Dedekind sums and the signature of f(x, y)+z N,II

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1 Sel. mth., New se. 5 (1999) /99/ $ /0 c Bikhäuse Velg, Bsel, 1999 Select Mthemtic, New Seies Dedekind sums nd the signtue o (x, y)+z N,II Andás Némethi Mthemtics Subject Clssiiction (1991). 32S. Key wods. Milno ibe, signtue, et invint, Dedekind sums, singulities. 1. Intoduction The pesent ppe hs two gols. Fist, we compute the signtue o the Milno ibe ssocited with singulity o type (x, y)+z N, nd we pove some conjectues o A. Duee. A. Duee in [4] listed some conjectues bout the signtue σ, the Milno numbe µ, nd the geometic genus p g o n isolted complete intesection singulity g :(C k+2, 0) (C k, 0). Nmely, he conjectued tht p g µ/6 (Conjectue 5.3 in [4], in the sequel ( )), which is equivlent to 3σ µ +3µ 0 (whee µ 0 is the nk o the kenel o the Milno lttice). In pticul ( ) implies weke conjectue (5.2 in [4]): the negtivity o the signtue. On the othe hnd, J. Whl in [17] constucted smoothing o non-complete intesection with positive signtue, showing the subtility o the poblem. But, even in the hypesuce cse, the conjectues still esist pesistent ttempts t poo (see [22]). In seies o ppes, the utho studied suspensions o hypesuce singulities (i.e., gems o type (x 1,...,x n )+z N n+1). In pticul, in [7], the signtue o +z N is computed in tems o et-invints η(; N) ssocited with (the vition stuctue o) nd the numbe N : σ( + z N )=η(;n) N η(; 1). In [10], the et-invint η(; N) o plne cuve singulity is expessed in tems o genelized Dedekind sums ssocited with the embedded esolution gph G o (we ecll this hee in 2.3). This is poweul eltion; o exmple, in the pticul cse o Bieskon polynomils, it is equivlent to the computtion o the lttice points in tethedon in tems o Dedekind sums, solved in pticul cses by Modell [6], nd in genel by Pommesheim [13]. This omul genelizes esult (Poposition (2.5) [12]) o W. Neumnn nd J. Whl, whee the signtue is computed when the link o g = + z N is n intege homology sphee. Using this

2 162 A. Némethi Sel. mth., New se. eltion (2.3) in [10] we pove tht o n ieducible gem, its et-invint nd the signtue o + z N e dditive with espect to the splice decomposition o the gph G. In pticul, we pove tht the inequlity ( ) is vlid o these type o gems. Unotuntely, dditivity esults s in [10] (o s in [12]) e not tue i is not ieducible. In this ppe, by dieent method, we ttck the sme poblem o bity. We pove ( )o+z N with educible nd with the ollowing dditionl estiction: gcd(m w,n) = 1 o ll multiplicities m w o the ieducible exceptionl divisos in the miniml embedded esolution o. Actully, we povide lge list o inequlities: we compe the signtue with the Milno numbe µ o (topologicl inequlity), with the multiplicity ν o (lgebic inequlity) nd the size o G (combintoil inequlity) (c. section 5). (Using ny o these inequlities one cn pove the negtivity o σ.) The method is the ollowing. We stt with the eltion which descibes the signtue in tems o Dedekind sums (c. 2.3). Using some popeties o these Dedekind sums, in section 3 we tnsom this eltion in some inequlities expessed in tems o the combintoics o the embedded esolution gph G o. These inequlities e veiied in section 4. Now we come to ou second point. This is less pecise but moe impotnt (o the utho). We e seching o n nswe to the ollowing question: wht is specil in the hypesuce cse? Why does the Milno ibe o hypesuce hve negtive signtue nd non-hypesuce mybe not? Fom the pesent ppe we len tht the nswe is in the pticul om o the embedded esolutions o hypesuce singulities. In the two dimensionl cse (i.e., when :(X, x) (C, 0), with (X, x) noml suce singulity) we cn be moe pecise: ll the combintoil eltions (which imply ou inequlities) poved in section 4 o plne cuve singulities (i.e., with X smooth) distinguish these hypesuce gems om the genel clss o gems o unctions deined on noml suce singulity (c. 4.25). The utho expects tht simil eltions e vlid in highe dimension, nd they e esponsible o the pticul behviou o hypesuce singulities. Something moe bout the eltion σ( + z N )=η(;n) Nη(; 1). Since η(; N) is peiodic in N, lim n (σ( + z N )/N )= η(; 1), nd ny estimte o η(; 1) will povide inequlities o σ when N is lge. In pticul, ny inequlity o type σ( + z N ) c N (c constnt) with c> η(; 1) is wek o lge N, but cn be vey diicult o smll N, becuse o the vey iegul behviou o η(; N) (i.e., o the Dedekind sums). Moeove, ny inequlity o type σ( + z N ) c I, (whee I = µ,oν o ny invint o ), cn be vey wek o with I lge, but vey shp o gems t the beginning o the clssiiction list. Fo gems with I smll, in ode to veiy n inequlity s bove, the peiodic tem η(; N) must be elly computed. This explins the om o ou inequlities in section 4 (nd the diiculties in thei poos). Finlly, we mention the ppes o Y. Xu nd S. S.-T. Yu [19], [20], whee

