Testing the Logarithmic Comparison Theorem for Free Divisors

Full Te ms & Condi ions o access and use can be ound a h ps://www. and online.com/ac ion/jou nalIn o ma ion?jou nalCode=uexm20 Expe imen al Ma hema ics ISSN: 1058-6458 (P in ) 1944-950X (Online) Jou nal homepage: h ps://www. and online.com/loi/uexm20 Tes ing he Loga i hmic Compa ison Theo em o F ee Di iso s F. J. Cas o-Jiménez & J. M. Ucha-En íquez To ci e his a icle: F. J. Cas o-Jiménez & J. M. Ucha-En íquez (2004) Tes ing he Loga i hmic Compa ison Theo em o F ee Di iso s, Expe imen al Ma hema ics, 13:4, 441-449, DOI: 10.1080/10586458.2004.10504553 To link o his a icle: h ps://doi.o g/10.1080/10586458.2004.10504553 Published online: 03 Ap 2012. Submi you a icle o his jou nal A icle iews: 16 View ela ed a icles Ci ing a icles: 6 View ci ing a icles Tes ing he Loga i hmic Compa ison Theo em o F ee Di iso s F. j. Cas o-jimenez and j. M. Ucha-En quez CONTENTS 1. In oduc ion 2. Spence Di iso s 3. How o Deduce ha LCT Holds 4. How o Deduce ha LCT Does No Hold 5. On he Regula i y o Loga i hmic 1>.Modules 6. Conclusions Acknowledgmen s Re e ences 2000 AMSSubjec Classi ica ion: P ima y 14F50; Seconda y 32C38, 32C35, 13Pxx, 68W30 Keywo ds: de Rham cohomology, Loga i hmic Compa ison Theo em, ee di iso s, G obne bases We p opose in his wo k a compu a ional c i e ion o es i a ee di iso DC C" e i ies he Loga i hmic Compa ison The- o em (LCT); ha is, whe he he complex o loga i hmic di e - en ial o ms compu es he cohomology o he complemen o D inC n• Fo Spence ee di iso s D == (I =0), we sol e a conjec- u e abou he gene a o s o he annihila ing ideal o 1/1 and make a conjec u e on he na u e o Eule homogeneous ee di- iso swhich e i y LCT. In addi ion, we p o ideexamples o ee di iso s de ined by weigh ed homogeneous polynomials ha a e no locally quasi-homogeneous. 1. INTRODUCTION Le Dbe a di iso (i.e., a hype su ace) in X := en. K. Sai o in oduced in [Sai o 80] he complex n-(log D) o holomo phic di e en ial o ms wi h loga i hmic poles along D. I is a sub complex o he me omo phic de Rham complex n-(*D) o me omo phic di e en ial o ms wi h poles along D. Le us deno e by io he inclusion mo - phism n-(logD) ~ n-(*D). G o hendieck's Compa i- son Theo em [G o hendieck 66] p o es ha he las com- plex calcula es he cohomology o he complemen o D in X. I was p o ed in [Cas o e al. 96] ha i Dis alocally quasi-homogeneous ee di iso hen he Loga- i hmic Compa ison Theo em (LCT) holds o D. We claim ha LeT holds o Di he mo phism iDis a quasi-isomo phism, i.e., i io induces an isomo phism on cohomology. Le us deno e by 0=0x he shea o holomo phic unc ions on X and ake xE X. Deno e by De (Ox) he Ox-module o C-de i a ions o Ox ( he elemen s in De (Ox) a e called ec o ields). This yields he shea De ( 0) o ec o ields on X. Following K. Sai o [Sai o 80], a ec o ield 8E. De (Ox) is said o be loga i hmic wi h espec o Di 8(/) = a] o some aE Ox, whe e 1is a local ( educed) equa ion o he ge m (D, x) c (X, x). The Ox-module ©A K Pe e s, l d. 1058-645812004$ 0.50 pe page Expe imen al Ma hema ics 13:4, page 441 442 Expe imen al Ma hema ics, Vol. 13(2004), No.4 o loga i hmic ec o ields (o loga i hmic de i a ions) wi h espec o Dis