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(1)Remarks on Maximally Elliptic Singularities By. Etsuo YOSHINAGA, Shigeki OHYANAGI" and Masayuki HIURA gl. Introduction. Let (V, P) be a weakly elliptic singularity with Stein V. In [2], Laufer proved that those weakly elliptic singulrities which satisfy the' minimality con-. ditions are equivalent to the Gorenstein singularities of p,=1, i.e., Gorenstein strongly elliptic singularities. Recently, Yau [3, 4] developed a theory ,of certain weakly elliptic singularities satisfying the maximality conditions, called,maximally elliptic singularities. Maximally elliptic singularities ,are Gorenstein, but their. geometric genera may be arbitrarily,large. Restricting them by P,==1, maximally elliptic singularities are,equivalent to Laufer's minimally elliptic singularities.. In this paper, we shall consider the elliptic sequence introduced by Yau [3, 4], and refer to the following remarks on weakly elliptic Gorenstein singularities.. REMARK I. The elliptic sequence is defined for a weakly elliptic singularity, and it consists a sequence of monotonously decreasing subvarieties Bo, Bi, ･･･, Bi. and their fundamental cycles. Each Bi will be blown-down into a weakly elliptic singularity. Suppose that (Vo,.po):==(V,p) is numerically Gorenstein, i.e., the canonical divisor on a resolution total space M of (V, p)'is given by a. cycle on the exceptional set A. Then, each (Vi,pD will be also numerically Gorenstein, where (Yi, Pi) is the weakly elliptic singularity obtained from Bi. In particular, the terminal (Vi, pt) is purely Gorenstein, i. e., (Vt, pi) is minimally. elliptic in the sense of Laufer. ･ - ' THEoREM I, Let (V,P) be a maxima,lly elliPtic singularity. Then, each (Vi, Pi) is also maximally elliptic.. Let T:(M, A)-->(V,P) be the minimal resolution of a maximally elliptic singularity, which satisfies that.each,irreducible component of A is nonsingular,. and each intersectiontamong the irreducible components of A satisfies the normal crossing conditions. Serre's duality gives Hi(M, O), as dual to Hk(M, 9), where 9 is the canonical sheaf on M) i.e., the sheaf of germs of holomorphic 2-forms. on M. By Laufer (Theorem 3.4, [2]), for suitable M, Hk(M, 9) may be identified. with HO(M-A, 9)/HO(M, 9). + Since Ov,p is a Gorenstein･local ring, there exists an tuEHO(M-A, 9) hav-. F of Mathematics, Faculty of Science, University of Tsukuba, Ibaraki 305 * InstitUte Japan..

