On irregular binomial D-modules

arXiv:1012.0618v2 [math.AG] 28 Dec 2010 ON IRREGULAR BINOMIAL D–MODULES MAR´ IA-CRUZ FERN ´ ANDEZ-FERN ´ ANDEZ AND FRANCISCO-JES ´ US CASTRO-JIM´ ENEZ Tuesday 18th June, 2013 ABSTRACT. We prove that a holonomic binomial D–module MA(I, β)is regular if and only if certain associated primes of Idetermined by the parameter vector β∈Cdare homogeneous. We further describe the slopes of MA(I, β)along a coordinate subspace in terms of the known slopes of some related hypergeometric D–modules that also depend on β. When the parameter βis generic, we also compute the dimension of the generic stalk of the irregularity of MA(I, β) along a coordinate hyperplane and provide some remarks about the construction of its Gevrey solutions. 1. INTRODUCTION Binomial D-modules have been introduced by A. Dickenstein, L.F. Matusevich and E. Miller in [DMM10]. These objects generalize both GKZ hypergeometric D-modules [GGZ87, GZK89] and (binomial) Horn systems, as treated in [DMM10] and [Sai02]. Here Dstands for the complex Weyl algebra of order n, where n≥0is an integer. Elements in Dare linear partial differential operators; such an operator Pcan be written as a finite sum P=X α,γ pαγxα∂γ where pαγ ∈C,α= (α1,...,αn), γ = (γ1,...,γn)∈Nnand xα=xα1 1···xαn n,∂γ= ∂γ1 1···∂γn n. The partial derivative ∂ ∂xiis just denoted by ∂i. Our input is a pair (A, β)where βis a vector in Cdand A= (aij)∈Zd×nis a matrix whose columns a1,...,anspan the Z-module Zd. We also assume that all ai6= 0 and that the cone generated by the columns in Rncontains no lines (one says in this case that this cone is pointed). The polynomial ring C[∂] := C[∂1,...,∂n]is a subring of the Weyl algebra D. The matrix A induces a Zd-grading on C[∂](also called the A-grading) by defining deg(∂i) = −ai. A binomial in C[∂]is a polynomial with at most two monomial terms. An ideal Iin C[∂]is said to be binomial is it is generated by binomials. We also say that the ideal Iis an A-graded ideal if it is generated by A-homogenous elements (equivalently if for every polynomial in Iall its A-graded components are also in I). The matrix Aalso induces a Zd-grading on the Weyl algebra D(also called the A-grading) by defining deg(∂i) = −aiand deg(xi) = ai. To the matrix Aone associates the toric ideal IA⊂C[∂]generated by the family of binomials ∂u−∂vwhere u, v ∈Nnand Au =Av. The ideal IAis a prime A-graded ideal. Partially supported by MTM2007-64509, MTM2010-19336 and FEDER, FQM333. MCFF supported by a grant from Iceland, Liechtenstein and Norway through the EEA Financial Mechanism. Supported and coordinated by Universidad Complutense de Madrid. 1 2 MAR´ IA-CRUZ FERN´ ANDEZ-FERN´ ANDEZ AND FRANCISCO-JES´ US CASTRO-JIM´ ENEZ Recall that to the pair (A, β)one can associate the GKZ hypergeometric ideal HA(β) = DIA+D(E1−β1,...,Ed−βd) where Ei=Pn j=1 aijxj∂jis the ith Euler operator associated with A. The corresponding GKZ hypergeometric D–module is nothing but the quotient (left) D–module MA(β) := D HA(β), [GGZ87], [GZK89]. Following [DMM10], for any A–graded binomial ideal I⊂C[∂]we denote by HA(I, β)the A-graded left ideal in Ddefined by HA(I, β) = DI +D(E1−β1,...,Ed−βd). The binomial D–module associated with