3 Vol. 5 (1999) Dedekind sums 163 they veiied the inequlity ( ) (ctully, even stonge vesion) in the cse o qusi-homogeneous gems g :(C 3,0) (C, 0). 2. σ N in tems o the embedded esolution gph o In this section we intoduce some nottions nd we ecll one o the min esults o [10]. Let :(C 2,0) (C, 0) be gem o n nlytic unction which deines n isolted singulity t the oigin. We conside n embedded esolution φ :(Y,D) (C 2, 1 (0)) o ( 1 (0), 0) (C 2, 0) (hee D = φ 1 (0))). Let E = φ 1 (0) be the exceptionl diviso nd let E = E w be its decomposition in ieducible divisos. I = is the ieducible decomposition o, thend=e S,wheeS is the stict tnsom o 1(0). Let G be the esolution gph o, i.e., its vetices V = W A consist o the non-owhed vetices W (coesponding to the ieducible exceptionl divisos), nd owhed vetices A (coesponding to the stict tnsom divisos o D). We will ssume tht no ieducible exceptionl diviso hs n utointesection nd W 0. Itwo ieducible divisos coesponding to v 1,v 2 Vhve n intesection point, then (v 1,v 2 )(=(v 2,v 1 )) is n edge o G. The set o edges is denoted by E. Since G is tee, one hs #W +#A=#E+1. (2.1) Fo ny w W,wedenotebyV w the set o vetices v V djcent to w. The gph G is decoted by the sel-intesection (o Eule) numbes e w := E w E w o ny w W. Fo ny v V,letm v be the multiplicity o φ long the ieducible diviso coesponding to v. In pticul, o ny A, one hs m = 1. The multiplicities stisy the ollowing eltions. Fo ny w W one hs e w m w + m v =0. (2.2) v V w These eltions detemine the multiplicities {m w } in tems o the sel-intesection numbes {e w } w. It is convenient to use the ollowing nottions: () o ny w W, we deine M w := gcd(m w,m v1,...,m vt ), whee V w = {v 1,...,v t };nd (b) o ny e =(v 1,v 2 ) E, we deine m e := gcd(m v1,m v2 ). Fo ny A, thee exists exctly one w W such tht (, w ) E. With these nottions one hs (c. lso [12], Poposition 2.5).

4 164 A. Némethi Sel. mth., New se Theoem. [10] Let :(C 2,0) (C, 0) be n isolted plne cuve singulity s bove. Then the signtue σ N o the Milno ibe o the suspension (x, y)+z N is σ N = η(; N) N η(;1), whee η(; K) =#A 1+ ( (K, me ) 1 ) ( (K, Mw ) 1 ) + e E +4 v V w m w (( kmv k=1 m w )) Notice lso tht K η(; K) is peiodic unction. (( kk m w )). 3. σ N vi the ecipocity lw o Dedekind sums In this section we will use the geneliztion o the ecipocity lw o Dedekind given by Rdemche [14], [21] in ode to ewite theoem (2.3). In the new expession ll the Dedekind sums will hve the intege N in the denominto. We septe the needed nottions nd cts bout the Dedekind sums in the ppendix. We stt with the ollowing esy emk. I g : V V Ris n bity unction, then g(w, v)+ g(, w )= ( ) g(u, v)+g(v, u). (3.1) v V w Since s(m w,n;m ) = 0 o ny A(becuse m = 1), (3.1) gives 4 [ s(m v,n;m w )=4 s(mv,n;m u )+s(m u,n;m v ) ]. (3.2) v V w e= Using the ecipocity lw (A.2), this expession is equl to [ 4s(m u,m v ;N) (N,m e )+ N2 m 2 e +m2 u (N,m v) 2 + m 2 v (N,m u) 2 ]. m u m v Using this equlity, the omul o the et-invint η(; N) given in (2.3) cn be tnsomed in η(; N) =#A 1 #E ( (N,Mw ) 1 ) 4 s(m u,m v ;N)+ N m 2 e m u m v m u (N,m v ) 2 + m (3.3) v(n,m u ) 2. m v m u