(2) 2 E. YosHiNAGA, S. OHyANAGi and M. HiuRA ing no zero near A. Now, the geometric genus of (V, p), i.e., dim.HO(M-A, 9)/ HO(M, 9), equals the length of the elliptic sequence l+1).2, since (V,P) is maximally elliptic. There exists tui, ･･･, cat in HO(M-A, 9) such that the images. of too:=w, toi, ･･･tut in HO(M-A, 9)/HO(M, 9) form a basis of this quotient vector space. Decompose the divisor (toi) into a positive part Li (i.e. defined by the zeros of toi) and a negative part Ki (i.e., defined by the poles of tui).. CoRoLLARy. For a maximally elliPtic singularity of P,=l+1, zve can taleea basis too, tui, ･･･,tut of HO(M-A, 9)/HO(M, 9) as follows;. (i) 1LilAIKil=￠ foralli, (ii) -dKi=ZBi+ZBi.,+'･'+ZB,-,+E forO,<..i!$l-1,and-Kt=E,. e. where E is the minimally elliPtic cycle on A. 4. REMARK II. Let (V, p) be a Gorenstein singularity of geometric genus nlll. Then, the exceptional set A of the minimal resolution of (V, p) satisfies the minimality conditions, i. e., any connected proper subvariety A'cA will be blown-. down into a normal surface singularity (Vi,P') of geometric genus 5n. Set n=2. There always exists such A'cA that P,(V',p')=1. Since a Gorenstein singularity of P,=2 is weakly elliptic, if any A'cA is of p,=O, then (V,p) satisfies the Laufer's minimally elliptic condition, so p,(V, p)==1; this is a contradiction,. But, in general, we cannot say that there always exists such A'cA that Pg(V', P')=Pg(V, P)'1} THEoREM II. Let (V, P) be a wealely elliPtic Gorenstein singularity of P,=. nlll. Then, there always exists such A'cA that p,(V', P')=p.(V, p)-1.. g2. Rremark I. For the minimal resolution (M, A) of a weakly elliptic singularity (V,P), which satisfies that each irreducible component Ai of A is nonsingular and each intersection among Ai's satisfies the conditions of normal crossing, there is uni-. t･. quely determined the minimally elliptic cycle on A; 9" ,. (1) p(E)==1, the virtual genus of the E, and. (2) for any O<D<E, p(D);SO.. '. ･ To a weakly elliptic singularity, the elliptic sequence will be defined as follows, using the informations of the above minimal resolution.. good" PekioNiLTtl'8nN. (2:"==(tYh2"agZ'vg]t'neY%tf 22 llieegkhPy eeX,;i$Pti'10gi2igi?Eri2S. t"en Elg':eii?a91. E･ZSObythede' nitionofthefundamentalcycleZ[2]. ･. If E･Z<Q, we say that the elliptic sequence is {Z}. Sup'pose that E･Z==O.. Let' Bi be the makimally ･'conned'ted subvariety of A such that IEIgBi and Ai･Z=O for a,ny AEBi. Leg Z, &,,be the fundamental cycle,on Bi. Also, in. general, E･ZB,;$O. ,.

(3) Remarks on Maximally Elliptic Singularities 3 If E･ZB,<O, then the elliptic sequence is {ZB,:==Z, ZB,}. Suppose E･ZB, =O. Let B2 be the maximally connected subvariety of B, such that IEIgB2 and Ai･ZB,=O for any AiSiB2. By the same arguments as above, continuing this process, we finally obtain Bt with E･ZB,<O. Then, we call {ZB,, ZB,, ･･･,ZB,} the elliptic sequence on A, and say that the length of the elliptic sequence is l+1.. Let KM be the canonical line bundle on M and KA be the numerically equivalent divisor on A to the Khr, i.e. KA=2leiAi with feiEQ such that Aj･KA ==Aj･KM for all AjC=A. One can easily find the KA by the adjunction formula Aj･Kif==-Aj･Aj+2gj-2, where gj is the genus of Aj as a Riemann surface. As all the fei in KA are integers, we say that (V, p) is numerically Gorenstein. One can see that Gorenstein singularities are numerically Gorenstein, `. but the converse is not always true.. In [4], the followings are proved. PRoposlTIoN 2.2. SuPPose that (V, P) is numerically Gorenstein, tveakly elliptic.. (1) the terminal szabvariety Be coinsides with the IEI,. (2) -K,,=Z.,+Z.,+･･･+ZB,-,+E, and. (3)P,(V,P):;ll+1. ,' ･. A (V, p) is said to be maximally elliptic, if it is a weakly elliptic, numeri-. callyGorensteinsingularitywithP,=l+1. ''' ' ･ In [4], it was proved that maximally elliptic singglarities are purely Gorenstein.. Also, Yau gave a criterion for weakly elliptic singula'rities to be maximally. elliptic in [3]. . ,. THEoREM 2.3 ([3]). Let (V, P) be a zvealely elliPtic, numerically Gorenstein. singularity with the elliPtic sequence {ZB,, ZB,, ･･･, ZBt-,, ZE}. Then, (V, P) is a maximally elliptic singularity if and only if HO(M, ec,.+E)->HO(M, ec,.) is sur]'ec-. '. tive for any O;Sl'fm{l and e(-Cj)/e(-Cj-E) is the sheaf of germs of sections of a trivial line bundle over (iEl, eE) for O.<..i511, where Cj: =.S] ZB,･ )N, ,. ' z=o. Let (V,p) be a weakly elliptic, numerically Gorenstein singularity with elliptic sequence {ZB,,･･･,ZB,}. Then, Bi==IEI and P(Z)==1 imply that B. ･･･ ,Bi are blown down into weakly elliptic singularities. So, we can define their elliptic sequences. One can easily check that the elliptic sequence on Bi is given by {ZBi, ZB,.,, ･'',ZE}･ Let Ui be a holomorphically convex neighborh6od of Bi, Kti, be the canonical line bundle on Ui, and Kb, be the canonical. divisor on Bi. Set i=1. For any Aj.gBb Aj･(Z+KA)=Aj･KA=Aj･KM== -A;･+2gj-2=Aj･Kb,. Since KA is a cycle on A, Kb, is numerically equivalent. to a cycle Z+KA on A. As we have -KA=ZB,+ZBI+'''+ZBi-,+E, Z+KA ==- (ZB,+･･･+ZBt-,+E), this is a cycle on Bi. Inductively, we can say that. each Bi is numerically Gorenstein. In particular, the terminal Bi==IEI is purely.