the triple (A, β, I)is, by definition, the quotient MA(I, β) := D HA(I,β). Notice that the ideal HA(IA, β)is nothing but the GKZ hypergeometric ideal HA(β). In [DMM10] the authors have answered essential questions about binomial D–modules. The main treated questions are related to the holonomicity of the systems and to the dimension of their holomorphic solution space around a non singular point. In particular, in [DMM10, Theorem 6.3] they prove that the holonomicity of MA(I, β)is equivalent to regular holonomicity when Iis standard Z-graded (i.e., the row-span of Acontains the vector (1,...,1)). However, it turns out that the final sentence in [DMM10, Theorem 6.3], stating that the regular holonomicity of MA(I, β)for a given parameter βimplies standard homogeneity of the ideal I, is true for binomial Horn systems but it is not for general binomial D–modules. This is shown by Examples 3.10 and 3.11. These two Examples are different in nature. More precisely, the system MA(I, β)considered in Example 3.10 is regular holonomic for parameters βoutside a certain line in the affine complex plane and irregular otherwise, while the system considered in Example 3.11 is regular holonomic for all parameters despite the fact that the binomial ideal Iis not homogeneous with respect to the standard Z–grading. This is a surprising phenomenon since it is not allowed neither for GKZ hypergeometric systems nor for binomial Horn systems. We further provide, in Theorem 3.7, a characterization of the regular holonomicity of a system MA(I, β)that improves the above mentioned result of [DMM10, Th. 6.3]. A central question in the study of the irregularity of a holonomic D-module Mis the computation of its slopes along smooth hypersurfaces (see [Meb90] and [LM99]). On the other hand, the Gevrey solutions of Malong smooth hypersurfaces are closely related with the irregularity and the slopes of M. More precisely, the classes of these Gevrey series solutions of Mmodulo convergent series define the 0-th cohomology group of the irregularity of M[Meb90, D´efinition 6.3.1]. In Section 4 we describe the L–characteristic variety and the slopes of MA(I, β)along coordinate subspaces in terms of the same objects of the binomial D–modules associated with some of the toral primes of the ideal Idetermined by β(see Theorem 4.3). The binomial D–module associated with a toral prime is essentially a GKZ hypergeometric system and the L–characteristic variety and the slopes along coordinate subspaces of such a system are completely described in [SW08] in a combinatorial way (see also [CT03] and [Har03, Har04] for the cases d= 1 and n=d+ 1). Gevrey solutionsof hypergeometric systems alongcoordinate subspaces are described in [Fer10] (see also [FC11], [FC08]). In Section 5 we compute the dimension of the generic stalk of the ON IRREGULAR BINOMIAL D–MODULES 3 irregularity of binomial D-modules when the parameter is generic (see Theorem 5.1). We finally give a procedure to compute Gevrey solutions of MA(I, β)by using known results in the hypergeometric case ([GZK89], [SST00] and [Fer10]). We are grateful to Ezra Miller for his useful suggestions and comments. 