On irregular binomial D-modules

a Xi :1012.0618 2 [ma h.AG] 28 Dec 2010 ON IRREGULAR BINOMIAL D–MODULES MAR´ IA-CRUZ FERN ´ ANDEZ-FERN ´ ANDEZ AND FRANCISCO-JES ´ US CASTRO-JIM´ ENEZ Tuesday 18 h June, 2013 ABSTRACT. We p o e ha a holonomic binomial D–module MA(I, β)is egula i and only i ce ain associa ed p imes o Ide e mined by he pa ame e ec o β∈Cda e homogeneous. We u he desc ibe he slopes o MA(I, β)along a coo dina e subspace in e ms o he known slopes o some ela ed hype geome ic D–modules ha also depend on β. When he pa ame e βis gene ic, we also compu e he dimension o he gene ic s alk o he i egula i y o MA(I, β) along a coo dina e hype plane and p o ide some ema ks abou he cons uc ion o i s Ge ey solu ions. 1. INTRODUCTION Binomial D-modules ha e been in oduced by A. Dickens ein, L.F. Ma use ich and E. Mille in [DMM10]. These objec s gene alize bo h GKZ hype geome ic D-modules [GGZ87, GZK89] and (binomial) Ho n sys ems, as ea ed in [DMM10] and [Sai02]. He e Ds ands o he complex Weyl algeb a o o de n, whe e n≥0is an in ege . Elemen s in Da e linea pa ial di e en ial ope a o s; such an ope a o Pcan be w i en as a ini e sum P=X α,γ pαγxα∂γ whe e pαγ ∈C,α= (α1,...,αn), γ = (γ1,...,γn)∈Nnand xα=xα1 1···xαn n,∂γ= ∂γ1 1···∂γn n. The pa ial de i a i e ∂ ∂xiis jus deno ed by ∂i. Ou inpu is a pai (A, β)whe e βis a ec o in Cdand A= (aij)∈Zd×nis a ma ix whose columns a1,...,anspan he Z-module Zd. We also assume ha all ai6= 0 and ha he cone gene a ed by he columns in Rncon ains no lines (one says in his case ha his cone is poin ed). The polynomial ing C[∂] := C[∂1,...,∂n]is a sub ing o he Weyl algeb a D. The ma ix A induces a Zd-g ading on C[∂](also called he A-g ading) by de ining deg(∂i) = −ai. A binomial in C[∂]is a polynomial wi h a mos wo monomial e ms. An ideal Iin C[∂]is said o be binomial is i is gene a ed by binomials. We also say ha he ideal Iis an A-g aded ideal i i is gene a ed by A-homogenous elemen s (equi alen ly i o e e y polynomial in Iall i s A-g aded componen s a e also in I). The ma ix Aalso induces a Zd-g ading on he Weyl algeb a D(also called he A-g ading) by de ining deg(∂i) = −aiand deg(xi) = ai. To he ma ix Aone associa es he o ic ideal IA⊂C[∂]gene a ed by he amily o binomials ∂u−∂ whe e u, ∈Nnand Au =A . The ideal IAis a p ime A-g aded ideal. Pa ially suppo ed by MTM2007-64509, MTM2010-19336 and FEDER, FQM333. MCFF suppo ed by a g an om Iceland, Liech ens ein and No way h ough he EEA Financial Mechanism. Suppo ed and coo dina ed by Uni e sidad Complu ense de Mad id. 1 2 MAR´ IA-CRUZ FERN´ ANDEZ-FERN´ ANDEZ AND FRANCISCO-JES´ US CASTRO-JIM´ ENEZ Recall ha o he pai (A, β)one can associa e he GKZ hype geome ic ideal HA(β) = DIA+D(E1−β1,...,Ed−βd) whe e Ei=Pn j=1 aijxj∂jis he i h Eule ope a o associa ed wi h A. The co esponding GKZ hype geome ic D–module is no hing bu he quo ien (le ) D–module MA(β) := D HA(β), [GGZ87], [GZK89]. Following [DMM10], o any A–g aded binomial ideal I⊂C[∂]we deno e by HA(I, β) he A-g aded le ideal in Dde ined by HA(I, β) = DI +D(E1−β1,...,Ed−βd). The binomial D–module associa ed wi h he iple (A, β, I)is, by de ini ion, he quo ien MA(I, β) := D HA(I,β). No ice ha he ideal HA(IA, β)is no hing bu he GKZ hype geome - ic ideal HA(β). In [DMM10] he au ho s ha e answe ed essen ial ques ions abou binomial D–modules. The main ea ed ques ions a e ela ed o he holonomici y o he sys ems and o he dimension o hei holomo phic solu ion space a ound a non singula poin . In pa icula , in [DMM10, The- o em 6.3] hey p o e ha he holonomici y o MA(I, β)is equi alen o egula holonomici y when Iis s anda d Z-g aded (i.e., he ow-span o Acon ains he ec o (1,...,1)). Howe e , i u ns ou ha he inal sen ence in [DMM10, Theo em 6.3], s a ing ha he egula holo- nomici y o MA(I, β) o a gi en pa ame e βimplies s anda d homogenei y o he ideal I, is ue o binomial Ho n sys ems bu i is no o gene al binomial D–modules. This is shown by Examples 3.10 and 3.11. These wo Examples a e di e en in na u e. Mo e p ecisely, he sys em MA(I, β)conside ed in Example 3.10 is egula holonomic o pa ame e s βou side a ce ain line in he a ine complex plane and i egula o he wise, while he sys em conside ed in Example 3.11 is egula holo- nomic o all pa ame e s despi e he ac ha he binomial ideal Iis no homogeneous wi h espec o he s anda d Z–g ading. This is a su p ising phenomenon since i is no allowed nei he o GKZ hype geome ic sys ems no o binomial Ho n sys ems. We u he p o ide, in Theo em 3.7, a cha ac e iza ion o he egula holonomici y o a sys em MA(I, β) ha imp o es he abo e men ioned esul o [DMM10, Th. 6.3]. A cen al ques ion in he s udy o he i egula i y o a holonomic D-module Mis he compu- a ion o i s slopes along smoo h hype su aces (see [Meb90] and [LM99]). On he o he hand, he Ge ey solu ions o Malong smoo h hype su aces a e closely ela ed wi h he i egula i y and he slopes o M. Mo e p ecisely, he classes o hese Ge ey se ies solu ions o Mmodulo con e gen se ies de ine he 0- h cohomology g oup o he i egula i y o M[Meb90, D´e ini ion 6.3.1]. In Sec ion 4 we desc ibe he L–cha ac e is ic a ie y and he slopes o MA(I, β)along coo di- na e subspaces in e ms o he same objec s o he binomial D–modules associa ed wi h some o he o al p imes o he ideal Ide e mined by β(see Theo em 4.3). The binomial D–module as- socia ed wi h a o al p ime is essen ially a GKZ hype geome ic sys em and he L–cha ac e is ic a ie y and he slopes along coo dina e subspaces o such a sys em a e comple ely desc ibed in [SW08] in a combina o ial way (see also [CT03] and [Ha 03, Ha 04] o he cases d= 1 and n=d+ 1). Ge ey solu ionso hype geome ic sys ems alongcoo dina e subspaces a e desc ibed in [Fe 10] (see also [FC11], [FC08]). In Sec ion 5 we compu e he dimension o he gene ic s alk o he ON IRREGULAR BINOMIAL D–MODULES 3 i egula i y o binomial D-modules when he pa ame e is gene ic (see Theo em 5.1). We i- nally gi e a p ocedu e o compu e Ge ey solu ions o MA(I, β)by using known esul s in he hype geome ic case ([GZK89], [SST00] and [Fe 10]). We a e g a e ul o Ez a Mille o his use ul sugges ions and