when d= 3 or β is simple (see also Section 2). Unfortunately, the presence of alternating signs in this formula do not yield a strong upper bound for the rank of MA(β); however, if βis simple, it quickly follows that the rank of MA(β)is at most (d−1)vol(A), see Corollary 2.2. We show that this bound is tight by constructing a sequence of examples for which the ratio rank(MA(β))/vol(A)tends to 2010 Mathematics Subject Classification. 13N10, 32C38, 33C70, 14M25. Key words and phrases. A–hypergeometric system, toric ring, D–module, holonomic rank. CB was partially supported by NSF Grant DMS 1661962. MCFF was partially supported by MTM2016-75024-P and FEDER. 1 arXiv:1907.08669v2 [math.AG] 3 Sep 2021 2 CHRISTINE BERKESCH AND MAR´ IA-CRUZ FERN ´ ANDEZ-FERN ´ ANDEZ d−1, see Theorem 3.1. In addition, we prove that the equality cannot hold for any example with simple parameter βand that our examples are minimal in certain sense, see Remark 3.4. Another interesting feature of these examples is that E(A)contains all the lattice points in the convex hull of the columns of Aand the origin. On the other hand, there are other known upper bounds for the holonomic rank of MA(β). In particular, rank(MA(β)) ≤(4d·vol(A)if IAis homogeneous [SST00], 4d+1 ·vol(A)otherwise [BFM18]. It was shown in [Fer13] that these upper bounds are qualitatively effective, i.e., there is some a > 1 such that for any d≥3, there is a (d×n)-matrix Adand a parameter βd∈Cdsuch that rank(MAd(βd)) ≥advol(Ad). However, the maximum possible value of d prank(MA(β))/vol(A)that has, up until now, appeared in the literature is 3 p7/5≈1.1187, see [Fer13, Example 2.6], which was first considered in [MW07]. The supremum of the value of d prank(MA(β))/vol(A)over the examples in the current note is 5 √4≈1.3195, i.e., d √d−1for d= 5. It is still an open problem to find the supremum of the set of values of d prank(MA(β))/vol(A)for variation among the set of full rank (d×n)-matrices Aand β∈Cd, for d≥3and n≥d+ 2. Acknowledgements. We are grateful to Laura Felicia Matusevich and Uli Walther for helpful discussions over the years on bounding the rank of an A-hypergeometric system. 1. LOWER BOUNDS FOR THE NORMALIZED VOLUME In this section, we recall the definition of normalized volume of an integer full rank matrix, see (1.1), and provide some lower bounds for it, see Lemma 1.2 and Corollary 1.5. These bounds will be used in the proof of Corollary 2.3. Fix a (d×n)-integer matrix A= (a1···an), where ai∈Zddenotes the ith column of A. With the convention that 0∈N, assume that ZA. .=Pn j=1 Zaj=Zdand that the affine semigroup NA. .=Pn j=1 Najis positive, meaning that NA∩(−NA) = {0}. We also assume for simplicity that all the columns of Aare distinct from each other and the origin. Identify Awith its set of columns, and for any subset Fof A, denote by ∆Fthe convex hull in Rd of the origin and F. We also identity Fwith its index set {j|aj∈F}. Given a lattice Λsuch that F⊆Λ⊆QF∩Zd, the normalized volume of Fin Λis the integer volΛ(F) = dim(RF)! ·volRF(∆F) [Zd∩QF: Λ],(1.1) where volRF(·)denotes Euclidean volume in RF. We write vol(A)for volZA(A) = volZd(A). By convention, Z∅ ={0}and vol{0}(∅) = 1. A subset Fof the columns of the matrix Ais a face of A, denoted FA, if R≥0Fis a face of the cone R≥0A. .