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(1),. Two-Dimensional Quasihomogeneous Isolated Singularities with Geometric Genus Equal to Two by }. i. l. Etsuo YOSHINAGA and Shigeki OHYANAGI. u･･. gl. Introduction. We shall call a hypersurface isolated singularity defined by a quasihomogeneous polynomial, with "a quasihomogeneous isolated singularity". Quasihomo-. geneous isolated singularities have been studied by several authors; Arnold [1,. 2], Orlik and Wagreich [4, 5], Yoshinaga and Suzuki [6], and Yoshinaga and Watanabe [7]. In [6], quasihomogeneous isolated singularities; non-degenerate quasihomogeneous polynomials, are classified under the conditions of their inner. modalities$4. Recently, Yoshinaga-Watanabe showed that the geometric genus P. of a quasihomogeneous isolated singularity is completely determined only by its weights. The quasihomogeneous isolated singularities with p,==O are the rational double points. Those ones with p.=1 are classified as a part of the minimally elliptic singularities by Laufer [3], and also as a part of the quasihomogeneous isolated singularities with inner modalities 1, 2, 3 or 4, by Yoshinaga-Suzuki [6]. In this paper, we classify the quasihomogeneous .isolated singularities with P.==2. In section 4, we list the minimal resolutions of our classified singularities. f'. t i. g2. Non-degenerate quasihemogeneous polynomials.. ei･ 1di･. In this section, we recall some definitions and theorems for quasihomogeneous polynomials. DEFINITIoN 2.1. Suppose that (ro, ri, ･･･,r.) are fixed positive numbers. A polynomial f(go, 2ri, ･･･, z.) is said to be quasihomogeneous of type -(ro,'ri, ･･･, rn). if it can be expressed as a linear combination of monomia!s x8'02iii ･･･ 21'n. for which ioro+iiri+ ･･･ +inrn=1. Let d denote the smallest Positive integer such that ro=qo/d, ri =qi/d, ･･･;. rn=qn/d,whereqi'sarein'tegers. ' ' ' ''' Then we have that. f(tqo2o, tql.7.1, ･･･, tgn2n)==td.f(zo, zb ･･･, 2n)..

(2) 2 E. YosHINAGA and S. OI-IyANAGI DEFiNiTIoN 2.2. A quasihomogeneous polynomial f is said to be non-degenerate if (O, O, ･･･ O) is an isolated critical point (that is, if its hypersurface has an isolated singularity at (O, O, ･･･, O).. In this paper, we consider 2-dimensional hypersurface isolated singularities. So, we have 3 variables zo, 2i and 22･. DEFINITIoN2.3. A quasihomogeneous, possibly degenerate, polynomial f(2o, 2i, 2i) is said to be of type I (II, ･-･,VII resp.) if there is a suitable per-. mutation z of {O, 1, 2} and non-zero complex numbers cro, ai, a2 such that. u. f(ao2.(o), crix.a), a2zrt(2)) is equal to. (I) 2go+2?i+2g2;ail.l2, (II)zgo+2?i+2i2g2;ao,ai>=2,. k .-. i. (III) 2oao+2?i2,+z,za22;a,l2, (IV)2go+2,z?i+2,2g2;a,>=2,. (V) zgo2i+z?ix,+z322o, (VI) zoao+zo2?i+2o2g2;ao>=2, (VII) 2g02,+2o2?i+Zo2g2･ THEoREM 2.4. (Orlife-PVagreich [4]) SuPPose h(zo, 2i, 22) is a non-degenerate. Polynomial. Then h(z)=f(z)+g(2) where f(z) is one of the (I), ･･･,(VII) tyPes above, and f and g have no common monomial. 