commen s. 2. PRELIMINARIES ON EULER–KOSZUL HOMOLOGY,BINOMIAL PRIMARY DECOMPOSITION AND TORAL AND ANDEAN MODULES We e iew he e some de ini ions, no a ions and esul s o [ES96], [MMW05], [DMM10] and [DMM210] ha will be used in he sequel. We will deno e R=C[∂]. Recall ha he A–g ading on he ing Ris de ined by deg(∂j) = −aj whe e ajis he j h-column o A. This A–g ading on Rcan be ex ended o he ing Dby se ing deg(xj) = aj. De ini ion 2.1. [DMM10, De ini ion 2.4] Le V=⊕α∈ZdVαbe an A-g aded R-module. The se o ue deg ees o Vis deg(V) = {α∈Zd:Vα6= 0} The se o quasideg ees o Vis he Za iski closu e in Cdo deg(V). Eule -Koszul complex K•(E−β;V)associa ed wi h an A-g aded R–module V. Fo any A–g aded le D–module N=⊕α∈ZdNαwe deno e degi(y) = αii y∈Nα. The map Ei−βi:Nα→Nαde ined by (Ei−βi)(y) = (Ei−βi−αi)ycan be ex ended (by C–linea i y) o a mo phism o le D–modules Ei−βi:N→N. We deno e by E−β he sequence o commu ing endomo phisms E1−β1,...,Ed−βd. This allows us o conside he Koszul complex K•(E−β, N)which is concen a ed in homological deg ees d o 0. De ini ion 2.2. [MMW05, De ini ion 4.2] Fo any β∈Cdand any A-g aded R–module V, he Eule -Koszul complex K•(E−β, V )is he Koszul complex K•(E−β, D ⊗RV). The i h Eule -Kozsul homology o V, deno ed by Hi(E−β, V ), is he homology Hi(K•(E−β, V )). Rema k 2.3. Recall ha we ha e he A–g aded isomo phism Hi(E−β, V )(α)≃ Hi(E−β+ α, V )(α) o all α∈Zd[MMW05]. He e V(α)is no hing bu Vwi h he shi ed A–g ading V(α)γ=Vα+γ o all γ∈Zd. Binomial p ima y decomposi ion o binomial ideals. We ecall om [ES96] ha o any subla ice Λ⊂Znand any pa ial cha ac e ρ: Λ →C∗, he co esponding associa ed binomial ideal is Iρ=h∂u+−ρ(u)∂u−|u=u+−u−∈Λi whe e u+and u−a e in Nnand hey ha e disjoin suppo s. The ideal Iρis p ime i and only i Λis a sa u a ed subla ice o Zn(i.e. Λ = QΛ∩Zn). We know om [ES96, Co olla y 2.6] ha any binomial p ime ideal in Rhas he o m Iρ,J := Iρ+mJ(whe e mJ=h∂j|j6∈ Ji) o some pa ial cha ac e ρwhose domain is a sa u a ed subla ice o ZJand some J⊂ {1,...,n}. Fo any J⊂ {1,...,n}we deno e by ∂J he monomial Qj∈J∂j. Theo em 2.4. [DMM210, Theo em 3.2] Fix a binomial ideal Iin R. Each associa ed binomial p ime Iρ,J has an explici ly de ined monomial ideal Uρ,J such ha I= Iρ,J ∈Ass(I)Cρ,J 4 MAR´ IA-CRUZ FERN´ ANDEZ-FERN´ ANDEZ AND FRANCISCO-JES´ US CASTRO-JIM´ ENEZ o Cρ,J = ((I+Iρ) : ∂∞ J) + Uρ,J , is a p ima y decomposi ion o Ias an in e sec ion o A– g aded p ima y binomial ideals. To al and Andean modules. In [DMM210, De ini ion 4.3] a ini ely gene a ed A-g aded R–module V=⊕Vαis said o be o al i i s Hilbe unc ion HV(de ined by HV(α) = dimCVα o α∈Zd) is bounded abo e. Wi h he no a ions abo e, a R–module o ype R/Iρ,J is o al i and only i i s K ull dimension equals he ank o he ma ix AJ(see [DMM10, Lemma 3.4]). He e AJis he subma ix o A whose columns a e indexed by J. In his case he module R/Cρ,J is o al and we say ha he ideal Iρ,J is a o al p ime and Cρ,J is a o al p ima y componen . I dim(R/Iρ,J )6= ank (AJ) hen he module R/Cρ,J is said o be Andean, he ideal Iρ,J is an Andean p