deno ed by De ( -log D)x and i is closed unde he b acke p oduc [-, -]. This yields a 0- module cohe en shea deno ed by De ( -log D), which is a submodule o he shea o ec o ields o e X. The di iso Dis said o be ee a he poin xED (see [Sai o 80]) i he Ox-module De ( -log D)x is ee (and, in his case, o ank n). The di iso Dis called ee i i is ee a each poin xED. Sai o's c i e ion ([Sai o 80]) says ha adi iso D == ( == 0) is ee a a poin xED i and only i he e exis s abasis {£5 1, ... , £5 n} o De ( -log D)x whose de e minan o coe icien s wi h espec o - he pa ial de i a i es is equal o U.: i, o some uEOx such ha u(x) =I o. Smoo h di iso s and no mal c ossing di iso a e ee. By [Sai o 80], any plane cu e DCC2is a ee di iso . As s a ed in [Cas o e al. 96], a di iso Dcen is called locally quasi-homogeneous i o each xED he e exis s a sys em o local coo dina es (Zl, ... ,zn) a ound xsuch ha he ge m (D, x) is de ined by a weigh ed homogeneous polynomial h(Zl, ... ,zn) wi h s ic ly pos- i i e weigh s o he a iables Zi. I a plane cu e DCC2 is de ined by a weigh ed homogeneous polynomial, hen Dis a locally quasi-homogeneous ee di iso -so LCT holds o such a plane cu e. In [Calde on e al. 02] a con- e se o his las esul is p o ed: i Dis a plane cu e and LeT holds o D, hen Dis locally quasi-homogeneous. I was also shown in [Calde on e al. 02] ha , in dimen- sion 3, he e a e ee di iso s ha e i y LCT and a e no locally quasi-homogeneous. Adi iso DcX is said o be Eule homogeneous i o each xED he e exis a local equa ion o he ge m I (D, x) and a ec o ield £5 E De (Ox) anishing a xsuch ha £5( ) == . In [Calde on e al. 02], i is p o ed ha LeT implies Eule homogenei y o place cu es. I is an open p oblem o desc ibe which ee di iso s e i y LCT in dimension g ea e han 2, and he e is an open conjec u e ela ed o his p oblem: Conjec u e 1.1. ([To elli 04]) LeT holds o he ge m (D, x) i and only i he annihila o ideal Ann x (1/ ) is gene a ed by elemen s o o de 1, whe e is a local equa ion o he ge m (D,x). He e Vx s ands o he ing o ge ms o linea di - e en ial ope a o s wi h coe icien s in he ing Ox and Ann x (1/ ) o he ideal o linea di e en ial ope a o s Pin i; such ha P(I/ ) == o. The o de o a nonze o elemen Po V x, P== is he in ege o d(P) == maxjjo] == a1 +... +a nI ao: =I O} (he e ao: E0x and 8i s ands o he pa ial de i a i e a~i). The p incipal symbol o Pis by de ini ion he ex- p ession a(P) == L ao:~O: , o:,lo:l=o d(P) iewed as an elemen o he ing o polynomials Ox [~] in n a iables, ~ == (~1, ... ,~n), wi h coe icien s in Ox. I V == Vxis he shea o ings o linea di e en ial ope a o s wi h holomo phic coe icien s, he s alk o V a xis Vx. The ing Vxis il e ed by he o de o i s elemen s. The associa ed g aded ing is deno ed by g (V x) . I is easy o p o e ha g (V x) is isomo phic o Ox[~]. One o he di icul ies in he s udy o which di iso s D e i y LCT is add essed by he examples. Gi en a di iso Di is ha d o p o e whe he he inclusion mo - phism io is a quasl-isomo phism". We p opose he e a compu a ional ool o es LCT in he ee case. An essen ial ing edien o ou me hod o co e he case o ee di iso s is a ecen esul o [Calde on and Na aez 04]: i Dis a ee di iso ha e i ies LCT hen Dis a Spence di iso . Roughly speaking, Dis a Spence ee di iso (see below) i i is ee and admi s a spe- cial ee esolu ion o he V-module V/ (De ( -log D)) whe e (De ( -log D)) is he le ideal o V gene a ed by De ( -log D). Wi h he help o his esul , i is enough o p o ide ac i e ion o es LCT o Spence - ee di iso s; he non-Spence ones do no e i y LCT. 2. SPENCER DIVISORS In o de o allow an easie eading o his a icle we e- iew he e he esul s o [Cas o and Ucha 02, Sec ions 3 and 4]. In [Calde on 99] he au ho associa es wi h a ee di i- so Dcen he so-called augmen ed Spence loga i hmic complex V ®o 1 . De ( -log D) -+ M10g D -+ o. I is a complex o V-modules and M10g D s ands o he quo ien (De (!'logD)). De ini ion 2.1. ([Cas o and Ucha 02, De ini ion 3.3]) We say ha a ee di iso Dis a Spence di iso i 1 The eade can conside , o example, he ad hoc explici p oo in [Calde on e al. 02] o show ha he di iso (xy(x+y)(xz+y) = 0) CC3 e i ies LCT. Cas o-Jimenezand Ucha-En quez: Tes ing he Loga i hmic Compa ison Theo em o F eeDi iso s 443 he V-module M10g Dis holonomic and he augmen ed .Spence loga i hmic complex is a (locally) ee esolu ion o M1ogD. Acohe en V-module Mon X is said o be holonomic i i s cha ac e is ic a ie y (see [Mebkhou 89, Chap e I, (2.2)]) has dimension n. De ini ion 2.2. ([Calde on 99, De ini ion 4.1.1]) The di- iso Dis said o be Koszul ee a he poin xED i i is ee a xand he e exis s a basis {81, ... ,8n}o De ( -log D)x such ha he sequence {O'(8 1) , ••• ,O'(8n)} o p incipal symbols is a egula sequence in he ing g (Vx) . The di iso Dis Koszul ee i i is Koszul ee a any poin o D. By [Sai o 80] and [Calde on 99, 4.2.2.]- any plane cu e DCC2is a Koszul ee di iso . By [Calde on 99, P opo- si ions 4.1.2 and 4.1.3] i Dis Koszul ee (in pa icula i Dis a plane cu e) hen i is a Spence di iso , bu he con e se is no ue, see [Calde on 99, Rema k 4.2.4] and [Cas o and Ucha 02, Sec ion 5.3]. Gi en a cohe en V-module M, he solu ion complexo Mis by de ini ion he complex R'Hom (M,O) {see, o example,. [Mebkhou 89, Chap e I, (2.6)]). This complex will be simply deno ed by Sol(M). The ollowing p oposi ion is a consequence o [Calde on 99, Theo em 4.2.1]. P oposi ion 2.3. I D is a Spence di iso hen he e exis s ana u al quasi-isomo phism om Sol(M1og) o n-(log D). Fo each xE X, we can conside he ideal Anng~ (1/ ) (he e, is a local equa ion o he ge m (D, x)) gen- e a ed by he di e en ial ope a o s PEVxsuch ha . P(I/ ) =0 and he o de o Pis equal o 1. In ac , such an ope a o Pmus ha e he o m 8+awhe e 8is aloga i hmic de i a ion in De ( -log D)x and 8( ) = a wi h aE Ox' Le us deno e by M10g D he quo ien V-module wi h s alks (M1og D) .- Vx x·- Anng~ (1/ ) . The module M10g Dadmi s in he Spence case a ee esolu ion comple ely analogous o he one o M1og D: V ®o / - De (-log D) ~ M10g D ~ 0, whe e De ( -log D) deno es he ee O-module o di e - en ial ope a o s ha can be w i en locally as 8+asuch ha 8E De ( -log D) and 8(/) = a] o some holomo - phic ge m a. The ollowing heo em is p o ed in [Cas o and Ucha 02, Theo em 4.3]: Theo em 2.4. Fo each Spence di iso DcXwe a ha e an isomo phism (M1og D)* ~ Mlog D . In his heo em (-)* means duali y in he sense o V- module heo y (see, o example, (Mebkhou 89, Chap e