111e h(z) is quasihomogeneous of tyPe (ro, ri, r2) then f(z) and g(2) are of type (ro, r,, r2). It should be noted that f(2) is not unique.. We say that h(2) is of class (*) if f(2) is of type (*).. DEFINITIoN 2.5. The geometric genus of a surface singularity (X, x)is the tv is a resolution of (X, x). We dimension over C of (Rig*Oi)., where g:X-->X denote this value by P,.. THEoREM 2.6. (Yoshinaga-Watanabe [7]) SuPPose that (X, x) is a hyPersurface isolated singularity dofned by a non-degenerate quasihomogeneous Polynomial of tyPe (ro, ri, r2). Then the geometric genzas of the (X, x) is given by. #{(2o, Rb R2) E! N3ld-(qo+qi+q,)l-ll2oqo+2iqi+R2q2} ,. 'b. where N is the set of all non-negative integers and ri =qi/d dojZned as before. f'. CoRoLLARy 2.7. A quasihomogeneous isolated singularity has P, =O if and only if d<qo+qi+q2･ CoRoLLARy 2.8. A quasihomogeneous isolated singzalarity has P,=1 ij and only if OSd-(qo+qi+q2)<min{qo, qi, q2}･ CoRoLLARy 2.9. A quasihomogeneous isolated singzalarity has P,==2 ijC and only ij qoS.d-(qo+qi+q2)<min{2qo, qi}, under the assumPtion qoS.qiSq2. Fortunately, we can write the weights of the types (I), ･･･,(VII) by ao, ai and a,. The type (I) has (1/ao, 1/a,, 1/a,), the type (II) has (1/ao, 1/ai, (ai-1)/aia2),. the type (III) has (1/a,,(a,-1)/(aia,-!), (ai-1)/(aia2--l)), the type (IV) has. ･(1/ao,(ao-1)/aoai,(aeai-ao+1)/aoaia2), the type (V) has ((aia2-a2+1)/A, <aoa2-ao+1)/A,(aoai-ai+1)/A),where A=aoaia2+1, the type (VI) has (1/ao, (ao-1)/aoai, (ao-1)/aoa2) and the type (VII) has ((ai-1)/(aoai-1), (ao-1)/(aoai-1), ai(ao-1)/a2(aoai-1)).. c,.

(3) Two-Dimensional Quasihomogeneous lsolated Singularities 3 tt g3. Quasihomogeneous isolated singularities with P. =2.. '. '. Let h(zo, 2i, 22) be a non-degenerate quasihomogeneous polynomial of class (I) (resp. (II), ･･･,(VII)), and denote the locus of it by (I)-(ao, ai, a2)(resp. (II)(ao, ab a2), ･･･, (VII)-(ao, ai, a2)),. THEoREM 3.1. ILIe (I)-(ao, ai, a2) has P.==2 then (ao, ai, a2) is one offollows: (2, 3, 12), (2, 3, 13), (2, 3, 14), (2, 3, 15), (2, 3, 16), (2, 3, 17), (2, 4, 8), (2, 4, 9), E. (2, 4, 10), (2, 4, 11), (2, 5, 7), (2, 5, 8), (2, 5, 9), (3, 3, 6), (3, 3, 7), (3, 3, 8) and. (3, 4, 5). '. PRooF. The weights of a polynomial of type (I) are given by (1/ao, 1/ai, 1/a2). Hence, by Corollary 2.9, we shall find all (ao, ai, a2) such that. 2SaoSaiSa2 and 1/a2$1-(1/ao+1/ai+1/a2)<min{2/a2, 1/ai}. Since 1/ao+1/ai+1/a2S.1/ao+1/ao+1/ao=3/ao, 1--(1/ao+1/ai+1/a2)>--1-3/ao. Now 1-(1/ao+1/ai+1/a2)<min{2/a2, 1/ai} .<.1/ao, thus ao<4. Namely ao is either 2 or 3.. (i) Let a,=2. Then 1/a2:Sl/2-(1/ai+1/a2)<min{2/a2,1/ai}. If 2aiEnv{ga2,. 1/a,gl/2--(1/a,+1/a2)<2/a2. We can write this as follows, 4ai+2a2$aaa2< 6ai+2a2. Thus, we have (ai-2)(a2-4)).8 and (ai-2)(a,-6)<12.. Therefore, (ai, a2) is one of the followings. . (3, 12), (3, 13), (3, 14), (3, 15), (3, 16), (3, 17), (4, 8), (4, 9), (4, 10) and (4, rl).. If 2ai>a2, 1/a2Sl/2-(1/ai+1/a2)<1/ai. 4ai+2a2gaia2<2ai+4a2. Then, we have (ai-2)(a2-4);-}i8 and (ai-4)(a2-2)<8. Thus (ai, a2) is one of the followings, <5, 7), (5, 8) and (5, 9).. `. (ii) Let a,=3. Then 1/a2;ll2/3-(1/ai+1/a2)<min{2/a2, 1/ai}. If 2ai:la2, 1/a2.<..2/3-(1/ai+1/a2)<2/a2. We have 6a2+3a2S.2aia2<9ai+3a2. Thus, (ai, a2) is one of the followings, (3, 6), (3, 7) and (3, 8). If 2ai>a,,. 1/a2$2/3-(1/ai+1/a2)<1/ai. So 6ai+3a2S.2aia2<3ai+6a2. Thus,(ai, a2) =(4, 5). ,. Q. E. D.. e. THEoREM 3.2. 