5 Vol. 5 (1999) Dedekind sums 165 Fo the lst sum we pply (3.1), (2.2) (nd m = 1), hence it cn be eplced by m v (N,m w )2 + m w (N,m ) 2 = ( e w )(N,m w ) 2 + m w. m w m v V w Using (2.1) nd the bove identity, (3.3) eds η(; N) = #W ( (N,Mw ) 1 ) 4 s(m u,m v ;N)+ [ N m 2 e + 1 w )(N,m w ) 3 m u m v ( e 2 + ] (3.4) m w. The peiodicity o η(; N) nd (3.4) pplied o N =1povide: 3.5. Theoem. [ ] σ N lim N N = η(;1)= #W + 1 m 2 e + 1 w 3 m u m v 3 ( e )+ m w. The expession σ N = η(; N) N η(; 1) o Theoem (2.3) tnsoms into the ollowing omul: 3.6. Theoem. σ N =(N 1) #W ( (N,Mw ) 1 ) 4 s(m u,m w ;N)+ 1 ( e w ) [ (N,m w ) 2 N 2] + 1 N2 m w. It is inteesting to conside some pticul cses: 3.7. Coolly. ) Assume tht (N,m w )=1o ny w W.Then σ N =(N 1) #W 4 s(m u,m v ;N)+ b) Assume tht N =2.Then 1 N 2 [ ( e w )+ σ 2 =# { } 1 :2 M w 2 ( e w )+ 2 m w m w ] m w..

6 166 A. Némethi Sel. mth., New se. Now, o (3.7.) we cn pply the inequlity (A.3) nd (2.1), nd we obtin: 3.8. Coolly. Assume tht (N,m w )=1o ny w W.Then [ σ N 1 N ( ) 1 N 2 #W+#E + 4#W #A+1+ ( e w )+ ] m w. N We would like to hve simil inequlity o bity N. We stt with the ollowing lemm: 3.9. Lemm. Thee exists one-to-one unction l : W Esuch tht l(w) = (w, v) o some v V. The poo is esy nd is let to the ede. Notice tht (A.4) povides o l(w) =(w, v) 4 s(m w,m v ;N)+ 1 ( (N,mw ) 2 N 2) 0. (3.10) Theeoe, by (3.10) nd (A.3) one hs 4 4 (u,v)/ l(w) s(m u,m v ;N)+ 1 ( (N,mv ) 2 N 2) s(m u,m v ;N) (N 1)(N 2) This inequlity nd (3.6) give: Coolly. Fo ny N the ollowing inequlity holds: σ N + 1 N N ( (N,Mw ) 1 ) + 1 #E + 1 N2 This inequlity simpliied eds s: Coolly. ) Fo ny N 1 (#A 1). ( e w 1) [ (N,m w ) 2 N 2] [ 3#W #A+1+ m w ]. σ N 1 N N #E + 1 N 2 [ 3#W #A+1+ m w ].

7 Vol. 5 (1999) Dedekind sums 167 b) I N 1 B() := [ ( e w 1)(m 2 w 1) ] /(3#E) then: σ N 1 N 2 [ 4#W #A+1+ ( e w )+ m w ]. Poo. Use e w 1 0, nd (N,m w ) 2 N 2 (m 2 w 1+(1 N 2 ). All ou inequlities ((3.8), (3.12), c. lso (3.5) nd (3.7.b)) give estimtes o type ( σ N )/(N 1) {combintoil expession in tems o G }. In the next section we will study these expessions. 4. Inequlities stisied by the miniml esolution gph o I. Peliminies Let :(C 2,0) (C, 0) be n isolted plne cuve singulity. Let G be its miniml embedded esolution, with the convention tht #W, (which in the sequel will be shotly clled esolution). We will keep the nottions o section 2 o the numeicl invints o G. Recll tht, o ny ow A,w Wis the unique vetex djcent with, ndm w is its multiplicity. In this section we will pesent some popeties o M := m w #W(G ) ( =2,3,4), which distinguish the plne cuve singulities mong the singul gems : (X, x) (C, 0) deined on noml suce singulity (X, x) (c. 4.32). We will compe M with the Milno numbe µ o (topologicl inequlity), with the multiplicity ν o (lgebic inequlity), nd with the size o G (combintoil inequlity). It is convenient to use the ollowing nottion: i two gems nd g hve the sme topologicl type, in pticul, the sme miniml embedded esolution gph G = G g,thenwewite g. (Notice tht cn hve lge modulity, but G depends only on the topologicl embedding ( 1 (0), 0) (C 2, 0).) We ecll the stuctue o G when is ieducible. Assume tht hs Newton pis (p i,q i ) s i=1 (p i 2, q i 1, q 1 >p 1 ). The miniml embedded esolution gph o y(x pi + y qi )(q i 1, p i 2) hs the ollowing om:

8 168 A. Némethi Sel. mth., New se. (y =0)...,u 1 i,u ti i,1. -,v i i,v 2 i (x pi +y qi =0),v 1 i whee u l i nd vl i (u0 i,v0 i 1, nd ul i,vl i 2ol>0) e given by the continuous ctions: p i q i = u 0 i 1 u 1 i u ti i ; q i = vi 0 p 1 i vi v i i. The gph G cn be econstucted (by splicing) om the gphs o y(x pi + y qi ) nd the numbes u 0 i s ollows ([5] Appendix o chp. 1, nd section 22):,u 1 1,u t1 s,1,u 0 s,1,u 1,u ts,1 s s ,u 0 2,1,u 1 2,u ts,1.,v 1 1,v 2 1.,v s,1 s,1,v 2 s,1. -,v s s,v 2 s,v 1 1,v 1 s,1,v 1 s In pticul I q i =1,then s #W(G )= #W(G y(x p i +y q i ) ). (4.1) i=1 #W(G y(x p i+y1 ))=p i. (4.2) Fo q i 2 we hve the ollowing estimte: 4.3. Lemm. Assume tht x p + y q o y(x p + y q ),with(p, q) =1,p 2, q 2.Then #W(G ) (p+q+1)/2. Poo. Fist notice tht G x p +y q nd G y(x p +y q ) die only by n ow, so the numbe o thei vetices gees. We will use the induction ove the pi (p, q). Assume tht q>p.iq=p+1,then 2#W(G )=p+q+ 1. Assume q p +2. Conside the blowing up (u, v) φ (x, y) =(uv, u). Then = φ = u p (v p +u q p ).

9 Vol. 5 (1999) Dedekind sums 169 Now, by the inductive step nd the bove emk: #W(G ) 1+(p+q p+1)/2 (p + q +1)/2. The unique ow o G coesponds to A = {}, nd whee s cn be computed inductively s ollows: m w = s p s, (4.4) 1 = q 1, nd o i 1: i+1 = q i+1 + p i p i+1 i. (4.5) Now, ssume tht is not ieducible: =,whee s e ieducible nd #A > 1(A := A(G )). It is convenient to ix n odeing o the set A. In the next lemms, we would like to compe the gphs G nd G s. Recll tht o ny gem g, G denotes its miniml embedded esolution gph. I we wnt to emphsize tht cetin invint is consideed in gph G, then we put G in penthesis ne the coesponding invint. I E is one o the ieducible exceptionl divisos o the embedded esolution φ :(Y,D) (C 2, 1 (0)) o 1 (0), nd g :(C 2,0) (C, 0) is ny gem, then m E (g, G ) denotes the multiplicity o g φ long E. In pticul, i E coesponds to w Winthe gph G,thenm E (,G )=m w (G )=m w. I (, A ), then (c. e.g. [3]): m E ( )=ν(, ), whee ν(, ) denotes the intesection multiplicity t the oigin. With these nottions one hs m w (G )= m E (,G )+2 ν(, ). (4.6) < Let G denote the miniml esolution gph o. By ou convention m w (G ) is the multiplicity o long the ieducible diviso (in G ) which intesects the stict tnsom o =0. We sy tht two gems e tngent i thei tngent cones t the oigin hve common line Lemm. Assume tht = g 0 nd = g 0 such tht 0 0 e topologiclly equivlent ieducible gems, nd 0 nd g e not tngent. Then () m E (,G ) 2#W(G ) m E (,G ) 2#W(G g) 2#W(G 0 )+2+2T +T,

10 170 A. Némethi Sel. mth., New se. (b) m E (,G )+2ν(g, 0 ) #W(G ) m E (,G )+2ν(g)ν( 0 ) #W(G g ) #W(G 0 )++(+2)T +2T, whee =3o 4, ndt =T =0, unless 0 is non-smooth (esp. smooth) component tngent with non-smooth component o g, in which cse T =1 (esp. T =1). Poo. Let φ 1 :(Y,E) (C 2,g 1 (0)) be the miniml esolution o g 1 (0) with exceptionl diviso E (nd W ). Let X 0 be the stict tnsom o 0 1 (0) vi φ 1. Let φ 2 be the miniml esolution o (Y,E X 0 )tthepointe X 0. Notice tht φ 2 cn be consideed s pt o the miniml esolution towe o (C 2,0 1 (0)). Set φ = φ 1 φ 2,ndletZ 0 (esp. Z ) be the stict tnsom o { 0 =0}(esp. o the ieducible component {g =0}o g) viφ. Assume tht Z 0 Z = o ny A g. Fist notice tht φ povides G, φ 1 povides G g, nd the numbe o ieducible exceptionl divisos o φ 2 is #W(G 0 ) 1. Now we clim tht the ollowing inequlities hold (with k =0): (i) m E (,G ) m E (,G )+T +2k (ii) #W(G ) #W(G g )+#W(G 0 ) 1 T +k (iii) ν(g, 0 ) ν(g)ν( 0 )+T +T +k. Indeed, in genel m E (,G ) m E (,G ), but i 0 is smooth tngent with g, thenm E ( 0,G ) = 1. But m E 0 ( 0,G ) 2, which gives (i). 0 Fo (ii) notice tht #W(G ) #W(G g )+#W(G 0 ) 1 (see bove), but i two non-smooth components e tngent, then they hve t lest two common ininitely ne points, so the miniml esolution gph stisies (ii); (iii) is obvious. Now ssume tht Z 0 Z = P o some A g. Let D be the ieducible exceptionl diviso o φ with P D. I the smooth gems Z 0 nd Z hve contct k (k 1), then we need k moe blow ups. The lst newly ceted exceptionl diviso is denoted by D. Then we clim tht the inequlities (i) (iii) e vlid. Indeed: m D (g )=m D (g )+k nd m D ( 0 )=m D ( 0 )+k, which poves (i), (ii) ollows by the sme gument s bove, nd inlly, (iii) ollows, o exmple, om Mx Noethe s theoem (c. e.g., [3]) which descibes the intesection multiplicity in tems o the multiplicity sequence. Finlly, notice tht (i) (iii) implies the lemm.