=Pn j=1 R≥0ajand F=A∩RF. The codimension of a nonempty face Fof Ais codim(F). .=d−dim(RF), with the convention that codim(∅) = d. A-HYPERGEOMETRIC RANK VERSUS THE VOLUME OF A3 Lemma 1.1. If τis a proper subset of Awith ∆τ∩A=τ, then there exists a column aof A\τ such that ∆τ∪{a}∩A=τ∪{a}. Moreover, if ∆τis not full dimensional, then amay be chosen so that dim(∆τ∪{a}) = dim(∆τ)+1. Proof. If ∆τ⊆Rdis full dimemsional, then choose any column a∈A\τ. Since a /∈∆τ, the vector ais a vertex of ∆τ∪{a}, and the rest of the vertices of ∆τ∪{a}are vertices of ∆τ. In particular, if there exists a vector a0∈(∆τ∪{a}∩A)\(τ∪{a}), then a0is not a vertex of ∆τ∪{a} and ∆τ(∆τ∪{a0}(∆τ∪{a}. Thus, acan be replaced by a0. Also, notice that (∆τ∪{a0}∩A)\(τ∪{a0})((∆τ∪{a}∩A)\(τ∪{a}). We can thus repeat this process of replacement of auntil the equality ∆τ∪{a00 }∩A=τ∪ {a00} holds for some a00 in A\τ. On the other hand, if ∆τ⊆Rdis not full dimensional, let a∈A\τbe such that dim(∆τ∪{a}) = dim(∆τ) + 1. Such a choice of aexists because the rank of Ais d. Since ∆τis a facet of ∆τ∪{a} and ∆τ∩A=τ, no point in (∆τ∪{a}∩A)\τis in the affine span of τ. Thus, if there exists a vector a0∈(∆τ∪{a}∩A)\(τ∪ {a}), we can replace aby a0and repeat the process until (∆τ∪{a}∩A) = τ∪{a}, in a similar way as in the full dimensional case.  Lemma 1.2. If FAis a face of A, then vol(A)≥volZd∩QF(F) + n−|F|−codim(F),(1.2) where |F|is the cardinality of F. In particular, vol(A)≥n−d+ 1. Proof. Since FAis a face of A,∆F∩A=F. By Lemma 1.1, there is a set σof codim(F) linearly independent columns of A\Fsuch that ∆F∪σis full dimensional and ∆F∪σ∩A=F∪σ. The normalized volume in the lattice Zdof F∪σis at least volZd∩QF(F). To see this, denote e F=Zd∩∆Fand notice that RF∩Zd=Ze Fand ∆e F= ∆F. Thus, using (1.1) we obtain volZd(F∪σ) = volZd(e F∪σ)≥volZ( e F∪σ)(e F∪σ)≥volZ e F(e F) = volZd∩RF(F), where the first inequality follows from the contaiment Z(e F∪σ)⊆Zdand the second one follows from [Wal07, Lemma 3.13]. Again by Lemma 1.1, there is a column aof A\(F∪σ)such that no other column of A\(F∪σ) lies in ∆F∪σ∪{a}, the convex hull of the codim(F) + |F|+ 1 points of F∪σ∪{a}and the origin. In fact, n−(codim(F) + |F|+ 1) more columns of A\(F∪σ)can be iteratively found in this way. Notice that each time a new point is added to F∪σusing Lemma 1.1, the normalized volume of the convex hull of the new set is increased at least by one. This proves the first statement. The second statement follows from the first one by taking F=∅. Remark 1.3. Notice that for any face FA,n− |F| − codim(F)≥0. If equality holds, we say that Ais a pyramid over F. By [SW12, Lemma 3.5], Ais a pyramid over Fif and only if Zd=ZF⊕Lj /∈FZaj. Further, if Ais a pyramid over F, then equality holds in (1.2) because Zd∩QF=ZFand vol(A) = volZF(F), see [SW12, Lemma 3.5]. The converse is not true; a counterexample is provided in Remark 3.4. On the other hand, if equality holds in (1.2), then all the lattice points in ∆A\∆Fare columns of A. Indeed, if there is a lattice point a∈∆A\∆Fwhich is not a column of A, then a matrix A0obtained by adding to Athe column awould have n+ 1 columns and vol(A) = vol(A0), so 4 CHRISTINE BERKESCH AND MAR´ IA-CRUZ FERN ´ ANDEZ-FERN ´ ANDEZ inequality (1.2) applied to A0shows that the inequality corresponding to Acannot be an equality in this case. Denote the toric ring associated to Aby SA. .