1]IC (II)-((ao, ai, a2) has P, =2 then (ao, ai, a2) is one of i'he. (2, 3, 8), (2, 3, 9), (2, 3, 10), (2, 3, 11), (2, 4, 6), (2, 4, 7), (2, 4, 8), (2, 5, 6),. (2, 5, 7), (2, 7, 4), (2, 8, 4), (2, 9, 4), (2, 10, 3), (2, 10, 4), (2, 11, 3), (2, 12, 3), (2, 13, 3), (2, 14, 3), (2, 15, 3), (3, 2, 6), (3, 2, 7), (3, 2, 8), (3, 3, 4), (3, 3, 5), (3, 4, 4), (3, 5, 3), (3, 6, 3), (3, 7, 3), (3, 9, 2), (3, 10, 2), (3, 11, 2), (3, 12, 2), (3, 13, 2), (3, 14, 2), (4, 2, 4), (4, 2, 5), (4, 6, 2), (4, 7, 2), (4, 8, 2), (4, 9, 2>,. (5, 2, 4), (6, 3, 2), (6, 4, 2), (7, 3, 2), (7, 4, 2), (8, 2, 2), (8, 3, 2), (9, 2, 2),. (10, 2, 2) and (11, 2, 2).. The weights of a polynomial of type (II) are given by (1/ao, 1/ai, (ai-1)/aiaij.. The proof of this theorem is similar to the proof of Theorem 3.1..

(4) 4 E. YosmNAGA and S. OHyANAGI THEoREM 3.3. 1]7C (III)-(ao, ai, a2) has P.=2 then (ao, ai, a2) is one of the follows: (2, 3, 7), (2, 3, 8), (2, 3, 9), (2, 3, 10), (2, 4, 6), (2, 4, 7), (2, 4, 8), (3, 2, 5),. (3, 2, 6), (3, 2, 7), (3, 3, 4), (3, 3, 5), (4, 2, 4), (4, 2, 5), (6, 2, 2), (7, 2, 2) and. (8, 2, 2).. '. The weights of a polynomial of type (III) are given by a. (1/ao, (a2-1)/(aia2-1), (ai-1)/(aia2-1)). , THEoREM 3.4. if (IV)-(ae, ai, a2) has P,==2 th2n (ao, ai, a2) is one of the follows:. K.. ". (2, 2, 6), (2, 2, 7), (2, 2, 8), (2, 4, 4), (2, 5, 3), (2, 5, 4), (2, 6, 3), (2, 7, 3), (3, 2, 4),'(3, 2, 5), (3, 4, 3), (3, 5, 3), (3, 6, 2), (3, 7, 2), (3, 8, 2), (3, 9, 2), (4, 2, 4), (4, 5, 2), (4, 6, 2), (4, 7, 2), (6, 3, 2), (7, 2, 2), (7, 3, 2), (8, 2, 2),. (8, 3, 2), (9, 2, 2) and (10, 2, 2).. The weights of a polynomial of type (IV) are given by (1/ao, (ao-!)/aoai, (aoai-ao+1)/aoaia2).. THEoREM 3.5. If (V)-(ao, ai, a2) has P,=2 then (ao, ai, a2) is one of the follozos: (2, 2, 5), (2, 2, 6), (2, 2, 7), (2, 3, 4) and (2, 3, 5).. The weights of a polynomial of type (V) are given by ((aia2-a2+1)/(aoaia2+1), (aoa2-ao+1)/(aoaia2+1,(aoai-ai+1)/(aoaia2+1)). THEoREM 3.6. There is no Polynomial of class (VI), (VII) zvhose locus has an isolaled singularity of P,==2, except for the above.. PRooF. The weights of a polynomial of type (VI) and (VII) are described by the former 3 monomials, i.e., (VI) and (VII) in Definition 2.3. Thus we find 1. the combinations (ao, ai, a2) which satisfies the condition of Corollary 2.9, similar. to Theorem 3.1. And then, we decide the following exponents (P, q);. Pr(ao-1)/aoai+q･(ao-1)/aoa2=i. Namely we make the polynomial of type. t. .4. (VI), or (VII) to be non-degenerate with the 4-th monomial 2?zg, P, q>=2.. For example, we have 2g+2o21+zo23+t212S(tlO) whose weights are (1/2, 1/4, 1/8) and satisfies Corollary 2.9. But this appears in Theorem 3.1. After all, we can see that there is no polynomial of type (VI) and (VII). which has ' P,=:2. ･. ''. '. tt. g4. Resolutions of quasihomogeneous isolated singuEarities with p,==2. Here we list the weighted dual graphs for our enumeration in section 3. There are exhausted 116 polynomials. One can easily check that there are some polynomials which have common weights. For example, (I)-(2, 3, 15) and (II)(2, 3, 10) coincide the weights (1/2, 1/3, 1/15). Such polynomials are combined.