I, (4.1) ]). These las wo esul s allow us o use V-module heo y in connec ion wi h he loga i hmic compa ison heo em as we will show in he ollowing sec ions. The ollowing ques ion is open: P oblem 2.5. Iden i y which ee di iso s a e Spence di iso s. An example o a ee di iso ha is no Spence is gi en in [Calde on and Na aez 04]. Rema k 2.6. Fo a gi en di iso D == (I =0) wi h 1ER= C[Xl,"" xn] he modules M10g D and M10g D can be ob ained by means o compu a ions o syzygies o polynomials using G abne bases: i o some mER, hen a 8 +...+an 8 - m = O. aXI aX n Since he inclusion o he ing o di e en ial ope a o s wi h coe icien s in R- he Weyl algeb a-in Vxis la , hese compu a ions in Rco e he analy ical se ing. 3. HOW TO DEDUCETHAT LeT HOLDS Le Dc C" be a ee di iso . We s a e he i s c i e ion o p o e ha LCT holds. In he p oo o i s co ec ness, we use some condi ions ha a e su icien o he com- plexes n-(log D) and n-(*D) o be quasi-isomo phic'[. Fo each ge m (D, x) c (C", x) de ined by a holomo - phic unc ion I, we deno e by bl he b- unc ion asso- cia ed wi h 1(see [Be ns ein 72]). Le us deno e by <PD :MlogD ~ O(*D) he na u al mo phism de ined .by <PD(P) =P(-}). Fo any cohe en V-module M he 2The symbol ~ be ween complexes s ands na u ally o quasi- isomo phisms om now on. 444 Expe imen al Ma hema ics, Vol. 13(2004), No.4 de Rham complex DR(M) associa ed wi h Mis by de i- ni ion he complex o shea es o C- ec o spaces V 1 V M lin 0 o ~ M ~ MQ90 n ~ " ... ~ Q90 ~G ~ , whe e ni s ands o he shea o holomo phic di e en ial i- o ms on X and 7 is he ex e io de i a i e de ined by 7(m Q9 w) == ( 7m) 1 w-(_I)deg(w)m Q9 dw. C i e ion 3.1. I Dis a Spence di iso , hen Ann (I/ ) == Anng) (I/ ) => LCT holds o D. P oo : Le us suppose ha Dis Spence . On he one hand, we ha e: • By P oposi ion 2.3, and (see [Mebkhou 89, page 41]) Sol(MlogD) DR((M1ogD)*). • By Theo em 2.4, we deduce ha On he o he hand, -1 is he smalles in ege oo o he b- unc ion b (see [To elli 04, P oposi ion 1.3]), and hen we ha e Fo he compu a ional aspec s, we no e: 1. The ee di iso D mus be Spence . This condi ion can be es ed using G abne bases in he Weyl al- geb a o compu e he modules o he syzygies ha appea , compu ing a " ee esolu ion o M10g D and checking whe he each module is he one equi ed by he Spence esolu ion p esen ed in De ini ion 2.1. We used he package o V-modules, Macaulay 2 ([G ayson e al. 99]), w i en by A. Leykin and H. Tsai. 2. The compa ison be ween Ann (I/ ) and Anng) (1/ ) needs he compu a ion o he an- nihila o . We ha e used [No o 02] in he examples. 3. By [To elli 04, P oposi ion 1.3], i Anng) (1/ ) == Ann (I/ ) hen he smalles .in ege oo o he b- unc ion b is -1. The global b- unc ion can be com- pu ed wi h he algo i hms o [Oaku 97] o [No o 02]. We ha e used o he examples he powe ul imple- men a ion o he la e in [No o e al. 00]. Wi h espec o he oo s o he b- unc ion o ee di- iso s, we can que y he ollowing: P oblem 3.2. Le D= ( == 0) be a ee di iso . Is -1 he smalles in ege oo o b ? In Figu e 1we gi e examples o ee di iso s wi h hei global b- unc ions and hei Eule ec o ields (i.e., o ype X == L Wi Xi 8x i wi h ui; 2:: 0 and x( ) == c . o some c EC {O}). C i e ion 3.1 applies o all o hem, so