(4) 4 E. YosriiNAGA, S. OHyANAGi and M. HiuRA Gorenstein.. Let denote the singularity obtained from Bi by (Vi,pi). Then, we have the following maintains.. (V, p) is numerically Gorenstein. >(Vi, Pi) is numerically Gorenstein. - >'･' l> (Ve"i, Pt-i) is numerically Gorenstein >(Vt, Pe) is purely Gorenstein.. o. In this section, we shall prove that;. (V, P) is maximally elliptic. s. >(Vi, Pi) is maximally elliptic > ･･･ >(Vt, Pt) is maximally elliptic.. REMARK. For all that (V, P) is purely Gorenstein, (Vi, Pi) is not always purely Gorenstein.. j -t either LEMMA 2.4. Let Dj:=,;,ZBi. Then,. '. HO(M, o(- Dj)/e(- Dj- E)) 2iiOE-!!t Hi(M, e(- Dj)/o(- Dj- E)),. HO(M,e(-D,)/e(D,-E))!;!!C!xHi(M,o(-Dj)/e(-D,-E)). PRooF. The following sequence of sheaves is exact.. O ---> eDj --> e.j.. ---> O(-Dj)/e(-Dd-E) -> O.. ' Taking the long exact cohomology sequence, H2(M, eDj)==0 and each cohomology. groupisfinitedimensional,sowehavethefollowingequation., ･ ' hO(M, e(-Dj)/e(-Dj-E))-hi(M, o(-Dj)/e(-Dj-E)). n. =-ho(M, e.?+hO(M, oD,.E)+hi(M, oD,)-hi(M, OD,+E). ==p(Dj)-p(Dj+E)=p(Dj)-p(Dj):--p(E)-Dj･E+1=-Dj･E=O, since IEIE.l!B,c ･･･cBi. Hence,. .` ho(M,o(-D,)/e(-D,-E))=hi(M,o(-D,)/e(-D,-E)). ･. Suppose that IEI--Ai. Then, Ai should be a nonsingular elliptic curve.. Now, e(-Dj)/o(-Dj-E)==o(-Dj)/e(-Dj-'A!)=o([-Dj]1A,). Since, AiCBL-i C'''CBb'c([-Dj]IA,)==-Dj･Ai=O. Hence, by the Riemann-Roch theorem, hO(M, e(-Dj)/e(-Dj-E))==1. Next, we suppose that IEI=VAi, 1$iStSn (t>=2). Then, all Ai are nonsingular rational curves. For this case, we can take the following･computation sequence of 'the fundamental cycle Z.. ` 'z,.,,o, z,, ･･･,Z,==E, ･･･,Z.==Z. .A. $.