2. PRELIMINARIES ON EULER–KOSZUL HOMOLOGY,BINOMIAL PRIMARY DECOMPOSITION AND TORAL AND ANDEAN MODULES We review here some definitions, notations and results of [ES96], [MMW05], [DMM10] and [DMM210] that will be used in the sequel. We will denote R=C[∂]. Recall that the A–grading on the ring Ris defined by deg(∂j) = −aj where ajis the jth-column of A. This A–grading on Rcan be extended to the ring Dby setting deg(xj) = aj. Definition 2.1. [DMM10, Definition 2.4] Let V=⊕α∈ZdVαbe an A-graded R-module. The set of true degrees of Vis tdeg(V) = {α∈Zd:Vα6= 0} The set of quasidegrees of Vis the Zariski closure in Cdof tdeg(V). Euler-Koszul complex K•(E−β;V)associated with an A-graded R–module V. For any A–graded left D–module N=⊕α∈ZdNαwe denote degi(y) = αiif y∈Nα. The map Ei−βi:Nα→Nαdefined by (Ei−βi)(y) = (Ei−βi−αi)ycan be extended (by C–linearity) to a morphism of left D–modules Ei−βi:N→N. We denote by E−βthe sequence of commuting endomorphisms E1−β1,...,Ed−βd. This allows us to consider the Koszul complex K•(E−β, N)which is concentrated in homological degrees dto 0. Definition 2.2. [MMW05, Definition 4.2] For any β∈Cdand any A-graded R–module V, the Euler-Koszul complex K•(E−β, V )is the Koszul complex K•(E−β, D ⊗RV). The ith Euler-Kozsul homology of V, denoted by Hi(E−β, V ), is the homology Hi(K•(E−β, V )). Remark 2.3. Recall that we have the A–graded isomorphism Hi(E−β, V )(α)≃ Hi(E−β+ α, V )(α)for all α∈Zd[MMW05]. Here V(α)is nothing but Vwith the shifted A–grading V(α)γ=Vα+γfor all γ∈Zd. Binomial primary decomposition for binomial ideals. We recall from [ES96] that for any sublattice Λ⊂Znand any partial character ρ: Λ →C∗, the corresponding associated binomial ideal is Iρ=h∂u+−ρ(u)∂u−|u=u+−u−∈Λi where u+and u−are in Nnand they have disjoint supports. The ideal Iρis prime if and only if Λis a saturated sublattice of Zn(i.e. Λ = QΛ∩Zn). We know from [ES96, Corollary 2.6] that any binomial prime ideal in Rhas the form Iρ,J := Iρ+mJ(where mJ=h∂j|j6∈ Ji) for some partial character ρwhose domain is a saturated sublattice of ZJand some J⊂ {1,...,n}. For any J⊂ {1,...,n}we denote by ∂Jthe monomial Qj∈J∂j. Theorem 2.4. [DMM210, Theorem 3.2] Fix a binomial ideal Iin R. Each associated binomial prime Iρ,J has an explicitly defined monomial ideal Uρ,J such that I=\ Iρ,J ∈Ass(I)Cρ,J 4 MAR´ IA-CRUZ FERN´ ANDEZ-FERN´ ANDEZ AND FRANCISCO-JES´ US CASTRO-JIM´ ENEZ for Cρ,J = ((I+Iρ) : ∂∞ J) + Uρ,J , is a primary decomposition of Ias an intersection of A– graded primary binomial ideals. Toral and Andean modules. In [DMM210, Definition 4.3] a finitely generated A-graded R–module V=⊕Vαis said to be toral if its Hilbert function HV(defined by HV(α) = dimCVαfor α∈Zd) is bounded above. With the notations above, a R–module of type R/Iρ,J is toral if and only if its Krull dimension equals the rank of the matrix AJ(see [DMM10, Lemma 3.4]). Here AJis the submatrix of A whose columns are indexed by J. In this case the module R/Cρ,J is toral and we say that the ideal Iρ,J is a toral prime and Cρ,J is a toral primary component. If dim(R/Iρ,J )6= rank (AJ)then the module R/Cρ,J is said to be Andean, the ideal Iρ,J is an Andean prime and Cρ,J is an Andean primary component. An A–graded R–module Vis said to be natively toral if there