LeT holds. The examples in ou lis ha e been selec ed because hey belong o a amily ha we a e abou o de ine. whe e O(*D) is he shea o me omo phic unc ions wi h poles along D and V. -} is i s sub-V-module gene a ed (locally) by he me omo phic unc ion 1. So we ob ain DR(Ann~(17 )) ~ DR(O(*D)) == ne(*D) as i is clea ha V· -} ~ Ann~(l/ ). The c ucial s ep is hen he compa ison be- ween he le ideals Anng) (1/ ) and Ann (I/ ): i Anng) (1/ ) == Ann (I/ ) we ob ain ha DR(M1ogD) == DR (V ) Anng)(1/ ) = DR (Ann;1/!)) ~ ne(*D). D De ini ion 3.3. We will say ha adi iso DcXis weakly locally quasi-homogeneous i o all xED he e a e local coo dina es (Zl, ... ,zn) on X, cen e ed a x, wi h espec o which D has ade ining equa ion h(Zl, ... ,zn) E0 such ha h(z 1, ••• ,z~n) is homogeneous o s ic ly pos- i i e weigh wi h he weigh s ui, posi i e o (no all) ze o. This is he case o ou examples in Figu e 1. They ha e weigh 0 o he a iable Z and he singula locus is he z-axis. I is clea ha he change Z ~ Z+Q (QE C) p oduces a equa ion ha admi s he same se o weigh s. We ha e ound many mo e examples o weakly locally quasi-homogeneous di iso s, and i - is appa en ha hey always e i y LCT. We da e o p opose he " nex conjec u e based on his expe imen al e idence. Cas o-Jimenezand Ucha-En quez: Tes ing he Loga i hmic Compa ison Theo em o F eeDi iso s 445 (Local) Equa ion Eule ope a o Global b- unc ion b xy(x + y)(xz +y) x8 x +y8 y(8 + 3/4){8 + 1/2)(8 + 5/4){8 + 1)3 xy(x + y)(x - y)(xz +y) x8 x +y8 y(8 + 1)3(8 + 6/5){8 + 3/5){8 + 2/5)(8 + 4/5) y(x 2+ y)(x 2z +y) x8 x+ 2y8 y(8 + 4/3){8 + 5/6){8 + 1)3 (8 o 1/2){8 + 2/3){8 + 7/6) (xz + y)(x 3_y3) x8 x +y8 y(8 + 3/4){8 + 1/2){8 + 5/4)(8 +1)3 (xz + y)(x 4 _y4) x8 x +y8 y(8 + 1)3(8 + 6/5){8 + 3/5){8 + 2/5){8 + 4/5) (xz + y)(x 7_y7) x8 x +y8 y(8 + 1/2){8 + 7/8){8 + 9/8){8 + 5/8) (8 + 3/4)(8 +1)3(8 + 3/8){8 + 1/4) (8 + 20/17)(8 + 7/17)(8 + 23/17)(8 + 16/17) xy(x 2+y3)(x2z +y3) 3x8 x+ 2y8 y(8 + 13/17){8 + 5/17){8 + 14/17){8 + 12/17) (8 + 10/17)(8 + 18/17){8 + 11/17)(8 + 1)3 (8 + 15/17){8 + 8/17){8 +21/17)(8 + 9/17){8 + 19/17) FIGURE 1. F ee di iso s ha e i y LeT. Conjec u e 3.4. I D is a weakly locally quasi- homogeneous ee di iso , hen LeT holds. P o ing LCT holds o locally quasi-homogeneous ee di iso s, makes s ong use o he ac ha such di iso s ha e a s uc u e o an analy ical p oduc a ound any poin o he di iso . Ou examples do no ha e his p op- e y, so he p oo o Conjec u e 3.4 has o be mo e sub le. We hink ha one o he ways in which he hypo heses o he iocally quasi-homogeneous case could be elaxed is ha some o he weigh s can be ze o. The condi ion on he exis ence o a local weigh ed equa ion seems o be mo e complex o elax; in he nex sec ion we will gi e examples o ee di iso s ha a e de ined by weigh ed ho- mogeneous equa ions (wi h all he weigh s posi i e) bu do no e i y LCT. Thus, by [Cas o e al. 96], hey a e no locally quasi-homogeneous (see Rema k 5.8). 