ime and Cρ,J is an Andean p ima y componen . An A–g aded R–module Vis said o be na i ely o al i he e exis a binomial o al p ime ideal Iρ,J and an elemen α∈Zdsuch ha V(α)is isomo phic o R/Iρ,J as A–g aded modules (see [DMM10, De ini ion 4.1]). P oposi ion 2.5. [DMM10, P oposi ion 4.2] An A–g aded R–module Vis o al i and only i i has a il a ion 0 = V0⊂V1⊂ ··· ⊂ Vℓ−1⊂Vℓ=V whose successi e quo ien s Vk/Vk−1a e all na i ely o al. Such a il a ion on Vis called a o al il a ion. Following [DMM10, De ini ion 5.1] an A-g aded R-module Vis said o be na i ely Andean i he e is an α∈Zdand an Andean quo ien ing R/Iρ,J o e which V(α)is o sion- ee o ank 1 and admi s a ZJ/Λ-g ading ha e ines he A-g ading ia ZJ/Λ→Zd=ZA, whe e ρis de ined on Λ⊂ZJ. Mo eo e , i Vhas a ini e il a ion 0 = V0⊂V1⊂ ··· ⊂ Vℓ−1⊂Vℓ=V whose successi e quo ien s Vk/Vk−1a e all na i ely Andean, hen Vis Andean (see [DMM10, Sec ion 5]). In [DMM210, Example 4.6] i is p o en ha he quo ien R/Cρ,J is Andean o any Andean p ima y componen Cρ,J o any A-g aded binomial ideal. We inish his sec ion wi h he de ini ion and a esul abou he so-called Andean a angemen associa ed wi h an A-g aded binomial ideal Iin R. Le us ix an i edundan p ima y decom- posi ion I= Iρ,J ∈Ass(I)Cρ,J as in Theo em 2.4. De ini ion 2.6. [DMM10, De ini ion 6.1] The Andean a angemen ZAndean(I)is he union o he quasideg ee se s qdeg(R/Cρ,J ) o he Andean p ima y componen s Cρ,J o I. F om [DMM10, Lemma 6.2] he Andean a angemen ZAndean(I)is a union o ini ely many in ege ansla es o he subspaces CAJ⊂Cn o which he e is an Andean associa ed p ime Iρ,J . F om [DMM10, Theo em 6.3] we ha e ha he binomial D–module MA(I, β)is holonomic i and only i −β /∈ ZAndean(I). ON IRREGULAR BINOMIAL D–MODULES 5 3. CHARACTERIZING REGULAR HOLONOMIC BINOMIAL D–MODULES Le Ibe an A–g aded binomial ideal and ix a binomial p ima y decomposi ion I=∩ρ,J Cρ,J whe e Cρ,J is a Iρ,J–p ima y binomial ideal. Le us conside he ideal Iβ:= −β∈qdeg(R/Cρ,J ) Cρ,J i.e., he in e sec ion o all he p ima y componen s Cρ,J o Isuch ha −βlies in he quaside- g ees se o he module R/Cρ,J . Rema k 3.1. No ice ha i −β /∈ ZAndean(I) hen R/Iβis con ained in he o al di ec sum M −β∈qdeg(R/Cρ,J ) R/Cρ,J and so i is a o al module. The ollowing esul gene alizes [DMM10, P oposi ion 6.4]. P oposi ion 3.2. I −β /∈ ZAndean(I) hen he na u al su jec ion R/I ։R/Iβinduces a isomo phism in Eule –Koszul homology Hi(E−β, R/I)≃ Hi(E−β, R/Iβ) o all i. In pa icula , MA(I, β)≃MA(Iβ, β). P oo . By [DMM10, P oposi ion 6.4] we ha e ha Hi(E−β, R/I)≃ Hi(E−β, R/I o al) o all i, whe e I o al deno es he in e sec ion o all he o al p ima y componen s o I. Thus, we can assume wi hou loss o gene ali y ha all he p ima y componen s o Ia e o al. The es o he p oo is now analogous o he p oo o [DMM10, P oposi ion 6.4] i we subs i u e he ideals I o al and IAndean he e by he ideals Iβand Iβ espec i ely, whe e Iβ= −β /∈qdeg(R/Cρ,J ) Cρ,J, and he Andean di ec sum LIρ,J Andean R/Cρ,J he e by he o al di ec sum M −β /∈qdeg(R/Cρ,J ) R/Cρ,J Finally, we can use Lemma 4.3 and Theo em 4.5 in [DMM10] in a simila way as [DMM10, Lemma 5.4] is used in he p oo o P oposi ion 6.4 o [DMM10].  