exist a binomial toral prime ideal Iρ,J and an element α∈Zdsuch that V(α)is isomorphic to R/Iρ,J as A–graded modules (see [DMM10, Definition 4.1]). Proposition 2.5. [DMM10, Proposition 4.2] An A–graded R–module Vis toral if and only if it has a filtration 0 = V0⊂V1⊂ ··· ⊂ Vℓ−1⊂Vℓ=V whose successive quotients Vk/Vk−1are all natively toral. Such a filtration on Vis called a toral filtration. Following [DMM10, Definition 5.1] an A-graded R-module Vis said to be natively Andean if there is an α∈Zdand an Andean quotient ring R/Iρ,J over which V(α)is torsion-free of rank 1 and admits a ZJ/Λ-grading that refines the A-grading via ZJ/Λ→Zd=ZA, where ρis defined on Λ⊂ZJ. Moreover, if Vhas a finite filtration 0 = V0⊂V1⊂ ··· ⊂ Vℓ−1⊂Vℓ=V whose successive quotients Vk/Vk−1are all natively Andean, then Vis Andean (see [DMM10, Section 5]). In [DMM210, Example 4.6] it is proven that the quotient R/Cρ,J is Andean for any Andean primary component Cρ,J of any A-graded binomial ideal. We finish this section with the definition and a result about the so-called Andean arrangement associated with an A-graded binomial ideal Iin R. Let us fix an irredundant primary decomposition I=\ Iρ,J ∈Ass(I)Cρ,J as in Theorem 2.4. Definition 2.6. [DMM10, Definition 6.1] The Andean arrangement ZAndean(I)is the union of the quasidegree sets qdeg(R/Cρ,J )for the Andean primary components Cρ,J of I. From [DMM10, Lemma 6.2] the Andean arrangement ZAndean(I)is a union of finitely many integer translates of the subspaces CAJ⊂Cnfor which there is an Andean associated prime Iρ,J . From [DMM10, Theorem 6.3] we have that the binomial D–module MA(I, β)is holonomic if and only if −β /∈ ZAndean(I). ON IRREGULAR BINOMIAL D–MODULES 5 3. CHARACTERIZING REGULAR HOLONOMIC BINOMIAL D–MODULES Let Ibe an A–graded binomial ideal and fix a binomial primary decomposition I=∩ρ,J Cρ,J where Cρ,J is a Iρ,J–primary binomial ideal. Let us consider the ideal Iβ:= \ −β∈qdeg(R/Cρ,J ) Cρ,J i.e., the intersection of all the primary components Cρ,J of Isuch that −βlies in the quasidegrees set of the module R/Cρ,J . Remark 3.1. Notice that if −β /∈ ZAndean(I)then R/Iβis contained in the toral direct sum M −β∈qdeg(R/Cρ,J ) R/Cρ,J and so it is a toral module. The following result generalizes [DMM10, Proposition 6.4]. Proposition 3.2. If −β /∈ ZAndean(I)then the natural surjection R/I ։R/Iβinduces a isomorphism in Euler–Koszul homology Hi(E−β, R/I)≃ Hi(E−β, R/Iβ) for all i. In particular, MA(I, β)≃MA(Iβ, β). Proof. By [DMM10, Proposition 6.4] we have that Hi(E−β, R/I)≃ Hi(E−β, R/Itoral) for all i, where Itoral denotes the intersection of all the toral primary components of I. Thus, we can assume without loss of generality that all the primary components of Iare toral. The rest of the proof is now analogous to the proof of [DMM10, Proposition 6.4] if we substitute the ideals Itoral and IAndean there by the ideals Iβand Iβrespectively, where Iβ=\ −β /∈qdeg(R/Cρ,J ) Cρ,J, and the Andean direct sum LIρ,J Andean R/Cρ,J there by the toral direct sum M −β /∈qdeg(R/Cρ,J ) R/Cρ,J Finally, we can use Lemma 4.3 and Theorem 4.5 in [DMM10] in a similar way as [DMM10, Lemma 5.4] is used in the proof of Proposition 6.4 of [DMM10].  