4. HOW TO DEDUCE THAT LCT DOES NOT HOLD We shall s a e he e a c i e ion gi ing a su icien condi ion o deduce ha LCT does no hold o a gi en ee di iso . This c i e ion is he con e se o C i e ion 3.1. C i e ion 4.1. I Dis a Spence di iso , hen Ann (l/ ) =1= Anng) (l/ ) => LCT does no hold o D. P oo : Le us conside he na u al sequence o V-module mo phism's o----+ K----+ M10g D ~ O( *D) whe e <PD(P) = P(l) o each PE M10g D and Kis he ke nel o <PD, 'We ha e K- Ann (l/ ) -Anng) (1/ ) · Suppose LCT holds o D. Then he inclusion map io : n-(logD) ~ n-(*D) = DR(O(*D)) is a quasi- isomo phism. On he o he hand, since Dis Spence , we ha e DR(MIOgD)~n-.(logD) (see [Cas o and Ucha 02, Theo em 4.3]). Then he composi ion mo phism 446 Expe imen al Ma hema ics, Vol. 13(2004), No.4 is a quasi-isomo phism. We ha e ha 'l/J == DR(¢D) by [Calde on and Na aez 04]; hus, by P oposi ion 4.2, ¢D mus be an isomo phism, con adic ing ha K i= (0). D P oposi ion 4.2. [Mebkhou 89, Chap e II, Theo em 4.1.5] Suppose ¢ : M--+ M' is ~ mo phism o holonomic V-modules. I DR(¢) : DR(M) --+ DR(M') is a quasi- isomo phism, hen ¢is an isomo phism. P oo : Taking he ke nel and coke nel o ¢, i is enough o show ha i Mis a holonomic V-module such ha DR(M) == 0, hen M == o. So, suppose we ha e DR(M) == 0 o a holonomic V-module M. By [Mebkhou 89, page 41] we ha e Sol(M*) ~ DR(M) and hen Sol(M*) == o. By applying [Mebkhou 89, Chap e II, Theo em 4.1.5], we ge VOO ® M* == 0 whe e VOO is he ing o linea di e en ial ope a o s o in ini e o de . Since VOO is ai h ully la o e V(see [Sa o e al. 73, Theo em 3.4.1.]), we ge M* == 0 and hen M == o. D C i e ia 3.1 and 4.1 gi e a necessa y and su icien con- di ion o decide i LCT holds o a Spence ee di iso . In pa icula , we ha e p o ed Conjec u e 1.1 o Spence di iso s: Theo em 4.3. Le D be a Spence ee di iso . LeT holds o he ge m (D, x) i and only i he annihila o ideal Ann x(1I ) is gene a ed by elemen s o o de 1, whe e is a local equa ion o he ge m (D, x). 5. ON THE REGULARITY OF LOGARITHMIC V-MODULES To apply C i e ion 4.1 we ha e chosen an al e na i e way ha , a he same ime, ea s he ollowing in e es ing p oblem: P oblem 5.1. A e he loga i hmic V-modules M10g D and M10g D egula holonomic o any ee di iso D? The answe is yes o plane cu es (see [Ucha 99] and [Cas o and Ucha 01]) and o all he examples we ha e s udied, including non-Eule homogeneous ex- amples. Mo eo e , o any Spence di iso D, since (M1og D)* is isomo phic o M10g D(see Theo em 2.4), he egula i y o M10g Dis equi alen o he one o M10g D. The module O(*D) is egula holonomic (see, o ex- ample, [Mebkhou 89, Chap. II, Th. 2.2.4]) so i is enough, in o de o p o e he egula i y o M10g D, o p o e he same p ope y o he ke nel Ko he na u al map M10g D ~ O(*D). P o ing he egula i y o a V-module, compu a ion- ally, is a e y di icul p oblem in gene al. We ha e ound many ac able examples due o a iendly p esen a ion o K ha we ha e ob ained o ou examples in Figu e 2. These p oposed ee di iso s a e pa icula cases o he amily {Dp,q == ((xz + y)(x p-yq) == 0) CC3} o p, q EN. They a e Koszul ee'', and he e o e Spence . C i e ion 4.1 applies, so hey do no e i y LCT. I is in- e es ing o poin ou ha e e y elemen o he amily ad- mi s an Eule ec o ield E == qx8 x +py8 y+ (p- q)z8 zE De ( -log Dp,q) wi h s ic ly posi i e weigh s o all he a iables when p>q. This ac means ha he de ining equa ion is a weigh ed homogeneous polynomial. Rema k 5.2. I would be in e es ing o desc ibe he b- unc ions o all he elemen s o