(5) Two-Dimensional Quasihomogeneous lsolated Singularities 5 by some parameters, for example, z3+2?+s･2}5+t･2riz50. After these process, we have 72 different systems of weights. For these weights, using the method in [4], we decide the minimal resolutions. We explain the notations we shall use in Tables. We shall employ Laufer's notations of unweighted dual graphs which he needed to describe minimally elliptic singularities [3].. O denotes a vertex with weight -3 and e denote a vertex with weight -2. These vertices are nonsingular rational curves, unless otherwise specified. ,. List I.. .. El. The vertex A* is a nonsingular. *. C tt. The. *. Ta. A* is a rational curve with a (2, 3)-type cusp point.. *--va--*. Tr. elliptic curve.. The. vertices are nonsingular rational curves which meet tangentially to first order.. The. vertices are nonsingular rational curves which. *･. meet transversaly at the same pomt. '+f. ¥,. `. A *,o. *-O. .. Al,****. Ai, *,o. *. *. *. Ag-,**,, *. *･. A7, *,o. *. *-. t-<) *.

(6) E. YosHINAGA and S. O}IyANAGI. 6. *. D4,*** *. D7,*,o *. *. ,Dg,*,o * ,. E6,** *. * s. E,, *. *. In order todescribe our enumeration, we need to introdece some special notations of weighted dual graphs. List II.. ,A, ee, A, eee. A, e. O, -4Am, O, AS O O.. A-,. The weighteddual graphs are described by giving values for the A*A* in the graphs in ListI. These A*A* are listed from left to right. The union of. subgraphs with theA, identified is indicated by +. Thus for example A*,o+. A*,o+A*,o+A*,o+A*,o with weights A*A* given by -2, -2, -2, -2, -2, denotes theweighted dual graph shown below.. * -2. -2* *-2 -2* *-2. ,. The simple underlined A* is attached to one graph from List II. The doubly (resp. the triply) underlined A* is attached to two (resp. three) graphs from List II. The attaching order is from left to right.. ･ ExAMpLE 1. In Table 3, (12), Tr+Ai+A-3and weights --2, -3, -3 denotes the fEOIiAOMWpinLgE gi.aPihn' Tabie 3, (s), Ei+A,+A,+A, and weight -3 denotes the. following graph.. ? * -3. .. -2 *. ?. -3. *-o. e-*-{ --3. -.