The following Lemma gives a description of the quasidegrees set of a toral module of type R/Cρ,J . E. Miller has pointed out that this result follows from Proposition 2.13 and Theorem 2.15 in [DMM210]. We will include here a slightly different proof of this Lemma. Lemma 3.3. For any Iρ,J –primary toral ideal Cρ,J the quasidegrees set of M=R/Cρ,J equals the union of at most µρ,J Zd–graded translates of CAJ, where µρ,J is the multiplicity of Iρ,J in Cρ,J . More precisely, for any toral filtration 0 = M0⊆M1⊆ ··· ⊆ Mwe have that 6 MAR´ IA-CRUZ FERN´ ANDEZ-FERN´ ANDEZ AND FRANCISCO-JES´ US CASTRO-JIM´ ENEZ the quasidegrees set of Mis the union of the quasidegrees set of all the successive quotients Mi/Mi−1that are isomorphic to Zd–graded translates of R/Iρ,J. Proof. Since Mis toral we have by [DMM10, Lemma 4.7] that dim(qdeg(M)) = dim M= rank AJ. Since Cρ,J is primary, any zero-divisor of Mis nilpotent. For all j∈Jwe have that ∂m j/∈Cρ,J ⊆Iρ+mJand so∂jis nota zero-divisor inMfor all j∈J. Thus, the true degrees set of Mverifies tdeg(M) = tdeg(M)−NAJ. This and the fact that dim(qdeg(M)) = rank AJ imply that there exists α1,...,αr∈Zdsuch that tdeg(M) = ∪r i=1(αi−NAJ)and (3.1) qdeg(M) = r [ i=1 (αi+CAJ) Consider now a toral filtration 0 = M0⊆M1⊆ ··· ⊆ M. We know that there are exactly µρ,J different values of isuch that Mi/Mi−1≃R/Iρ,J(γi)for some γi∈Zd. For the other successive quotients Ml/Ml−1≃R/Iρl,Jl(γl)we have that Iρl,Jlis a toral prime which properly contains Iρ,J . In particular, we have that rank AJl= dim R/Iρl,Jl<dim R/Iρ,J = rank AJ. Since qdeg(R/Iρl,Jl) = CAJlhas dimension rank AJl<rank AJand qdeg(M) = Siqdeg(Mi/Mi−1)we have by (3.1) that the quasidegrees set of any Mi/Mi−1is contained in the quasidegrees set of some Mj/Mj−1≃R/Iρ,J (γj). In particular r≤µρ,J and each affine subspace (αi+CAJ)in (3.1) is the quasidegrees set of some Mj/Mj−1≃R/Iρ,J (γj). Remark 3.4. Notice that HA(Iρ,J , β) = DHAJ(Iρ, β) + D(∂j:j /∈J). In addition, if Iρ,J is toral then the DJ–module MAJ(Iρ, β)is isomorphic to the hypergeometric system MAJ(β)via an A–graded isomorphism of DJ–modules induced by rescaling the variables xj,j∈J, using the character ρ. Thus we can apply most of the well-knows results for hypergeometric systems to MA(Iρ,J , β)(with Iρ,J a toral prime) in an appropriated form. Lemma 3.5. If Iρ,J is toral and −β∈qdeg(R/Iρ,J )the following conditions are equivalent: i) Hi(E−β, R/Iρ,J)is regular holonomic for all i. ii) H0(E−β, R/Iρ,J)is regular holonomic. iii) Iρ,J is homogeneous (equivalently AJis homogeneous). Proof. i)⇒ii)is obvious, ii)⇒iii)follows straightforward from [SW08, Corollary 3.16] and iii)⇒i)is a particular case of the last statement in [DMM10, Theorem 4.5] and it also follows from [Hot98, Ch. II, 6.2, Thm.].  