(5) Remarks on Maximally Elliptic Singularities 5 For the above sequence, Aij･Zj-i==1 if 7' 7Sk, and Ai,･Zle-i==2.. Consider the following exact sequences. o ----> e(- Dj- Zi)/o(- Dj- E) ----> o(- Dj- Z.,.,)/o(- Dj- E). ---->O(-DjHZi-i)/O(-Dj-Zi)----->O, for1,<,.iSk'1. Consider their long exact cohomology sequences and sheaf isomorphisms. e(-Dj---Zim,)/e(-DrZi)!-:te([-Dj-Zi-,]]A,), where Zi==Zi-i+A*. Via the Riemann-Roch theorem, we can say that;. HO(M,e(-Dj-Zi)/e(-DrE))ptN-O, forISi,<..fe-1,. ' Hi(M,e(-Dj-Zi)/e(-DrE))cr-C, for1$iSle-1, HO(M, o(-Dj)/0(-Dj-Z,))!-;:C. '. and. Hi(M, e(-D,)/0(-Dj-Z,))2-:O, Hence, we have the following exact sequence. O - HO(M, e(-Dj-Z,)/e(-Dj-E)) --> HO(M, e(-Dj)/o(-Dj-E)) -> Ho(M, o(-Dj)/o(---Dj-Z,)) --> Hi(M, o(-Dj-Z,)/e(-Dj-E)). --> Hi(M, e(-D,)/e(-D,-E)) -> Hi(M, o(-D,)/e(-D,-Zi)) .O; o ---> O -> HO(M, e(-Dj)/o(-Dj-E)) ---> C D C ---> Hi(M, e(-Dj)/e(-Dd-E)) -> O t---> O･ Therefore, we have proved our lemma. Suppose that (V, p) is maximally elliptic. Then, s. HO(M, o(-C,)/o(-C,- E)) cr- C, ,. since o(--Cj)/0(-Cj-E)!-iite(L). for a trivial Iine bundle L on (IEI, eE).. o. HO(M, e(-C,-E)) ---> HO(M, e(-C,)) ----> C ----> O, so HO(M, e(-Cj-E))->HO(M, e(-Cj)) is not isomorphic.. There are natural inclusions and their commutative diagram with exact rows.. O o e(-Cj- E) o o(-Cj) . o(-Cj)/o(-Cj- E) .O. iS J. O o 0(-Dr E) e e(-Dj) . O(-Dj)/e(-Dr E) ---> O Then, we have the following cohomology exact sequence..

(6) 6 E. YosmNAGA, S. OHyANAGi anP M. HiuRA p '. O ----> HO(M, e(-Cj-E)) -. flO(M, O(-Cj)). t ,J O. HO(M, e(-Dj-E)) -> HO(M, O(-Dj)) -> HO(M, e(-Cj)/e(-Cj-E)) 2! C . ･･･. J ---i> HO(M, e(-Dj)/e(-Dj-E)) - ･･･. 'G. By Lemma 2.4, HO(M, e(-D,)/O(-D,-E)):N:O, or !!!C. Suppose that. HO(M, e(-Dj)/o(-Dj-E))cr-o.. '. s. Then, pN is isomorphic. But, P:PNIHo(M,e(-cj)ie(.-cj-E)) is notisomorphic. Hence,. HO(M, o(- Dj)/e(- Dj- E)) !!! C; Hi(M, e(- Dj)/e(- Dj- E)). The surjectivity of HO(M, eDj+E)-HO(M, ODj) is implied from the surjectivity of HO(M, ocj+E).HO(M, ecj). Therefore, (Vi, Pi) is maximally elliptic. Then, we can inductively show that (Vi, Pi) is maximally elliptic for all i.. CoRoLLARy 2.5. For a maximally elliPtic singularity (V, P) of P.==l+1, there exist too, toi, ･･･, tutEHO(M-A, 9) satieflrying (1) and (2) below, which form. a basis for the HO(M-A, 9)/HO(M, 9). Let denote the divisor of toi by (tui), and decompose it into a positive part Li and a negative part Ki; (wi)=Li+Ki with 1Li1A1Kil==(discrete points).. (1) lLilAIKil=￠ foralli,. te t /t. ' (2) -Ki=ZBi+ZBi+,+'''+ZB,-.,+E forallO;:;li;$l-1,and-Kt=E. ' ' ' ' ' ' g3. Remark II. Let (V, P) be a normal surface singularity and (M, A) be the minimal resolution of it. For a connected proper subvariety A' of the exceptional set A, Xlg 32a,iA,d9,n,Ot?,fih,e.?..Or,'e.S,P,O.n,dl:g(e?'pm/gi surface singuiarity by (v', p'). Then,. s(V, p): =max{p,(V', p'): A' runs over all subvarieties}.. In general, we have that s(V,p)S.p,(V,p). In particular, if (V,P) is a. Gorensteinsingularity,thens(V,P).<.P,(V,P). , ･. . In this section, we shall consider that question: What singularities satisfy. s(V, P)=p,(V, p)-1? t Suppose P,(V, P)==2. Then, any Gorenstein singularity is weakly elliptic: [5]. If s(V, P)=:O, then (V, p) satisfies the Laufer's criterion to be minimally elliptic. So, the geometric gen' us must equal 1. This is a contradiction. Hence,. tr. l.