=C[∂]/IA∼ =C[NA]. Proposition 1.4. If vol(A) = n−d+ 1, then SAis normal. Proof. Let Hbe an affine hyperplane such that τ. .=H∩Asatisfies that ∆τis full dimensional and all the columns of Anot in τbelong to the open half space determined by Hnot containing the origin; such a hyperplane exists because NAis positive and 0is not a column of A. For any σ⊆τ of cardinality dsuch that ∆σis a full dimemsional simplex and ∆σ∩A=σ,volZd(σ) = 1 because otherwise, by adding a point of A\σusing Lemma 1.1 and taking the convex hull iteratively, the volume would increase by at least one in each step and the normalized volume of Awould be larger than n−d+ 1. Now, since volZd(σ)=1,σforms a basis in the lattice Zdand Nσ=Zd∩R≥0σ. Since R≥0Aequals the union of the cones R≥0σfor simplices σ⊆τwith τas above, it follows that R≥0A∩Zd=NA, and hence, SAis normal.  Corollary 1.5. If SAis not Cohen–Macaulay, then d≥2,n≥d+ 2, and vol(A)≥n−d+ 2. Proof. If either d= 1 or n−d= 1, then under our hypotheses on A,SAis Cohen–Macaulay. On the other hand, if vol(A)< n−d+2, then SAis normal by Lemma 1.2 and Proposition 1.4, which implies that SAis Cohen–Macaulay by [Hoc72, Theorem 1].  The inequality in Corollary 1.5 is sharp; for any d≥2and n≥d+ 2, there is a pointed matrix A as above with vol(A) = n−d+ 2 such that SAis not Cohen–Macaulay. To see this, notice first that for d= 2 and n=d+ 2 = 4, the matrix A=1 1 0 0 0 1 2 3  satisfies that vol(A) = 4 and SAis not Cohen–Macaulay. On the other hand, in order to produce examples with n≥5, it is enough to modify this example by adding the columns (0, k)tfor k= 4, . . . , n −1, and this operation keeps SAinvariant up to isomorphism. To construct more examples with the same value of n−dbut larger d, it is enough to consider a pyramid over the previous example. This alters SAby tensoring over Cwith a polynomial ring in a number of variables equal to the increment of d. 2. RANK VERSUS VOLUME IN THE SIMPLE CASE 2.1. Combinatorics of the rank. In this subsection, we recall some notations and results from [Ber11] and a formula for the rank of an A-hypergeometric system in a particular case, see (2.1), proved in [Oku06]. For a face FA, consider the union of the lattice translates Eβ F. .=Zd∩(β+CF)r(NA+ZF) = G b∈Bβ F (b+ZF), where Bβ F⊆Zdis a set of lattice translate representatives. As such, |Bβ F|is the number of translates of ZFappearing in Eβ F, which is by definition equal to the difference between [Zd∩QF:ZF] and the number of translates of ZFalong β+CFthat are contained in NA+ZF. A-HYPERGEOMETRIC RANK VERSUS THE VOLUME OF A5 Given the set J(β). .={(F, b)|FA, b ∈Bβ F}, the ranking lattices of Aat βare defined to be Eβ. .