Remark 3.6. Recall from [DMM10, Theorem 4.5] that for any toral module Vwe have that −β /∈qdegVif and only if H0(E−β, V ) = 0 if and only if Hi(E−β, V ) = 0 for all i. In particular, since the D–module 0is regular holonomic it follows that conditions i) and ii) in Lemma 3.5 are also equivalent without the condition −β∈qdeg(R/Iρ,J ). Theorem 3.7. Let I⊆Rbe an A-graded binomial ideal such that MA(I, β)is holonomic (equivalently, −β /∈ ZAndean(I)). The following conditions are equivalent: i) Hi(E−β, R/I)is regular holonomic for all i. ii) MA(I, β)is regular holonomic. iii) All the associated toral primes Iρ,J of Isuch that −β∈qdeg(R/Cρ,J)are homogeneous. ON IRREGULAR BINOMIAL D–MODULES 7 Proof. The implication i)⇒ii)is obvious. Let us prove ii)⇒iii). For any toral primary component Cρ,J of Iwe have I⊆Cρ,J and so there is a natural epimorphism MA(I, β)։ MA(Cρ,J, β). Since MA(I, β)is regular holonomic then MA(Cρ,J, β)is also regular holonomic. Take a toral filtration of M=R/Cρ,J ,0⊆M1⊆ ··· ⊆ Mr=M. We claim that (3.2) Hj(E−β, Mi/Mi−1)and H0(E−β, Mi−1)are regular holonomic for all i, j. Let us prove (3.2) by decreasing induction on i. For i=r, we have a surjection from the regular holonomic D–module H0(E−β, Mr) = MA(Cρ,J, β)to H0(E−β, Mr/Mr−1)and so it is regular holonomic too. By Remark 2.3, Lemma 3.5 and Remark 3.6 we have that the D-module Hj(E−β, Mr/Mr−1)is regular holonomic for all j. Since H1(E−β, Mr/Mr−1)−→ H0(E−β, Mr−1)−→ H0(E−β, Mr) is exact we have that H0(E−β, Mr−1)is regular holonomic. Assume that (3.2) holds for some i=k+ 1 ≤rand for all j. We consider the exact sequence 0−→ Mk−1−→ Mk−→ Mk/Mk−1−→ 0 and the following part of the long exact sequence of Euler-Koszul homology (3.3) ···H1(E−β, Mk/Mk−1)→ H0(E−β, Mk−1)→ H0(E−β, Mk)։H0(E−β, Mk/Mk−1). By induction hypothesis H0(E−β, Mk)is regular holonomic. This implies that H0(E− β, Mk/Mk−1)is regular holonomic by (3.3). Applying Remark 2.3, Lemma 3.5 and Remark 3.6 we have that Hj(E−β, Mk/Mk−1)is regular holonomic for all j. Thus, by (3.3) we have that H0(E−β, Mk−1)is regular holonomic too and we have finished the induction proof of (3.2). Assume that −β∈qdeg(R/Cρ,J ). By Lemma 3.3 there exists isuch that −βlies in the quasidegrees set of Mi/Mi−1≃R/Iρ,J (γi)and we also have by (3.2) that H0(E−β, Mi/Mi−1)≃ H0(E−β+γi, R/Iρ,J )(γi) is a nonzero regular holonomic D-module. Thus, by Lemma 3.5 we have that Iρ,J is homogeneous. Let us prove iii)⇒i). By Proposition 3.2 we just need to prove that MA(Iβ, β)is regular holonomic. We have that all the associated primes of Iβare toral and homogeneous. In particular M=R/Iβis a toral module and for any toral filtration of Mthe successive quotients Mi/Mi−1are isomorphic to some Zd–graded translate of a quotient R/Iρi,Jiwhere Iρi,Jiis toral and contains a minimal prime Iρ,J of Iβ. Such minimal prime is homogeneous by assumption and so AJis homogeneous. Since Ji⊆Jwe have that AJiand Iρi,Jiare homogeneous too. Now, we just point out that that the proof of the last statement in [DMM10, Theorem 4.5] still holds for V=Mif we don’t require Ato be homogenous but all the primes occurring in a toral filtration of Mto be homogeneous.  Remark 3.8. Theorem 3.7 shows in particularthat the property ofa binomialD-moduleMA(I, β) of being regular (holonomic) can fail to be constant when −βruns outside the Andean arrangement. This phenomenon is forbidden to binomial Horn systems MA(I(B), β)(see [DMM10, 8 MAR´ IA-CRUZ FERN´ ANDEZ-FERN´ ANDEZ AND FRANCISCO-JES´ US CASTRO-JIM´ ENEZ Definition 1.5]) since the inclusion I(B)⊆IAinduces a surjective morphism H0(E−β, I(B)) ։MA(β) and then regular holonomicity of H0(E−β, R/I(B)) implies regular holonomicity