The ollowing Lemma gi es a desc ip ion o he quasideg ees se o a o al module o ype R/Cρ,J . E. Mille has poin ed ou ha his esul ollows om P oposi ion 2.13 and Theo em 2.15 in [DMM210]. We will include he e a sligh ly di e en p oo o his Lemma. Lemma 3.3. Fo any Iρ,J –p ima y o al ideal Cρ,J he quasideg ees se o M=R/Cρ,J equals he union o a mos µρ,J Zd–g aded ansla es o CAJ, whe e µρ,J is he mul iplici y o Iρ,J in Cρ,J . Mo e p ecisely, o any o al il a ion 0 = M0⊆M1⊆ ··· ⊆ Mwe ha e ha 6 MAR´ IA-CRUZ FERN´ ANDEZ-FERN´ ANDEZ AND FRANCISCO-JES´ US CASTRO-JIM´ ENEZ he quasideg ees se o Mis he union o he quasideg ees se o all he successi e quo ien s Mi/Mi−1 ha a e isomo phic o Zd–g aded ansla es o R/Iρ,J. P oo . Since Mis o al we ha e by [DMM10, Lemma 4.7] ha dim(qdeg(M)) = dim M= ank AJ. Since Cρ,J is p ima y, any ze o-di iso o Mis nilpo en . Fo all j∈Jwe ha e ha ∂m j/∈Cρ,J ⊆Iρ+mJand so∂jis no a ze o-di iso inM o all j∈J. Thus, he ue deg ees se o M e i ies deg(M) = deg(M)−NAJ. This and he ac ha dim(qdeg(M)) = ank AJ imply ha he e exis s α1,...,α ∈Zdsuch ha deg(M) = ∪ i=1(αi−NAJ)and (3.1) qdeg(M) = [ i=1 (αi+CAJ) Conside now a o al il a ion 0 = M0⊆M1⊆ ··· ⊆ M. We know ha he e a e ex- ac ly µρ,J di e en alues o isuch ha Mi/Mi−1≃R/Iρ,J(γi) o some γi∈Zd. Fo he o he successi e quo ien s Ml/Ml−1≃R/Iρl,Jl(γl)we ha e ha Iρl,Jlis a o al p ime which p ope ly con ains Iρ,J . In pa icula , we ha e ha ank AJl= dim R/Iρl,Jl<dim R/Iρ,J = ank AJ. Since qdeg(R/Iρl,Jl) = CAJlhas dimension ank AJl< ank AJand qdeg(M) = Siqdeg(Mi/Mi−1)we ha e by (3.1) ha he quasideg ees se o any Mi/Mi−1is con ained in he quasideg ees se o some Mj/Mj−1≃R/Iρ,J (γj). In pa icula ≤µρ,J and each a ine subspace (αi+CAJ)in (3.1) is he quasideg ees se o some Mj/Mj−1≃R/Iρ,J (γj). Rema k 3.4. No ice ha HA(Iρ,J , β) = DHAJ(Iρ, β) + D(∂j:j /∈J). In addi ion, i Iρ,J is o al hen he DJ–module MAJ(Iρ, β)is isomo phic o he hype geome ic sys em MAJ(β) ia an A–g aded isomo phism o DJ–modules induced by escaling he a iables xj,j∈J, using he cha ac e ρ. Thus we can apply mos o he well-knows esul s o hype geome ic sys ems o MA(Iρ,J , β)(wi h Iρ,J a o al p ime) in an app op ia ed o m. Lemma 3.5. I Iρ,J is o al and −β∈qdeg(R/Iρ,J ) he ollowing condi ions a e equi alen : i) Hi(E−β, R/Iρ,J)is egula holonomic o all i. ii) H0(E−β, R/Iρ,J)is egula holonomic. iii) Iρ,J is homogeneous (equi alen ly AJis homogeneous). P oo . i)⇒ii)is ob ious, ii)⇒iii) ollows s aigh o wa d om [SW08, Co olla y 3.16] and iii)⇒i)is a pa icula case o he las s a emen in [DMM10, Theo em 4.5] and i also ollows om [Ho 98, Ch. II, 6.2, Thm.].  