(7) Two-Dimensional Quasihomogeneous Isolated Singularities. 7. Table 1. The weighted dual graph for our enumeration whose. fundamental'cycles Z have ZZ== -1. :. weights. ,. dual graph. 1. (1) (2) (3) (4) (5) (6) (7) (8) (9). t. .. (10). r. 1/2,. 1/3,. 1/12. 1/2,. 1/3,. 1/13. 1/2,. 1/7,. 3/14. 1/3,. 1/5,. 4/15. 1/2,. 1/3,. 2/27. 1/2,. 1/3,. 1/14. 1/2,. 1/3,. 1/15. 1/2, 1/3,. 2/33. 1/2,. 1/3,. 1!17. 1/2,. 1/3,. 1/16. l. I Ta+Ai , Tr+Ai Ab **** + Al. 1 E,,..+A, 1. l Es,.+Ai l j D4,...+A,. I 1/2, 1/10, 3/10 1/2, 1/11, 1/4, 1/6,. 10/33. 5/12. i. F. E. i. Ta+A-s. 1/2,. 1/5,. Ta+ Ai El+ A,+ Ai. 1/7. 1/2, 1/4, 1/9 112,. l Ta + Ai+ Ai. 5/23, 3/23. 11/36. (10). l 1/2,. (11). E 1/2, 1/4, 3/28. I. 1/8, 7/32. (12). 1/2, 1/13, 4/13. (13). 1/2, 1/5, 2/15. (14). 1/2, 114, 1/10. (15). 1/2,. 1 1. (17) (18). l. l I. l. l. 1/2, 1/5, 4/35 1/2, 1/15,. 14/45. (19). 1/2, 1/5, 1/9. (20). 1/2, 7/31,. 3/31. (22). 1/4, 1/11 E 1/2, 1/2, 1/4, 3/32. (23). l 1/2, 1/14, 13/42. (21). F. 1/2, 115,. 1/8. (24). l. (25). , 1/2, 1/10, 9/40 j. A2 A-3 A2 Ai+ Ai. Al, **** + Al. l. Al,****+Al+Al A*,o+A*,o+A*,o. 1/9, 2/9. 1/2, 9/29, 2/29. Ta+ Tr+ Tr+ Tr+. J Al, **** + Ar3. .. 1. El+ AS. 7/23, 2/23. 1/2, 1/12,. (16). El+A., Cu+A-,. 1/2,. 1/2, 1/4, 1/8 E. El+Ai Cu+A, EI+A, El+A,. A.A.. i. -1. =1 =Ll. : -2, -3. -2, -2, -3. l. I modality. I3 134 ls I3 I34 !44 14 I. E =1.. l. i inner. -2, -2, -2, -3. : -2, -3. L. I. =3.. -2, -2, -3. 1. The weighted dual graphs whose Z have ZZ= -2.. Table 2.. (1) (2) (3) (4) (5) (6) (7) (8) (9). r I I l. l. I. +A*,o+A*,o+A2. I. E6,.. + A-3. [. A*,o + D7,*,o + Ai. I. Es,.+A-, Dg,, + A,. 1. A*,o+Dz*,o+A2 E,,., + Al+ Al. l. D,,...+Al+Al. l D4,*** + A-3 l D4,*.* + Al i A*,o+Ag,**,e+A2. -1. 3. =1. 3. -1 -2, -2, -2 -3, -2,. l. -3 -3. 4 3. :. -3 -4. ,. -2,.-2, -=-L3. F. [,. 4 4 3. -2, -2, -3, -2,. 4. I =2 L -2, -3, l l -2, -2, -2, i l -2,. 4. -2, -3 -2, -4 -2, -3 -2, -2,. =3.. 5 5. 6. l l E. -2 .3 -3{,-3 -2, -3 -3, -3 -2, -2. !. 4. 4. -2 -2, -3 i -2, -2. l. 4. -2, -3, -3 -2, -3, -3 -2, -2, -2, -2,. l. l I. 4. v. 6. 4 6 6 5. t. 5. l. 4 5 6. 1. i. l. E l t.

(8) E. YosHiNAGA and S. OHYANA.JE. 8. Table 3. The weighted dual graphs whose Z have ZZ= -3. .(1) (2) (3) (4) (5) (6) (7) (8) (9). [. E. 1. 1/3, 1/9, 914 113,. 1/10, 9/20. l -1. 3. Cu + A-4. =1. 3. 1/3, 2/21, 19/42. Ta+A-4. -2, -3. 4. 1/7, 217, 317. El+ A,+ A3. .T2. 4. 1/3, 1/3, 1/6. 1/8, 7/16, 9/32. El+A,+A,+Ai Ta+Ai+A-3. 4. 1/4, 3/20, 17/40. Ta+ A6. E!3 =,3L, -3 .?-, r4'. 1/6, 5/18, 13/36. Ta+Ai+A2. -5, =--2. 5. Tr+ Ai. -3 -2, -2, -3 =2, -3, -3 -2, ri, -3. 1/3,. 1/4, 1/5. (10). 1/3, 1/11, 5/11. Tr+A-4. (11). 1/4, 1/7, 3/7. (12). 3/25, 7/25, 11/25. (13). 1/3, 2/11, 3/11. (14). 4/25, 7/25, 9/25. Tr+ Tr+ Tr+ Tr+. (15). 1/3,. 5. 4. =L2, -2,. AS Ai+ A-3 A3 A2+ Ai. =2, -2,. 