he amily. The oo s seems o ollow a pa e n ela ed o he weigh s o he a iables, in a way somewha analogous o he isola ed (quasi-homogeneous) singula i y case. Le us explain how we ha e s udied he egula i y o he ke nel K o he di iso s o he amily {Dp,q == (xz + y)(x p-yq) == O)}. To ob ain a p esen a ion o he quo ien K == Ann (11 ) Anng)(ll ) he ollowing p ocedu e is well known: 1. Ge a se o gene a o s {9I, , 9 } o Ann (1I ) and a se o gene a o s {lI, , ls} o Anng) (II )· 2. Compu e he V-module So syzygies among 91, ... ,9 , lI, ... ,ls· 3. Fo e e y gene a o SE +s o S dele e i s las s componen s o ob ain SED", In his way, i S == (SI, ... ,S ) hen we ha e K~ __ __; (SI, ... ,S ) ha is, using a ma ix o ows and columns (each column is a gene a o Si). As soon as p and qg ow o Dp,q ( o p, q 2: 8), he compu a ions o annihila o s, b- unc ions, and ke nels be- come huge and he examples in ac able wi h he imple- men a ion we ha e used. Howe e , in he ac able cases 3An a gumen abou he dimension o he cha ac e is ic a ie y o M10g Dp,q can be used o he whole amily. Cas o-Jimenezand Ucha-En quez: Tes ing he Loga i hmic Compa ison Theo em o F eeDi iso s 447 (Local) Equa ion Global b- unc ion bl (xz + y)(x 4_ y3) (8 + 19/15)(8 + 2/3}(8 + 1/2)(8 + 1)3(8 + 13/15) (8 + 17/15)(8 + 4/3)(8 + 16/15)(8 + 14/15)(8 + 5/4) (s + 7/15)(s + 3/4)(8 + 8/15)(s + 11/15) (xz + y)(x 5_ y3) (s + 23/18)(s + 11/18)(s + 19/18){8 + 1/2)(s + 8/9) (8 + 5/4)(s + 5/9)(s + 17/18)(8 + 13/18)(8 +1)3 (s + 25/18)(8 + 3/4){s + 7/9)(s + 11/9)(s + 4/9)(8 + 10/9) (8 + 17/24)(s + 4/3)(8 + 2/3)(s + 23/24){s + 19/24) (XZ + y)(x 7_ y3) (8 + 13/24){s + 5/4){s + 5/12)(8 + 29/24)(s + 25/24) (s + 1)3(s + 7/12)(s + 7/6)(8 + 1/2)(s + 31/24) (s + 5/6)(s + 11/12){s + 35/24)(8 + 3/4)(8 + 13/12) (xz + y)(x 3 .- y4) (s + 13/15)(s + 11/15)(8 + 16/15)(s + 19/15)(8 +1)2 (s + 17/15){s + 2/3)(8 + 4/3)(s + 8/15)(s + 7/15)(8 + 14/15) (XZ + y)(x 3_ y5) (s + 4/9)(8 + 23/18)(s + 25/18)(8 + 13/18)(8 + 11/9) (s + 10/9)(8 + 19/18)(s + 1)2(8 + 17/18) (s + 7/9)(8 + 8/9){8 + 5/9)(s + 11/18) FIGURE 2. F ee di iso s ha do no e i y LeT. he ke nels u n ou o be ep esen ed by ma ices wi h a special s uc u e. We ha e used [G ayson e al. 99] and [No o e al. 00]4. Example 5.3. In he case p=4 and q=3, he ke nel K can be ep esen ed (using some elemen a y simpli ica- ions) by he ma ix whe e P= -360;, 48 2 2 3204 2216 Q= -Sz OyOZ +125 zOxoz -25Z0yOZ 17397 417 + '1"25 0xoz -1250y· ( Xy zOz +8 o0 0 o0 0 o0 0 ~Ox 1 o o -1 960 ; P) o0 10' o1 Lemma 5.5. I he V-module Kis ep esen ed by he ma ix Example 5.4. In he case p=5 and q=3, he ke nel can be ep esen ed by M= o0 o0 o0 o o o 1 0 o1 o0 o o 1 ( x~ yzoz +5P Q) o0 1 0 , o0 0 1 4Mo e p ecisely, we ha e compu ed Ann (l/ ) compu ing he annihila o o /8 in 1'[8] wi h he command AnnFs o [G ayson e al. 99] and eplaced sby -1, p o ided ha Risa/ Asi (wi h he command b c ) has shown ha he smalles in ege oo o b/ is -1. o any a ECand PI, ... .P; E V, hen K ~ V/(x, y, zoz +a). P oo : I is enough. o de ine he isomo phism cp V + 1/N ~ V/(x,y,zoz +a), 448 Expe imen al Ma hema ics, Vol. 13(2004), No.4 whe e Nis he submodule gene a ed by he columns o M. I is de ined as ollows: and 81, ... ,8nis he dual basis o De ( -log D), hen and iI n-1(V 0,Oe n)n ~ iI n-1(V 0,Ol (log D )). and i s in e se maps I o e1. o The mo phism d1 can be ead now as Lemma 5.6. I he V-module Kadmi s ap esen a ion as in Lemma 5.5, hen i is egula holonomic. iI n-1 (V 0, Oe n) [g] iI n-1(V 0, Oe n)n ([81.g], ... , [8 n.g]) P oo : Ob ious om he ep esen a ion we ha e ob ained o K ~ V/(x, y, z8 z+a), clea ly a egula holonomic V- module. 