11 Vol. 5 (1999) Dedekind sums Lemm. Let = be the ieducible decomposition o. Then () m E (,G ) 2#W (G ) [ mw (G ) 2#W(G ) ] +2(#A 1) + 2T + T, (b) m w (G ) #W(G ) [ mw (G ) #W(G ) ] + (#A 1)+ 2 < ν( ) ν( )+(+2)T +2T, whee =3o 4, ndt=t =0unless hs non-smooth ieducible component tngent to some othe non-smooth (esp. smooth) component, in which cse T =1(esp. T =1.) Poo. The poo is ove induction. We eplce step by step ll the ieducible components o (stting with the smooth ones) by such tht nd the tngent cone o is in geneic position with espect to the tngent cone o the othe components. The inductive step is povided by (4.7). With the nottion o (4.7), notice tht #W(G )=#W(G g )+#W(G 0 ) 1. (4.9) Theeoe, (4.7.) eds m E (,G ) 2#W(G ) m E (,G ) 2#W(G )+2T +T. (4.10) Moeove, using (4.6) nd (4.9), (4.7.b) eds m w (G ) #W(G ) m w (G ) #W(G )+(+2)T +2T. (4.11) So induction cn be pplied. Finlly, notice tht i the components o hve ll distinct tngent cones, then ν(, )=ν( ) ν( ), m E (,G )=m w (G ), nd #W(G ) = #W(G ) (#A 1). Hence the lemm ollows (gin by 4.6).

12 172 A. Némethi Sel. mth., New se. II. Exmples We e inteested in the ollowing expessions: E top := m w 2#W(G ) #A +1 (µ 2), E lg := m w 3#W(G ) (ν 2 2ν 3), E com := m w 4#W(G ) #A +1. The ollowing tble gives these invints o the ollowing plne cuve singulities: x (i.e. smooth), x 2 + y 2k+1,(A 2k,k 1), x 3 + y 3k+1, (E 6k, k 1), x 3 + y 3k+2,(E 6k+2, k 1), x 2 + y 2k,(A 2k 1,k 1), (x 2 + y 2k+1 )(y 2 + x 2l+1 ), (A 2k,2l, k 1, l 1), y(x 2 + y 2k+1 ), (D 2k+3, k 1). #A m w #W E top E lg E com ( ew ) smooth A 2k 1 2(2k +1) k+2 0 k 1 6 2k+4 E 6k 1 3(3k +1) k+3 k 1 6k 6 5k 9 2k+7 E 6k+2 1 3(3k +2) k+3 k 6k 3 5k 6 2k+7 A 2k 1 2 4k k 2 k+3 1 2k 1 A 2k,2l 2 4l +4k+12 k+l+3 0 k+l+2 1 2k+2l+7 D 2k+3 2 4k +7 k+2 1 k+1 2 2k+4 The Milno numbe in the cse o A 2k,2l is 2k +2l+ 7, in othe cses is exctly the coesponding index. III. The topologicl inequlity Theoem. I G is the miniml embedded esolution gph o, ndµis the Milno numbe o, then () E top := m w 2#W(G ) #A +1 (µ 2) 0. I A n (n>1), A 2k,2l (k 1, l 1), D 2k+3 (k 1), E 6,then (b) ( e w ) µ. m w 4#W(G ) #A +1+ Poo. Assume tht is ieducible with s = 1 nd Newton pi (p, q). By (4.3) nd (4.4), m w 2#W pq (p + q +1)=µ 2. Assume tht s 2 (c. I). Let (l) be gem with Newton pis (p i,q i ) l 1=1,(1 1 s). Notice tht µ( (l) )=( l 1)(p l 1) + p l µ( (l 1) ). (4.13)