=[ (F,b)∈J (β) (b+ZF). Note that the ranking lattices of Aat βis precisely the union of those sets (b+ZF)contained in Zd\NAsuch that β∈(b+CF). This is closely related to the set of holes of the affine semigroup NA, namely the set (Zd∩R≥0A)\NA. Definition 2.1. A rank jumping parameter βis simple (for a face GA) if the set of maximal pairs (F, b)in J(β)with respect to inclusion on b+ZFall correspond to a unique face GA. The main result in [Ber11] states that the rank of MA(β)can be computed from the combinatorics of Eβand ∆A. An explicit formula for the rank is given when the rank jumping parameter βis simple for a face GA(see also [Oku06] for this particular case); in this case, rank(MA(β)) = vol(A) + |Bβ G|·(codim(G)−1) ·volZG(G).(2.1) 2.2. Some upper bounds for the rank. In this subsection, we use (2.1) to provide upper bounds for the rank of an A-hypergeometric system MA(β)when the parameter βis simple, see Corollaries 2.2 and 2.3. We also prove that rank(MA(β)) <2·vol(A)if βlies outside an affine subspace arrangement of codimension at least three, see Theorem 2.4. Corollary 2.2. If β∈Cdis simple for the face FA, then rank(MA(β)) ≤codim(F)·vol(A). In particular, if d≥3and β∈ E(A)is simple, then rank(MA(β)) ≤(d−1) ·vol(A). Proof. The first statement follows from (2.1) and the definition of normalized volume in (1.1). Indeed, |Bβ F|·volZF(F)≤[Zd∩QF:ZF]·volZF(F) = volQF∩Zd(F)≤vol(A).(2.2) We can assume without loss of generality that vol(A)≥2, since otherwise Ais a simplex and E(A) = ∅. For the second statement, notice first that if codim(F) = d, then volZF(F) = 1 = |Bβ F|and rank(MA(β)) = vol(A) + d−1≤(d−1) ·vol(A),(2.3) since d≥3and vol(A)≥2. Thus, we can assume that codim(F)≤(d−1) and the second upper bound follows from the first one.  We can improve the bound in Corollary 2.2 as follows. Corollary 2.3. If d≥3and β∈Cdis simple for the face FA, then rank(MA(β)) ≤codim(F)·vol(A)−(codim(F)−1)(n−|F|−codim(F)).(2.4) In particular, if β∈ E(A)is simple, then rank(MA(β)) vol(A)<(d−1).(2.5) 6 CHRISTINE BERKESCH AND MAR´ IA-CRUZ FERN ´ ANDEZ-FERN ´ ANDEZ Proof. From (2.1) and the first inequality in (2.2), rank(MA(β)) ≤vol(A) + volQF∩Zd(F)(codim(F)−1).(2.6) Now volQF∩Zd(F)in (2.6) can be bounded using (1.2) in order to obtain (2.4). For (2.5), notice first that if codim(F) = 1, then rank(MA(β)) = vol(A)by (2.1); otherwise, (codim(F)−1) ≥1. By [MMW05, Corollary 9.2], E(A) = ∅is equivalent to SAbeing Cohen– Macaulay. Thus, for the case when codim(F) = d, it is enough to use that the inequality in (2.3) is in fact strict, because vol(A)≥4by Corollary 1.5. For the remaining cases, the second part in the inequality (2.4) is bounded above by (d−1)vol(A)−(n−|F|−codim(F)). Thus, it is enough to see that n− |F| − codim(F)≥1. By way of contradiction, assume that n−|F|−codim(F) = 0 (i.e., Ais a pyramid over F), so that any β∈Cdcan be written uniquely as β=βF+βFwith βF∈CF,βF∈CFfor F:= A\Fand rank(MA(β)) = rank(MF(βF)),(2.7) see [SW12, Lemma 3.7]. It follows from (2.7) and Remark 1.3 that β∈ E(A)if and only if βF∈ E(F). Notice also that if FGA, then Eβ G=Eβ0 Gfor any β0∈β+CF. If βis simple for F, the generic vectors β0∈β+CFare also simple for Fand rank(MA(β0)) = rank(MA(β)). Thus, rank(MF(β0 F)) = rank(MF(βF)) >volZF(F). It follows that for generic γ∈CF,rank(MF(γ)) >volZF(F), which is a contradiction, as this should be equality by [Ado94].  