of MA(β), which is equivalent to the standard homogeneity of IAby [Hot98, SST00, SW08]. Definition 3.9. The non-regular arrangement of I(denoted by Znon−regular(I)) is the union of the Andean arrangement of Iand theunion of quasidegreessets of the quotients of Rby primary components Cρ,J of Isuch that Iρ,J is not homogeneous with respect to the standard grading. So, we have Znon−regular(I) = ZAndean(I)∪ [ Iρ,J non homogeneous qdeg(R/Cρ,J) . Example 3.10. Consider the ideal I=h∂2 1∂2−∂2 2, ∂2∂3, ∂2∂4, ∂2 1∂3−∂2 3∂4, ∂2 1∂4−∂3∂2 4i. It is A-graded for the matrix A=1220 1202 but Iis not standard Z-graded. We have the prime decomposition I=I1∩I2∩I3where I1=h∂2, ∂3, ∂4i,I2=h∂2 1−∂2, ∂3, ∂4iand I3=h∂2, ∂2 1−∂3∂4iare toral primes of I. In particular ZAndean(I) = ∅) and by the proof of [DMM10, Proposition 6.6] we have that Zprimary(I) = {0}(see [DMM10, Definition 6.5] for the definition of the primary arrangement Zprimary(I)). Using [DMM10, Theorem 6.8] we have that MA(I, β)is isomorphic to the direct sum of MA(Ij, β)for j= 1,2,3if β6= 0. Moreover, qdeg(R/Ij) = C1 1for j= 1,2and qdeg(R/I3) = C2. Thus, for generic parameters (more precisely for β∈C2\C1 1) we have that MA(I, β)is isomorphic to MA(I3, β)that is a regular holonomic by Lemma 3.5. On the other hand, there is a surjective morphism from MA(I, β)to MA(I2, β)and if β∈C1 1 we have that MA(I2, β)is an irregular D-module because s= 2 is a slope along x2= 0. Thus we conclude that MA(I, β)is regular holonomic if β∈C2\C1 1and it is an irregular holonomic D-module when β∈C1 1. In particular, Znon−regular(I) = C1 1⊂C2. It can also be checked that the singular locus of MA(I, β)is {x1x2x3x4(x2 1−4x3x4) = 0}when β∈C1 1 and {x3x4(x2 1−4x3x4) = 0}otherwise. Example 3.11. The primary binomial ideal I=h∂1−∂2, ∂4 3, ∂3 4, ∂3 3−∂2 4iis A–graded with respect to the matrix A= (1 1 2 3). Note that Iis not homogeneous with respect to the standard Z-grading. However, its radical ideal √I=h∂1−∂2, ∂3, ∂4iis homogeneous. Thus, by Theorem 3.7 we have that MA(I, β)is regular holonomic. 4. L–CHARACTERISTIC VARIETY AND SLOPES OF BINOMIAL D–MODULES Let Lbe the filtration on Ddefined by a weight vector (u, v)∈R2nwith ui+vi=c > 0for some constant c > 0. This includes in particular the intermediate filtrations pF +qV between the filtration Fby the order of the linear differential operators and the Kashiwara-Malgrange filtration Valong ON IRREGULAR BINOMIAL D–MODULES 9 a coordinate subspace. The filtrations pF +qV are the ones considered when studying the algebraic slopes of a coherent D–module along a coordinate subspace [LM99]. We will consider the L–characteristic variety ChL(N)of a finitely generated D–module Non Cndefined as the support of grLNin T∗Cn(see e.g. [Lau87], [SW08, Definition 3.1]). We recall that in fact for L=pF +qV this is a global algebraic version of Laurent’s microcharacteristic variety of type s=p/q + 1 [Lau87, §3.2] (see also [SW08, Remark 3.3]). The L-characteristic variety and the slopes of a hypergeometric D-module MA(β)are controlled by the so-called (A, L)–umbrella [SW08]. Let us recall its definition in the special case when vi>0for all i. We denote by ∆L Athe convex hull of {0, aL 1,...,aL n}where aL j=1 