13 Vol. 5 (1999) Dedekind sums 173 Fist conside the expession P (l) := p l l µ( (l) ). Then P (1) 4ndol 2: P (l) p l P (l 1) = p l + q l 1 2. Theeoe P (l) P (l 1) + 2, hence by induction P (l) 2+2l o ny 1 l<s. Now, by (4.1): E top (l) Etop (l 1) = p l l p l 1 l 1 2#W(G y(x p l+y q l ) ) µ( (l) )+ µ( (l 1) ). I q l 2 then by (4.3), (4.5) nd (4.13): E top (l) Etop (l 1) (p l 1)P (l 1) 2 P (l 1) 2. The sme inequlity is vlid in the cse q l = 1 (use (4.2) insted o (4.3)). Theeoe E top (l) Etop (l 1) 2l 2 o ny 2 l s, which gives: o ieducible one hs: E top s(s 1). (4.14) Now ssume tht =. Fist ecll tht µ() = µ( )+2 < ν(, ) #A+1. (4.15) Now, (4.6), (4.8.) nd (4.15) gives: E top E top +2T +T (c. (4.8) o nottions). (4.16) Theeoe () ollows om (4.14) nd (4.15). In ode to pove (b) we need to veiy tht E top + ( e w ) 2#W 2, (4.17) excepting the ou cses given in the hypothesis. Conside the invint I(G) = w ( e w) 2#W ssocited with ny gph G. Ate blow up, #W inceses 1, nd ( e w ) with 2 o 3. Theeoe, o ny G, I(G) 1, but i we blow up t lest one node o the exceptionl diviso, then I(G) 0. Theeoe, i hs t lest one cto with ν( ) 2, then I(G ) 0; nd I(G )=0 A 2k. (4.18) Assume tht (4.17) is not tue o. I is ieducible, then ν() 2, hence by (4.14) nd (4.18) s =1. Foν() 4, one hs I(G ) 2, hence by (II) A 2k o E 6. (In ct, i s =1,thenI(G )=#W [q/p] 2.) Assume tht #A > 1, nd let S be the numbe o smooth components. Since E top smooth = 1, one hs S 2. I S = 2, then by (4.18) we cn hve no othe component, hence A 2k 1. I S 1, then by (4.18) I(G ) 0, hence ll the tngent lines e distinct (by 4.16). In this cse I(G )= I(G )+#A 1. (4.19) So, i S = 1, then (4.18), (4.19) nd E top smooth = 1 implies D 2k+3. IS=0,then similly: A 2k,2l.

14 174 A. Némethi Sel. mth., New se. IV. The lgebic inequlity Theoem. I G is the miniml embedded esolution gph o nd ν is the multiplicity o, then () E lg := m w 3#W(G ) (ν 2 2ν 3) 0, (b) m w 3#W(G ) #A Poo. The poo is simil s the poo o (4.12). Fist ecll tht i is ieducible, then ν = p 1...p s (ecll p 1 <q 1 ). I s +1, then E lg 0by(II)ip 1 = 2 o 3, nd by (4.3) i p 1 4. I s 2, then E lg (l) E lg (l 1) by simil gument s in (4.12). I =, then by (4.8.b) E lg Elg. This lst inequlity, togethe with tble (II), shows (b) in the cse ν 3. I ν>3, then ν 2 2ν 3 #A 1. (The detils e let to the ede.) V. The combintoil inequlity Theoem. Let G be the miniml embedded esolution gph o. () Assume tht A n (n 1), E 6, E 8, A 2k,kl (k 1, l 1), D 2k+3 (k 1). Then E com := m w 4#W(G ) #A (b) E com 6o ny, nde com #A(#A 1) 4 i #A 2. Poo. Fo ieducible with s =1:E com 6, nd o p 1 4: E com tble (II) nd (4.3)). By simil inductive step s in (4.12) E com (l) Theeoe 0 (use the E com (l 1) +6. E com 6(s 2) o ieducible. (4.22) I #A 2, wite = g 0 with 0 ieducible. With the nottion o (4.7.b), i 0,thenm E (,G )=m E (,G g), hence by (4.6) nd (4.7.b): E com E com g + E com 0 +2ν(g) ν( 0 )+3+6T +2T. (4.23) Notice tht Esmooth com = 3. Let S be the numbe o smooth components o. Assume #A =2. IS=2,then A 2k 1. (4.22) nd (4.23) implies tht i S = 1, then eithe E com 0o D 2k+3,ndiS= 0, then eithe E com 0o A 2k,2l. In ll cses E com 2. Assume tht #A > 2. Notice tht E com 0 +2ν(g)ν( 0 ) 2#A 5, hence (4.23) gives E com Eg com +2(#A 1), which povides (b).