Rema k 3.6. Recall om [DMM10, Theo em 4.5] ha o any o al module Vwe ha e ha −β /∈qdegVi and only i H0(E−β, V ) = 0 i and only i Hi(E−β, V ) = 0 o all i. In pa icula , since he D–module 0is egula holonomic i ollows ha condi ions i) and ii) in Lemma 3.5 a e also equi alen wi hou he condi ion −β∈qdeg(R/Iρ,J ). Theo em 3.7. Le I⊆Rbe an A-g aded binomial ideal such ha MA(I, β)is holonomic (equi alen ly, −β /∈ ZAndean(I)). The ollowing condi ions a e equi alen : i) Hi(E−β, R/I)is egula holonomic o all i. ii) MA(I, β)is egula holonomic. iii) All he associa ed o al p imes Iρ,J o Isuch ha −β∈qdeg(R/Cρ,J)a e homoge- neous. ON IRREGULAR BINOMIAL D–MODULES 7 P oo . The implica ion i)⇒ii)is ob ious. Le us p o e ii)⇒iii). Fo any o al p ima y componen Cρ,J o Iwe ha e I⊆Cρ,J and so he e is a na u al epimo phism MA(I, β)։ MA(Cρ,J, β). Since MA(I, β)is egula holonomic hen MA(Cρ,J, β)is also egula holonomic. Take a o al il a ion o M=R/Cρ,J ,0⊆M1⊆ ··· ⊆ M =M. We claim ha (3.2) Hj(E−β, Mi/Mi−1)and H0(E−β, Mi−1)a e egula holonomic o all i, j. Le us p o e (3.2) by dec easing induc ion on i. Fo i= , we ha e a su jec ion om he egula holonomic D–module H0(E−β, M ) = MA(Cρ,J, β) o H0(E−β, M /M −1)and so i is egula holonomic oo. By Rema k 2.3, Lemma 3.5 and Rema k 3.6 we ha e ha he D-module Hj(E−β, M /M −1)is egula holonomic o all j. Since H1(E−β, M /M −1)−→ H0(E−β, M −1)−→ H0(E−β, M ) is exac we ha e ha H0(E−β, M −1)is egula holonomic. Assume ha (3.2) holds o some i=k+ 1 ≤ and o all j. We conside he exac sequence 0−→ Mk−1−→ Mk−→ Mk/Mk−1−→ 0 and he ollowing pa o he long exac sequence o Eule -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 induc ion hypo hesis H0(E−β, Mk)is egula holonomic. This implies ha H0(E− β, Mk/Mk−1)is egula holonomic by (3.3). Applying Rema k 2.3, Lemma 3.5 and Rema k 3.6 we ha e ha Hj(E−β, Mk/Mk−1)is egula holonomic o all j. Thus, by (3.3) we ha e ha H0(E−β, Mk−1)is egula holonomic oo and we ha e inished he induc ion p oo o (3.2). Assume ha −β∈qdeg(R/Cρ,J ). By Lemma 3.3 he e exis s isuch ha −βlies in he quasideg ees se o Mi/Mi−1≃R/Iρ,J (γi)and we also ha e by (3.2) ha H0(E−β, Mi/Mi−1)≃ H0(E−β+γi, R/Iρ,J )(γi) is a nonze o egula holonomic D-module. Thus, by Lemma 3.5 we ha e ha Iρ,J is homoge- neous. Le us p o e iii)⇒i). By P oposi ion 3.2 we jus need o p o e ha MA(Iβ, β)is egula holonomic. We ha e ha all he associa ed p imes o Iβa e o al and homogeneous. In pa - icula M=R/Iβis a o al module and o any o al il a ion o M he successi e quo ien s Mi/Mi−1a e isomo phic o some Zd–g aded ansla e o a quo ien R/Iρi,Jiwhe e Iρi,Jiis o al and con ains a minimal p ime Iρ,J o Iβ. Such minimal p ime is homogeneous by assump ion and so AJis homogeneous. Since Ji⊆Jwe ha e ha AJiand Iρi,Jia e homogeneous oo. Now, we jus poin ou ha ha he p oo o he las s a emen in [DMM10, Theo em 4.5] s ill holds o V=Mi we don’ equi e A o be homogenous bu all he p imes occu ing in a o al il a ion o M o be homogeneous.  