vjaj. The (A, L)-umbrella is the set ΦL Aof faces of ∆L Awhich do not contain 0. The empty face is in ΦL A. One identifies τ∈ΦL Awith {j|aL j∈τ}, or with {aj|aL j∈τ}, or with the corresponding submatrix Aτof A. By [SW08, Corollary 4.17] the L-characteristic variety of a hypergeometric D–module MA(β) is (4.1) ChL(MA(β)) = [ τ∈ΦL A Cτ A where Cτ Ais the Zariski closure in T∗Cnof the conormal space to the orbit Oτ A⊂T∗ 0Cn=Cn corresponding to the face τ. In particular ChL(MA(β)) is independent of β. By definition we have the equality Oτ A:= (C∗)d·1τ Awhere 1τ A∈Nnis defined by (1τ A)j= 1 if j∈τand (1τ A)j= 0 otherwise. The action of the torus is given with respect to the matrix A. If the filtration given by Lequals the F-filtration (i.e. the order filtration) then this description of the F–characteristic variety coincides with a result of [Ado94, Lemmas 3.1 and 3.2]. Proposition 4.1. If Mis a Iρ,J –coprimary toral module and −β∈qdeg(M)then the L– characteristic variety of H0(E−β, M)is the L–characteristic variety of MA(Iρ,J ,0). In particular, the set of slopes of H0(E−β, M)along a coordinate subspace in Cncoincide with the ones of MA(Iρ,J ,0). Proof. Since Mis Iρ,J –coprimary there exists m≥0such that Im ρ,J annihilates M. Consider a set of A–homogeneous elements m1,...,mk∈Mgenerating Mas R–module. This leads to a natural A–graded surjection Lk i=1 R/Im ρ,J(−deg(mi)) ։M. In particular, there is a surjective morphism of D-modules k M i=1 H0(E−β, R/Im ρ,J(−deg(mi))) ։H0(E−β, M) inducing the inclusion: ChL(H0(E−β, M)) ⊆ V(inL(Im ρ,J ), Axξ) = V(inL(Iρ), AJxJξJ, ξj:j /∈J). Here (x, ξ)stands for the coordinates in the cotangent space T∗Cn,xξ = (x1ξ1,...,xnξn)and Vis the zero set in T∗Cnof the corresponding ideal. The equality ChL(MA(Iρ,J ,0)) = V(inL(Iρ), AJxJξJ, ξj:j /∈J)follows from [SW08, (3.2.2) and Corollary 4.17]. Thus, (4.2) ChL(H0(E−β, M)) ⊆ChL(MA(Iρ,J ,0)) 16 MAR´ IA-CRUZ FERN´ ANDEZ-FERN´ ANDEZ AND FRANCISCO-JES´ US CASTRO-JIM´ ENEZ Since Cρ,J is primary and its radical ideal is Iρ+mJ=h∂3 x−∂2 y, ∂z, ∂ti, we have that MA(Cρ,J, β)is an irregular binomial D-module for all parameters β∈C(see Theorem 3.7) and that it has only one slope s= 3/2along its singular locus {y= 0}. We are going to compute the Gevrey solutions of MA(Cρ,J , β)corresponding to this slope. By the previous argument and using that n=h∂4 z, ∂2 ti ⊆ Bρ,J we obtain that any Gevrey solution of MA(Cρ,J , β)along {y= 0}can be written as f=X γ,k λγ,kzγztγtφk(β−2γz−2γt) where λγ,k ∈C,γ= (γz, γt),γz∈ {0,1,2,3},γt, k ∈ {0,1}and φk(β−2γz−2γt) = X m≥0 ((β−3k)/2−γz−γt)3m (k+ 2m)2m x(β−3k)/2−γz−γt−3myk+2m is a Gevrey series of index s= 3/2along y= 0 at any point p∈ {y= 0} ∩ {x6= 0}if (β−3k)/2−γz−γt/∈N. We just need to force the condition ∂x∂t(f) = ∂2 z(f)in order to obtain the values of λγ,k such that fis a solution of MA(Cρ,J , β). In this example, we obtain the conditions λ(2,1),k =λ(3,1),k = 0 for k= 0,1and λ(γz+2,0),1=((β−3k)/2−γz) (a+ 1)(a+ 2) λ(γz,1),k for k, γz= 0,1. In particular we get an explicit basis of the space of Gevrey solutions of MA(Cρ,J, β)along y= 0 with index equal to the slope s= 3/2and we have that the dimension of this space is 8. 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