(7) Remarks on Maximally Elliptic Singularities 7 s(V, p) =1.. Accordingly, we are going to consider the above question with weakly elliptic, numerically Gorenstein singularities.. Let (V, p) be a weakly elliptic, numerically Gorenstein singularity of P.= g+1. Proposition 2.2 says that g+1 must be less than, or equal to the length of the elliptic sequence l+1. Let tuo, wi, ･･･,to. be a basis of HO(M-A, 9)/. '. HO(M, 9), (tui)=Li+Ki:decomposition as in section 2. Here, we may assume that IKo[DIKilD･･･DIKgl without loss of generality. Then, we have that {IKol,IKil, ･･･,IK.i}C{Bo=A, Bb ･･･,Bi} and. {-K,,-K.･･･,-K.}c{ :E) ZB,+E, 2 ZB,+E,･･･,ZB,.,+E,E}･ O$iSl-1 ISiSl-1. `. In particular, -Kg==E, since we assumed (V, P) to be numerically Gorenstein, Furthermore, if (V,p) is purely Gorenstein, then -Kb= Z ZB,+E.. 0Si$l-1. tv On A:=IKil, the Kb, i.e. wo, cannot remain alive, since ]Kb) coinsides with the whole A. On A, tui, ･･･,tog can survive and form a basis of HO(M'-A, 9')/HO(Mt,9t> where M' is a strictly pseudoconvex neighborhood of A in M. So any connected A' such thattvAgA'cA, the associated normal surface singularity, not necessarily numerically Gorenstein, will be of p,(V',p')=g. Hence, we have. rv tv. the following.. THEoREM 3.1. Let (V,P) be a weakly elliPtic Gorenstein singularity and (M, A) be the minimal resolution. Then, there alwaysexists such connected Proper. subvarietyA'cAthat.･･ . ' ' Pg(V',P')==Pg(V,P)-1, i･e"S(V,P)=Pg("V,P)-1･. Moreover, we can see that. Let P(A'):={KL,IS.i:.{g:IKilcA' or A's. IKil}. Theri, -. CoRoLLARy 3.2. P,(V', p')==#(P(A')) holds.. g4. Examples. ,. In this section, we shall see some examples on the ground of our remarks.. ExAMpLE4.1. Let V:=={(x,y,z)(!iC3:z3+yiO+yx3=:O},P==(O,O,O). For this (V,p), we can calculate that P.==1 and p.==3. The minimal good resolution of this has the following weighted dual graph A. Each vertex corresponds to a nonsingular rational curve. A vertex nothing written in it denotes of selfintersection number -2. The elliptic sequence on the A is as follows, of length 3..

(8) E. YosHINAGA, S. OHYANAGI and M. HIuRA. 8. 11 22 Z,,--Z==12333333321, Z,,==12333321OOO, 22 11. h. 1 2. IB,-- E=12321Ooooo o. 2 1. Our Corollary 2.5 says that there are following too, toi, to2 in HO(M-A, 9) forming a basis. Here, each n' unmber denotes the order of the pole of tei on the component, and number in parenthesis denotes the order of the zero of toi on the wave-1,ike non-compact Iine,. ( (Do). 3. 3 .. 6. 6 9. 8. 6. ,. 7. 3 6. 5. 4 3 2 1.