Notice that the difference between (2.5) and the second statement of Corollary 2.2 is that (2.5) is a strict inequality. Theorem 2.4. The set E2(A). .={β∈Cd|rank(MA(β)) ≥2·vol(A)} is an affine subspace arrangement of codimension at least three in Cd. Proof. The exceptional arrangement E(A)is known to be a finite union of translates of linear subspaces CGfor faces GAof codimension at least two [MMW05, Corollary 9.4 and Porism 9.5]. Moreover, it is shown in [MMW05, Theorem 2.6] that rank of MA(β)is upper-semicontinuous as a function of βwith respect to the Zariski topology. Thus, on each irreducible component C of E(A)the rank of MA(β)is constant outside a Zariski closed subset of C, of codimension at least one in C(i.e., of codimension at least three in Cd). Moreover, this codimension three set is also an affine subspace arrangement; see the argument after Definition 4.7 in [Ber11]. It is thus enough to find, for any codimension two component C, a set of parameters β∈Csuch that rank(MA(β)) <2·vol(A)and whose Zariski closure is C. Indeed, if Chas codimension two, we have that C=b+CGfor some face GFof codimension two and some b∈Cd. Notice that for any proper face G0Anot containing G, the intersection C∩(Zd+CG0)is at most a countably and locally finite union of translates of the linear space CG∩CG0of codimension at least three. Since there are only finitely many such faces G0, the Zariski closure of the set C0:= C\[ G6=G0A (Zd+CG0) A-HYPERGEOMETRIC RANK VERSUS THE VOLUME OF A7 is C. Moreover, for G0as above and β∈C0, we have that Zd∩(β+CG0) = ∅, hence Eβ G0=∅. In particular, the only possible faces involved in J(β)are Gand the two facets containing G. Thus, [Ber11, Section 5.3 and Example 6.21] yield the inequality rank(MA(β)) ≤vol(A) + |Bβ G|(codim(G)−1)volZG(G), where equality holds if βis simple for G, as in Definition 2.1. From (2.4) applied to the codimension two face G, we obtain that rank(MA(β)) ≤2·vol(A)−(n−|G|−2) <2·vol(A)(2.8) where n− |G| − 2≥1holds by the same argument as in the proof of Corollary 2.3. Thus, rank(MA(β)) <2·vol(A)for β∈C0, where the Zariski closure of C0is C. 3. A SEQUENCE OF EXAMPLES IN THE SIMPLE CASE In this section, we prove that for any d≥3, the strict inequality (2.5) from Corollary 2.3 is sharp for simple parameters βas in Definition 2.1, see Theorem 3.1. Theorem 3.1. There is a sequence of full rank (d×(2d−1)) integer matrices {Ad,b}∞ b=2 for which there is a simple parameter β∈Cdindependent of bfor which lim b→∞ rank(MAd,b (β)) vol(Ad,b)=d−1. In fact, the set of simple parameters β∈Cdthat maximize the ratio rank(MAd,b (β))/vol(Ad,b)is a line through the origin. Consider the following (d×(2d−1))-matrix with d≥3: Ad,b = (a1a2···a2d−1). .=Id−1Id−10d−1 0t d−11t d−1b,(3.1) where b≥2is an integer, Id−1denotes the identity matrix of rank d−1,1d−1is the column vector consisting of d−1entries of 1, and 0d−1is the zero column vector of length d−1. Note that ZAd,b =Zd. We now compute the normalized volume of Ad,b in this lattice. To do this, for j∈Z, set h(j). .