5. 4 4. -:- Ii. 5. =L2, -3,. =4. 5. mz:3L-. 5. Tr+Ai+Ai+Ai. mh3, =3., .-. (16). Al,**** + A-4. 1/3, 1/6., 5/18. Al,**** + A3. (18). 1/3, 1/3, 2/15. Al,.,.. + Al+ A,+ Al. (19). 1/9, 4/9, 5/18. Al,**** + Ai + A3. (20). 1/4, 1/6, 3/8. Al,**** + Al+ Al. (21). 1/3, 1/7, 217. Ai,*,o+Ai,*,o+Ai,*,o. -3 -3 -2, =.-u3, =3, -3 -2, -2, -=3, -3 -2, -2, "=L2, -5 .z,Z, -2, - 2, -2. 4. (17). Ai,*,o+A7,*,o+A3. -2, -2. 8. A*,o + A*,o + A*,o. -2, -2, -2,. 6. 1/7, 2/7, 5/14. + A*,o+ A*,o+ AS A*,o+A*,o+A*,o. -3, -2, -2,. 1/3, 1/13, 6113. D4,***+A-4. l. (22). +Ai,*,o+A3. (23). I. j 1/3, 2/15, 13/45 ] 1/4, 1/8, 7/16. i (24). -2, -2,. E f. E. I. l (25) (26). L. E. (27) (28). i ]. 1/3,. 1/4, 3/16. D4,...+Al+A-,. 1/4, 3/8, 5/32. D4,.*. + Al+ Al. 1/3, 1/8. D4,..*+Ai+A,+Ai. (29). 1/3,. 114, 1/9, 4/9. A*,o + Ag,**,o + AS. (31). 4/31, 9/31, 11/31. A*,o+ Ag,**,o+ A2+ Ai. (32). 1/3j 2/27, 25/54. E6,** + A-,,. l. A*,o+D7,*,o+Ai+Ai Es,.+A-4 A*,o+D7,*,o+A2+Ai. f. 1/7, 3/8. (33). 1/4,. 1/3, 1/14, 13/28. (35). 118,. (36). 7/24, 17/48 1/4, 3/28, 25/56. (37). 1/10, 9/20, 11/40. A*,o+D7,*,o+A5 E6,., + Al+ A-3. 6 5 6. 8. l l. 7. -2, -2. (30). (34). 6. -2, -2. D4,*** + Al. 3/29, 8/29, 13/29. 3,. :. +A*,o+A*,o+Ai+A2. s. - 2,. -=2r, -3, '-. lp. 3. 113, 1/7 1/3, 1/12, 11/24. 1. t. El+ A.,. I. i. -2, -2, -2, -2, -3, -2, -2, -2, -3,. -2, -2, -3, -3, -3, -2, -2, -3 -2. -3 -5 -3 -4 -3 -2 -3. 4 4 5 6. 6 6. 7 4 7. =3 -2, -3 -2, -2 r i -3, -3. 4 7 6 I. 5. '.

(9) Two-Dimensional Quasihomogeneous lsolated Singularities 9 References [1]. ARNoLD, V.I.: Normal forms of functions near degenerate critical points, the Weyl group Ale, Dk, Eh and Lagrange singularities. Funct. Anal. Appl. 6 (1972). [2]. ARNoLD, V.I.: Normal forms of functions in neighborhood of degenerate critical points. Uspehi Mat. Nauk 29 (1974) (10-50). LAuFER, H.B.: On minimally elliptic singularities. Amer. J. Math. 99 (1977). 254-274.. [3]. 1257-1295.. v. [4] ,[ 5] t. ORLiK, P. and WAGREicH, P.: Isolated singularities of algebraic surfaces with C* action. Ann. Math. 93 (1971) 205-228. ORLiK, P. and WAGREicH, P.: Singularities of algebraic surfaces with C* action. Math. Ann. 227 (1971) 183-193.. [6]. YosHiNAGA, E. and SuzuKi, M.: Normal forms of nondegenerate quasihomogeneous. ,[ 7]. YosHiNAGA, E. and WATANABE, K.: On the geometric genus and inner modality. functions with inner modality$4. Inv. math. 55 (1979) 185-206. of quasihomogeneous isolated singularities. Sci. Rep. Yokohama National Univ. 25 (1978) 46-53.. Etsuo Yos}iiNAGA Shigeki OHyANAGi. DepartmentofMathematics InstituteofMathematics Faculty of Education Faculty of Science YokohamaNationalUniversity UniversityofTsukuba. Kanagawa 240 Japan Ibaraki 305 Japan. '. ..

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