15 Vol. 5 (1999) Dedekind sums 175 VI. Remks In the pevious inequlities the minimlity o the esolution gph G is necessy. With some dditionl blow ups, we my keep the invints µ, ν, m w constnt, but incese #W bitily high I we wnt to omulte in vey simple om the bove esults poved o plne cuve singulities, then we cn sy tht m w is lge with espect to the othe invints. On the othe hnd, o genel gems :(X, x) (C, 0) (whee (X, x) is noml suce singulity) it is no longe tue tht m w is lge. In pticul, ll the inequlities pesented in this section e speciic o plne cuve singulities, nd the negtivity o σ( + z N ) is povided exctly by the existence o this type o eltions in G. We expect tht simil eltions exist o the embedded esolution o highe dimensionl hypesuce singulities. To see tht in genel m w cn be smll, conside the ollowing digm with mw = 4, but with #W bity high. -2 (1) ::: (2) (2) (1) (1) 5. Inequlities stisied by σ N Fo the convenience o the ede we ecll the nottions: #W (esp. #E) denotes the numbe o vetices (esp. edges) o the miniml embedded esolution gph G o, A is the numbe o ieducible components o, ν is its multiplicity nd µ is its Milno numbe. I. Topologicl inequlities Let ɛ be 1 i A n (n 1), A 2k,2l (k 1, l 1), D 2k+3 (k 1), E 6, othewise ɛ = Theoem. () σ( + z 2 ) 1 2( µ +#A 1 ). (b) Fo ny nd N: σ ( + z N) 1 N N #E + 1 N 2 ( µ #W 2 ).

16 176 A. Némethi Sel. mth., New se. (c) I N B()+1 (c b), then σ ( + z N) 1 N 2 ( ) µ 3ɛ. (d) I (N,m w )=1o ny w W(G ),then σ ( +z N) 1 N N ( ) 1 N 2 ( ) #W+#E + µ 3ɛ nd with the sme ssumption (below p g is the geometic genus): σ( + z N ) µ(+z N )/3 p g ( + z N ) µ( + z N )/6. o, equivlently: Notice tht in the inequlity (5.1.) the coeicient 1/2 oµis the sme s in the inequlity σ µ/2 µ 0 o Tomi [16]. Poo. By (2.2) thee e t lest two vetices w such tht m w is odd, hence (3.7.b) nd (4.12.) give (). (b c) ollows om (4.12) nd (3.12); nd the ist pt o (d) om (4.12) nd (3.8). Recll the stndd eltions (o gem g :(C 3,0) (C, 0): µ = µ 0 +µ + +µ, σ = µ + µ nd 2p g = µ 0 +µ +. The hypothesis (N,m w ) = 1 o ny w implies tht µ 0 ( + z N ) = 0 (use, e.g. A Cmpo s theoem bout the chcteistic polynomil o in tems o m w s). Theeoe σ µ/3 is equivlent to p g µ/6. We ledy know om the ist pt o (d) tht these inequlities e vlid o A n, A 2k,2l, D 2k+3, E 6. In the sequel we veiy these cses. Let g :(C 3,0) (C, 0) be n isolted singulity with Milno lttice L g.letg t be deomtion o g such tht g 0 = g nd g t,ot 0 smll, hs k singul points with Milno lttices L 1,...,L k. Then thee is ntul embedding i L i L g.ic is the codimension o this embedding, then σ(g) c + i σ(l i). In the suspension cse, ny deomtion t o povides n embedding in L( + z N ). Now we veiy the exceptionl cses. I A n, then σ µ/3 ollows om (5.1.). Actully, (5.1.) implies tht ( ) σ(x 2 + y 2k+1 + z N ) k(n 1). I D 2k+3, then i we put the components in geneic position, we obtin n embedding A 2k A 1 A 1 D 2k+3. Theeoe σ(d 2k+3 + z N ) σ(a 2k + z N )+ σ(a 1 +z N )+(N 1) k(n 1) 2(N 1) + (N 1) <µ/3 (hee we used ( )). I A 2k,2l, one hs the embedding A 2k A 2l 4A 1 A 2k,2l. Recll tht µ(a 2k,2l )=2k+2l+7. Now( ) gives esily the esult i k + l 4, othewise we use σ(x 2 + y 3 + z N ) 4(N 1)/3 (povided tht (N,2) = 1) which cn be esily veiied by Bieskon s omul [2]. The cse E 6 is gin Bieskon type nd its veiiction is let to the ede.

17 Vol. 5 (1999) Dedekind sums 177 II. Algebic nd combintoil inequlities, nd the limit lim n σ N /N Similly, s in the topologicl cse, we cn give list o inequlities coesponding to the dieent cses. In the next theoem we pesent only ew; the inteested ede cn omulte ll the othes Theoem. Fo ny nd N: σ( + z N ) 1 N N #E + 1 N 2 (ν2 2ν 3 #A +1). σ(+z N ) 1 N N σ( + z N ) 1 N N #E 0. #E+1 N2 (#W +#A(#A 1) 6). Poo. Use (3.12), (4.20) nd (4.21). Ou ist inequlity in (5.2) is simil to the min esult o [1] Theoem. The limit lim N ( σ N )/N = η( ;1) stisies η(;1) (µ +#W+#A 3+B)/3, whee B := ( e w) 2#W 1;ndlso: η(;1) (3#W +#A 8)/3. Poo. Use (3.5) nd (4.12). III. Conjectues Fo ny :(C 2,0) (C, 0) nd N conjectully, the ollowing inequlities hold: 1. σ( + z N ) N+1 µ(+zn ). 2. η(;1)+η(;n) η(;n +1) 0, (which would imply the monotonity: σ N+1 σ N ). 3. η(;1)+η(;n) η(;n +1) µ /3, (which would imply, o exmple, the inequlity σ µ/3 o+z N ).