0 Theo em 5.7. The elemen s o he amily {Dp,q} wi h ke nels as in Lemma 5.5 do no e i y LeT. In addi ion, he co esponding V-modules M10g Dp ,q and M10g Dp,q a e egula holonomic. P oo : Simply apply C i e ion 4.1; he co esponding ke - nels a e no null so LCT does no hold. The egula i y is a consequence o Lemma 5.6. 0 Rema k 5.8. In [Calde on and Na aez 02, P oblem 6.5] he au ho s ask whe he a ee di iso de ined by a quasi- homogeneous polynomial (wi h s ic ly posi i e weigh s) is locally quasi-homogeneous. The answe o his ques- ion is nega i e: by Theo em 5.7 he i s h ee examples p o ided in Figu e 2 a e ee di iso s de ined by quasi- homogeneous polynomials (wi h s ic ly posi i e weigh s) ha do no e i y LCT. Since locally quasi-homogeneous ee di iso s e i y LCT (see [Cas o e al. 96]), hese di- iso s a e no locally quasi-homogeneous. We hink ha e e y elemen o he amily wi h p>q(lcm(p, q) =1) a e in he same si ua ion, bu we don' ha e a p oo o his gene al esul . In [Cas o e al. 96] i was no ed ha i LCT holds hen . he mo phism is injec i e,whe e Vis a S ein neighbo hood (su icien ly small) o O. In dimension 2, his condi ion is equi alen o LCT (see [Calde on e al. 02]). The examples o Figu e 2 show ha he condi ion on d1is no su icien in dimension g ea e han 2. I {W1,.'.'W n}is a ee basis o n1(log D) as O -module The space iI n-1(V O, Oe n)is isomo phic o he space So Lau en se ies, con e gen o all ~ = (Xl,".' Xn) wi h ~ =1= 0 and whose nonze o coe icien s a e hose wi h s ic ly nega i e indices in all a iables. I is clea ha i he e exis s an elemen 81= L~=l WiXi8i wi h all he ui; 20 in he basis o De ( -log D) i ollows ha d1is injec i e, as in he i s h ee examples o Figu e 2. 6. CONCLUSIONS Many examples ha e been ea ed wi h he explici me h- ods we ha e p oposed in his wo k o s udy he Loga i h- mic Compa ison Theo em (LCT) on ee di iso s. The esul s on he weakly locally quasi-homogeneous exam- ples we ha e ied has lead us o conjec u e ha hey e i y LCT. As a consequence, hey would be Spence di iso s. We ha e p o ed ha LCT holds o a Spence ee di- iso D == ( =0) i and only i Ann (I/ ) is gene a ed by di e en ial ope a o s o o de 1. On he o he hand, we ha e gi en examples ha an- swe a ques ion p oposed in [Calde on and Na aez 02]: whe he he e exis ee di iso s de ined by weigh ed homogeneous polynomials ha a e no locally quasi- homogeneous. We ha e p o ed ha o hese examples he loga i hmic V-modules a e egula . ACKNOWLEDGMENTS Bo h au ho s we e pa ially suppo ed by BFM-2001-3164 and FQM-333. We a e e y g a e ul o P o esso s L. Na aez- Maca o and Z. Mebkhou o hei use ul commen s. Du ing he p epa a ion o he inal e sion o his wo k, he i s au- ho was is ing he Ecole No male Supe ieu e (Pa is). He is g a e ul o he Depa e nen de Ma hema iques e Applica- ions o i s hospi ali y.