= (0,...,0, j)t. Lemma 3.2. The normalized volume of Ad,b in (3.1) is b+d−1. Proof. The polytope ∆Ad,b can be decomposed as the union of two polytopes in Rdthat intersect in a common facet. One of these polytopes is the convex hull of the origin, the first 2(d−1) columns of Ad,b, and the lattice point h(1). This is a prism with height 1and base equal to a unit (d−1)-simplex, so its normalized volume in Zdis d. The second polytope is the convex hull of h(1) and the last dcolumns of Ad,b, which is a d-simplex. This d-simplex is the lattice translation by h(1) of the d-simplex that is the convex hull of the origin, the first (d−1)-columns of Ad,b, and h(b−1); therefore, its normalized volume in Zdis b−1. Remark 3.3. The last column of Ad,b is b·ed, where edis the dth standard basis vector in Cd. The face Fb. .={a2d−1}  Ad,b has normalized volume 1in the lattice ZFband Zd∩CFb= b−1 G k=0 h(k)+ZFb 8 CHRISTINE BERKESCH AND MAR´ IA-CRUZ FERN ´ ANDEZ-FERN ´ ANDEZ consists of btranslated copies of ZFb. Remark 3.4. The normalized volume of Fb. .={a2d−1}in the lattice Zd∩QFbis b. In particular, equality holds in (1.2) for A=Ad,b and F=Fb. Proposition 3.5. The exceptional arrangement of Ad,b is a finite union of lines parallel to Ced. Proof. The first d−1columns of Ad,b and its last column are linearly independent, and their nonnegative hull is precisely the first orthant Rd ≥0. Thus, R≥0Ad,b =Rd ≥0and R≥0Ad,b ∩Zd=Nd. To determine the set of holes of NAd,b, given by Nd\NAd,b, which can be written as a finite union of lattice translates of NFb=Nbed, notice first that the affine semigroup S⊆Ndgenerated by the first 2(d−1) columns of Ad,b is normal, and their lattice span is Zd. Note also that, for Fb. .={a2d−1}, NAd,b ∩CFb=NFb=Nbed. In order to complete the description of Nd\NAd,b, denote by ∆bthe simplex given by the convex hull of the following points in NAd,b: 0, bad=b(e1+ed), bad+1 =b(e2+ed), . . . , ba2d−2=b(ed−1+ed), an=bed. Since NAd,b =S+Nbed, the set of holes of NAd,b is the union of the sets c+Nbed, where cruns through the lattice points: Zd∩∆b\ R≥0(e1+ed)∪R≥0(e2+ed)∪ ··· ∪ R≥0(ed−1+ed)∪ bed+ d−1 X k=1 R≥0ek!!. It now follows from [MMW05, Ber11] that the exceptional arrangement of Ad,b is E(Ad,b) = d−1 [ k=1 b−2 [ m=0 (mek+CFb). By the proof of Proposition 3.5, if b≥3, then all the lattice points in ∆Ad,b belong to E(Ad,b). Lemma 3.6. If Fb. .={a2d−1}, then the function β∈Cd7→ rank(MAd,b (β)) reaches its maximum exactly when β∈CFb, and this maximum value is (d−1)b+ 1. Proof. We first show that if β∈CFb, then the rank of MAd,b (β)is (d−1)b+ 1. In this case, the ranking lattices at βare Eβ= b−1 [ j=1 h(j)+ZFb. Thus, by (2.1), the rank jump at βis equal to rank(MAd,b (β)) −vol(Ad,b) = |Bβ Fb|·volZFb(Fb)·(codim(Fb)−1), where codim(Fb) = d−1,volZFb(Fb) = 1, and |Bβ Fb|=b−1by Remark 3.3, since ZFb⊆(NAd,b +ZFb)∩(β+CF)∩Zd. Thus, rank(MAd,b (β)) = volZd(Ad,b) + (b−1)(d−1), which gives the desired equality by Lemma 3.2. In order to prove that this is the maximum value of rank(MAd,b (β)), it is enough to observe that when βlies in a component of the form (mek+CFb)⊆ E(Ad,b)with m6= 0, then the computation A-HYPERGEOMETRIC RANK VERSUS THE VOLUME OF A9 of the rank jump is analogous to the previous case, but the number |Bβ Fb|will be smaller. 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