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´
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ANDEZ AND FRANCISCO-JES´
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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. No ice ha 8 = 4 ·2is he expec ed dimension (see Theo em 5.1) since µρ,J = 4 and he
dimension o he co esponding space o MA(Iρ,J , β)is 2(see [FC11, FC08]).
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CENTRE OF MATHEMATICS FOR APPLICATIONS (UNIVERSITY OF OSLO)AND DEPARTMENT OF ALGEBRA
(UNIVERSITY OF SEVILLA).
E-mail add ess:[email p o ec ed]
DEPARTMENT OF ALGEBRA (UNIVERSITY OF SEVILLA).
E-mail add ess:[email p o ec ed]