Rema k 3.8. Theo em 3.7 shows in pa icula ha he p ope y o a binomialD-moduleMA(I, β) o being egula (holonomic) can ail o be cons an when −β uns ou side he Andean a ange- men . This phenomenon is o bidden o binomial Ho n sys ems MA(I(B), β)(see [DMM10, 8 MAR´ IA-CRUZ FERN´ ANDEZ-FERN´ ANDEZ AND FRANCISCO-JES´ US CASTRO-JIM´ ENEZ De ini ion 1.5]) since he inclusion I(B)⊆IAinduces a su jec i e mo phism H0(E−β, I(B)) ։MA(β) and hen egula holonomici y o H0(E−β, R/I(B)) implies egula holonomici y o MA(β), which is equi alen o he s anda d homogenei y o IAby [Ho 98, SST00, SW08]. De ini ion 3.9. The non- egula a angemen o I(deno ed by Znon− egula (I)) is he union o he Andean a angemen o Iand heunion o quasideg eesse s o he quo ien s o Rby p ima y componen s Cρ,J o Isuch ha Iρ,J is no homogeneous wi h espec o he s anda d g ading. So, we ha e Znon− egula (I) = ZAndean(I)∪ [ Iρ,J non homogeneous qdeg(R/Cρ,J) . Example 3.10. Conside he ideal I=h∂2 1∂2−∂2 2, ∂2∂3, ∂2∂4, ∂2 1∂3−∂2 3∂4, ∂2 1∂4−∂3∂2 4i. I is A-g aded o he ma ix A=1220 1202 bu Iis no s anda d Z-g aded. We ha e he p ime decomposi ion I=I1∩I2∩I3whe e I1=h∂2, ∂3, ∂4i,I2=h∂2 1−∂2, ∂3, ∂4iand I3=h∂2, ∂2 1−∂3∂4ia e o al p imes o I. In pa icula ZAndean(I) = ∅) and by he p oo o [DMM10, P oposi ion 6.6] we ha e ha Zp ima y(I) = {0}(see [DMM10, De ini ion 6.5] o he de ini ion o he p ima y a angemen Zp ima y(I)). Using [DMM10, Theo em 6.8] we ha e ha MA(I, β)is isomo phic o he di ec sum o MA(Ij, β) o j= 1,2,3i β6= 0. Mo eo e , qdeg(R/Ij) = C1 1 o j= 1,2and qdeg(R/I3) = C2. Thus, o gene ic pa ame e s (mo e p ecisely o β∈C2 C1 1) we ha e ha MA(I, β)is isomo phic o MA(I3, β) ha is a egula holonomic by Lemma 3.5. On he o he hand, he e is a su jec i e mo phism om MA(I, β) o MA(I2, β)and i β∈C1 1 we ha e ha MA(I2, β)is an i egula D-module because s= 2 is a slope along x2= 0. Thus we conclude ha MA(I, β)is egula holonomic i β∈C2 C1 1and i is an i egula holonomic D-module when β∈C1 1. In pa icula , Znon− egula (I) = C1 1⊂C2. I can also be checked ha he singula locus o MA(I, β)is {x1x2x3x4(x2 1−4x3x4) = 0}when β∈C1 1 and {x3x4(x2 1−4x3x4) = 0}o he wise. Example 3.11. The p ima y binomial ideal I=h∂1−∂2, ∂4 3, ∂3 4, ∂3 3−∂2 4iis A–g aded wi h espec o he ma ix A= (1 1 2 3). No e ha Iis no homogeneous wi h espec o he s anda d Z-g ading. Howe e , i s adical ideal √I=h∂1−∂2, ∂3, ∂4iis homogeneous. Thus, by Theo em 3.7 we ha e ha MA(I, β)is egula holonomic. 4. L–CHARACTERISTIC VARIETY AND SLOPES OF BINOMIAL D–MODULES Le Lbe he il a ion on Dde ined by a weigh ec o (u, )∈R2nwi h ui+ i=c > 0 o some cons an c > 0. This includes in pa icula he in e media e il a ions pF +qV be ween he il a ion Fby he o de o he linea di e en ial ope a o s and he Kashiwa a-Malg ange il a ion Valong ON IRREGULAR BINOMIAL D–MODULES 9 a coo dina e subspace. The il a ions pF +qV a e he ones conside ed when s udying he algeb aic slopes o a cohe en D–module along a coo dina e subspace [LM99]. We will conside he L–cha ac e is ic a ie y ChL(N)o a ini ely gene a ed D–module Non Cnde ined as he suppo o g LNin T∗Cn(see e.g. [Lau87], [SW08, De ini ion 3.1]). We ecall ha in ac o L=pF +qV his is a global algeb aic e sion o Lau en ’s mic ocha ac- e is ic a ie y o ype s=p/q + 1 [Lau87, §3.2] (see also [SW08, Rema k 3.3]). The L-cha ac e is ic a ie y and he slopes o a hype geome ic D-module MA(β)a e con- olled by he so-called (A, L)–umb ella [SW08]. Le us ecall i s de ini ion in he special case when i>0 o all