(9) Remarks on Maximally. Elliptic Singularities. 9. 2. ( toi). 2. 4. 4. 6. U, 5. 4. 4. 2 3. 2. '. (1). ll. -d --. -L-.-l. ,. --- --de. l. ･s. -- -- lt -･ -･ -t. 1. 1. ((02). 2. 2 3. U, 2I. 2 1. 1. l. 1. ' -"-tnt-e-t---1. .vvvvvNl /x. ll 11. (1) 1-- L--+.l ･1 l. '. "--t-de--1 1. ･i. ' ExAMpLE 4.2. Next, we shall consider the following quasi-simple elliptic weighted dual graph. Let R denote the embedding point of the elliptic curve, and P denote the intersection point of the elliptic curve and the rational curve,. The notations for the relations between P and R are introduced in [6]. "Max. ell." means that the weighted dual graph corresponds to a maximally elliptic singularity.. '.

(10) E. YosHINAGA, S. OHyANAGI and M HiuRA. 10. ××××××. N t"N; ())--o--<>-o O-CHC>-<:> eHc>.( ts /---< [1]. [1]. sd. [1]. Pg==4. Pg==3. Pg==2. Max. ell.. Max. ell.. Max. ell.. Wo, tu1, (D2, (1)3. too, (lt)1, Q)2. (Do, Wl. Pg==3. Pg--2. Pg==1. not Max. ell.. not Max. ell.. not Max. ell.. but Gorenstein. but Gorenstein. not Gorenstein. (Do, W2, to3. (1)o, (V2. wo. Pg=2. Pg=1. Pg=1. 2P it 2R. not Max. ell.. not Max. ell.. not Max. ell.. 3P=3R. but Gorenstein. not Gorenstein. not Gorenstein. toO, W3. too. (Oo. Pg=2. Pg=2. Pg==1. 2P==2R. not Max. ell.. not Max. ell.. not Max. ell.. 3Pt3R. not Gorenstein. but Gorenstein. not Gorenstein. (OO, C02. COO,'C02. wo. Pg= 1. Pg=1. Pg=1. 2Pi2R. not Max. ell.. not Max. ell.. not Max. ell.. 3P iL3R. not Gorenstein. not Gorenstein. not Gorenstein. too. (Do. (oo. P== R. P ik R. 2P=2R 3P=3R P 7E R. PiER. P# R. N --. 's. o o (to,)=---3 -2 -1 o (to,)=-4 -3 -2 -1 (te,)==-1 O o. (to,)==--2 -1 O. ExAMpLE 4.3. Let Vi:= {(x, y, 2) EC3:22+y6+x9==O} and V2:= {(x, y, 2) Ei C3: 24+y`+x5==O}. Then, we can see that Pf==P.==2 and Pg=4 for (Vi, (O,O,O)) and Pf=P.=3 and P, =4 for (V2, (O, O, O)). Their weighted dual graphs of the minimal resolutions are described below. Both Vi and V2 are Gorenstein, but not weakly elliptic. The connected proper subgraphs of them are as follows.. ,. .-.

(11) 11. Remarks on Maximally Elliptic Singularities. -t-.... -(V,,O). (V2,O). -4-4 -4-4. weighteddualgraphof theminimalresolution. -3-1. -3. [1]. 4nonsingularrational curves, transversalymeetingat. .apolnt. '. Pg -. connectedpropersubgraphs. -1-3. [1]----------. o. -4. S2---t -' 1. [1]-----------. 1. 4-4. --------------- -----"---. -4-4. ----o. ([}). Pg. o. "-------------({]). -------o. .. 22. s(v,p). 1. References [1] LAuFER, H.B.: On rational singularities, Amer. J. Math. 94 (1972), 597-608. : On minimally elliptic singularities, Amer. J. Math. 99 (1977),1257-1295. [2] [3] YAu, S.S.T.: Normal two-dimensional elliptic singularities, Trans. A.M.S. 254 (1979), 117-134. ,. : On maximally elliptic singularities, Trans. A.M.S. 257 (1980), 269-329. [4] YosHiNAGA, E. and OHyANAGi, S.: A criterion for 2-dimensional normal singu[5] larities to be weakly elliptic, Sci. Rep. Yokohama National Univ. 26 (1979), 5-7.. [6]. and :Someexamplesofweaklyellipticsingularities,Sci.Rep. Yokohama National Univ. 27 (1980), 11-18,. [7]. and :Two-dimensionalquasihomogeneoussingularitiesofPg=3, Sci. Rep. Yokohama National Univ. 28 (1981), 23-33.. ..

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