18 178 A. Némethi Sel. mth., New se. Appendix In this ppendix we septe some popeties o the genelized Dedekind sums [14], [21]. They e deined o bity non-zeo integes, b, c by s(b, c; ) = k=1 (( kb )) (( )) kc, whee ((x)) is the usul unction deined vi the ctionl pt {x} s { {x} 1/2 i x/ Z ((x)) = 0 othewise. Below ( 1,..., n ) denotes the getest common diviso o these numbes. By simil gument s in [10] (A.1), one cn pove tht ( ) b(, b, c) s(b, c; ) =(, b, c) s (, b)(b, c), c(, b, c) (, c)(b, c) ; (, b, c). (A.1) (, b)(, c) The mous geneliztion o the ecipocity lw o Dedekind, given by Rdemche [14] (see lso [21]), ssets tht i, b, c e stict positive, mutully copime integes, then s(b, c; )+s(c, ; b)+s(b, ; c) = 1/4+( 2 +b 2 +c 2 )/12bc. Using (A.1), this eltion o bity, stict positive integes eds s(b, c; )+s(c, ; b)+s(b, ; c) = (, b, c) (b, c) 2 + b 2 (, c) 2 + c 2 (b, ) 2. (A.2) 12bc The inequlity (A.3) o [10] togethe with (A.1) o this ppe povide: Coolly. Fo ny integes, b, c ( >0) one hs: ( 1)( 2) s(b, c; ). 12 (A.3) s(b, c; ) 2 (, b) (A.4) Reeences [1] T. Ashikg. The Signtue o the Milno ibe o Complex Suce Singulities on Cyclic Coveings. Pepint (1995). [2] E. Bieskon. Beispiele zu Dieentiltopologie von Singulitäten. Invent. Mth. 2 (1966), 1 14.

19 Vol. 5 (1999) Dedekind sums 179 [3] E. Bieskon nd H. Knöe. Plne lgebic cuves. Bikhäuse, [4] A. Duee. The Signtue o Smoothings o Complex Suce Singulities. Mth. Ann. 232 (1978), [5] D. Eisenbud nd W. Neumnn. Thee-Dimensionl Link Theoy nd Invints o Plne Cuve Singulities. Ann. o Mth. Studies 110. Pinceton Univesity Pess, [6] L.J. Modell. Lttice points in tethedon nd genelized Dedekind sums. J. Indin Mth. 15 (1951), [7] A. Némethi. The equivint signtue o hypesuce singulities nd etinvint. Topology 34 (1995), [8] A. Némethi. The et-invint o vition stuctues I. Topology nd its Applictions 67 (1995), [9] A. Némethi. The el Seiet om nd the spectl pis o isolted hypesuce singulities. Compositio Mth. 98 (1995), [10] A. Némethi. Dedekind sums nd the signtue o (x, y) +z N. Select Mthemtic, New se. 4 (1998), [11] W. Neumnn. Splicing Algebic Links. Advnced Studies in Pue Mth. 8 (1986), , (Poc U.S.-Jpn Semin on Singulities 1984). [12] W. Neumnn nd J. Whl. Csson invint o links o singulities. Comment. Mth. Helv. 65 (1991), [13] J.E. Pommesheim. Toic vieties, lttice points nd Dedekind sums. Mth. Ann. 295 (1993), [14] H. Rdemche. Geneliztion o the Recipocity omul o Dedekind sums. Duke Mth. Jounl 21 (1954), [15] H. Rdemche nd E. Gosswld. Dedekind sums. The Cus Mth. Monogphs 16 (1972). [16] M. Tomi. The inequlity 8p g µ o hypesuce two-dimensionl isolted double points. Mth. Nch. 164 (1993), [17] J. Whl. Smoothings o noml suce singulities. Topology 20 (1981), [18] Y. Xu nd S.S.-T. Yu. The inequlity µ 12p g 4 o hypesuce wekly elliptic singulities. Contempoy Mth. 90 (1989), [19] Y. Xu nd S.S.-T. Yu. Duee s conjectue nd coodinte ee chcteiztion o homogeneous singulities. J. Di. Geomety 37 (1993), [20] Y. Xu nd S.S.-T. Yu. A shp estimte o the numbe o integl points in tethedon. J. eine ngew. Mth. 423 (1992), [21] D. Zgie. Highe dimensionl Dedekind sums. Mth. Ann. 202 (1973), [22] C. Webe. (Ed.) Noeuds, tesses et singulités. Monogphie No. 31 de L Enseignement Mthémtique, 1983, (Poblem C. o W. Neumnn). A. Némethi The Ohio Stte Univesity Dept. o Mthemtics 231 West 18th Avenue Columbus, OH USA e-mil: nemethi@mth.ohio-stte.edu