i. We deno e by ∆L A he con ex hull o {0, aL 1,...,aL n}whe e aL j=1 jaj. The (A, L)-umb ella is he se ΦL Ao aces o ∆L Awhich do no con ain 0. The emp y ace is in ΦL A. One iden i ies τ∈ΦL Awi h {j|aL j∈τ}, o wi h {aj|aL j∈τ}, o wi h he co esponding subma ix Aτo A. By [SW08, Co olla y 4.17] he L-cha ac e is ic a ie y o a hype geome ic D–module MA(β) is (4.1) ChL(MA(β)) = [ τ∈ΦL A Cτ A whe e Cτ Ais he Za iski closu e in T∗Cno he cono mal space o he o bi Oτ A⊂T∗ 0Cn=Cn co esponding o he ace τ. In pa icula ChL(MA(β)) is independen o β. By de ini ion we ha e he equali y Oτ A:= (C∗)d·1τ Awhe e 1τ A∈Nnis de ined by (1τ A)j= 1 i j∈τand (1τ A)j= 0 o he wise. The ac ion o he o us is gi en wi h espec o he ma ix A. I he il a ion gi en by Lequals he F- il a ion (i.e. he o de il a ion) hen his desc ip ion o he F–cha ac e is ic a ie y coincides wi h a esul o [Ado94, Lemmas 3.1 and 3.2]. P oposi ion 4.1. I Mis a Iρ,J –cop ima y o al module and −β∈qdeg(M) hen he L– cha ac e is ic a ie y o H0(E−β, M)is he L–cha ac e is ic a ie y o MA(Iρ,J ,0). In pa - icula , he se o slopes o H0(E−β, M)along a coo dina e subspace in Cncoincide wi h he ones o MA(Iρ,J ,0). P oo . Since Mis Iρ,J –cop ima y he e exis s m≥0such ha Im ρ,J annihila es M. Conside a se o A–homogeneous elemen s m1,...,mk∈Mgene a ing Mas R–module. This leads o a na u al A–g aded su jec ion Lk i=1 R/Im ρ,J(−deg(mi)) ։M. In pa icula , he e is a su jec i e mo phism o D-modules k M i=1 H0(E−β, R/Im ρ,J(−deg(mi))) ։H0(E−β, M) inducing he inclusion: ChL(H0(E−β, M)) ⊆ V(inL(Im ρ,J ), Axξ) = V(inL(Iρ), AJxJξJ, ξj:j /∈J). He e (x, ξ)s ands o he coo dina es in he co angen space T∗Cn,xξ = (x1ξ1,...,xnξn)and Vis he ze o se in T∗Cno he co esponding ideal. The equali y ChL(MA(Iρ,J ,0)) = V(inL(Iρ), AJxJξJ, ξj:j /∈J) ollows om [SW08, (3.2.2) and Co olla y 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 p ima y and i s adical ideal is Iρ+mJ=h∂3 x−∂2 y, ∂z, ∂ i, we ha e ha MA(Cρ,J, β)is an i egula binomial D-module o all pa ame e s β∈C(see Theo em 3.7) and ha i has only one slope s= 3/2along i s singula locus {y= 0}. We a e going o compu e he Ge ey solu ions o MA(Cρ,J , β)co esponding o his slope. By he p e ious a gumen and using ha n=h∂4 z, ∂2 i ⊆ Bρ,J we ob ain ha any Ge ey solu ion o MA(Cρ,J , β)along {y= 0}can be w i en as =X γ,k λγ,kzγz γ φk(β−2γz−2γ ) whe e λγ,k ∈C,γ= (γz, γ ),γz∈ {0,1,2,3},γ , k ∈ {0,1}and φk(β−2γz−2γ ) = X m≥0 ((β−3k)/2−γz−γ )3m (k+ 2m)2m x(β−3k)/2−γz−γ −3myk+2m is a Ge ey se ies o index s= 3/2along y= 0 a any poin p∈ {y= 0} ∩ {x6= 0}i (β−3k)/2−γz−γ /∈N. We jus need o o ce he condi ion ∂x∂ ( ) = ∂2 z( )in o de o ob ain he alues o λγ,k such ha is a solu ion o MA(Cρ,J , β). In his example, we ob ain he condi ions λ(2,1),k =λ(3,1),k = 0 o k= 0,1and λ(γz+2,0),1=((β−3k)/2−γz) (a+ 1)(a+ 2) λ(γz,1),k o k, γz= 0,1. In pa icula we ge an explici basis o he space o Ge ey solu ions o MA(Cρ,J, β)along y= 0 wi h index equal o he slope s= 3/2and we ha e ha he dimension o his space is 8. 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