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(1)On the Classification of Quartic Surfaces with a Triple Point Part I By. Tadashi TAKAHASHI*, Kimio WATANABE** and Teiichi HIGUCHI*** gl. Introduction. Over seventy years ago, the classification of complex projective quartic surfaces. by their singularities were given by Jessop [1]. The, need for a further account. may be questioned. In this paper, we consider those quartic surfaces which have singular triple points in P3, and classify them by their singularities. We shall need a recognition principle to identify certain types of singularities for functions. not in normal form, since we use the results of Arnold ([2], [3]) and, Bruce and Wall ([4]) in normal forms.. For a triple point P of a quartic surface V in P3, we may choose homogerieous coordinates [X, Y, Z, W] with P at [O, O, O, 1]. The defining equation. of V takes the following form. f,(X, Y, Z)W+f,(X, Y, Z)=O where fi denotes a homogeneous polynomial of degree i (i=3, 4), and the local affine coordinates at P will be given by setting J7V=1. Then, we can see that V is rational. Because there exists a birational mapping:. [X, Y, Z]e[Xf,(X, Y, Z), vr,(X, Y, Z), q,(X, Y, Z), -f,(X, Y, Z)]. v. In [6], it is proved that a quartic surface with a triple point has only one minimally elliptic triple point and other singularities of V are all rational double. pomts. ,.. Our first classification invariant is the type of the cubic form f3. We list. the types of complex projective plane cubic curves: Nodal Curve (denoted by NC), Cuspidal Curve (CC), Conic and Chord (QC), Conic and Tangent (QT), 3 General Lines (TG), 3 Concurrent Lines (TC), Multiple and Single Lines (MS), Triple Line (TL) and Non-singular Elliptic Curve (NS).. *. Graduated school of Education, University of Tsukuba.. ** Institute of Mathematics,'University of Tsukuba. *** Departmeng of Mathematics, Faculty ofl.Education, Yokohama. National University..

(2) T. TAKAHASHI, K. WATANABE and T. HiGucHi. 48. CC. NC. (>-< o. <) TL. MS. TC. TG. Q[D. QC. NS. Fig. 1,. From now, assuming that Y is normal (i.e., V has only isolated singularities), we shall consider those initial 4 cases which have curves of degree >=2 in turn.. There are good tables of typical defining equations for minimally elliptic singularities in [5], we shall use these tables in the next section.. g2. Classification.. Case NC. Thus P is a T3,3,le (k>=4) singularity. And we may choose. coordinates so. that f,(X, Y, Z)=XYZ+Y3+z3. We will say that [1, O, O] is a Nodal point in this case.. in E2 I!iO,o]. zto. y=o f4=O. x=o Fig. 2. LEMMA 1. Let. F=(XYZ+Y3+Z3)VV+f,(X, Y, Z), G==(XYZ+Y3+Z3)VV+g,(X, Y, Z). where A, gi is homogeneous of degree i,'. f,(X, Y, Z) =C,X4+C,X3Y+C,X3Z+C,X2Y2+C,X2YZ+C,X2Z2. +C,XY3+C,XY2Z+C,XYZ2+C,XZ3+C,,Y4+C,,Y3Z +C,,Y2Z2+C,,YZ3+C,,Z4..

(3) Classification of Quartic Surfaces with a Triple Point, I 49 (a) F=O and G= O are projectively equivalent quartic surfaces, the equivalent fixing P, if and only if f,(-03-￠3, 02￠, 0ip2) and g,(-03-ip3, 02ip, 0￠2) are. equivalent as a homogeneous polynomial of degree 12. (b) Singularities of F=O other than that at P correspond to multiple inter-. sections of f3(X, Y, Z)=XYZ+Y3+Z3=:O with f,(X, Y, Z)==O away from [1, O, O] in p2.. (c) A le-tuple intersection of f3==O with f4==O away from [1, O, O] corresponds to an Ale-i singula' rity in F=O. (d) If f4(1, O, O)40, we have'a T3,3,, singularity at P. If f4(1, O, O)=O, we have a T3,3,4+le singularity at P for le =-1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. where k is the multiplicity with the one of two branches, and the multiplicity with the other branch is equal to one. If the multiplicities with the two branches are both at least 2, then F=O has a non-isolated singularity at P. (e) The nodal point [1, O, O, O] is not a singular point.. PRooF. (a) If F=O and G==O are projectively equivalent, the change of coordinates will take the 12 !ines through P on F=O to those through P on G==O, preserving multiplicities. Let these 12 lines are Li, L2, ･･･,Li2, Ll, LE,. ,L(2, then there is a matrix (aij) (ISi, 1':;l12), non-singular, such that (Lb L2, ･･･,Li2) (aij)=(Ll, LS, ･･･,Ll2). Since Lb L2, ･･･,Li2, Ll, LS, ･･･, Ll2 are the solutions of f3(X, Y, Z)=:O with ft(X, Y, Z)=O and f,(X, Y, Z)=O. with '''. g4(X,Y,Z)=O, f4(0,g)=(aiO+big)(a20+b2g)･･･(ai20+bi2g), where Li= aiO+bisD, g4(0,go)=(alO+bl{p)(aEO+bEgD)･･･(al20+bl2sp), where L;･=aEO+b;･sD. Therefore f4(0, g) and g4(0, g) are linearly equivalent as a 12 degree equation. Conversely if f4(0, g) and g4(0, g) are linearly equivalent, the change of coordinates on [0, g] in Pi induces an automorphism of the Nodal curve. Therefore for arbitrary g5(X, Y, Z), f,(X, Y, Z)==sgg(X, Y, Z)+(XYZ+Y3'. +Z3)L for some non-zero s, linear form L. By a change of coordinates of the form J7V'=M7-L, f,(X, Y, Z)=sg6(X, Y, Z)+(XYZ+Y3+Z3)(J7V'-W). And substitute to F==O. Consequently F==I7V'(XYZ+Y3+Z3)+sg,(X, Y, Z). Q.E.D.. (b) Let Q be a singular point on V which is not P. Byachange of coordinates we may assume Q=[O, 1, -1, O] (when 0=1, g=-1). And moreover let Coeff.(GI, 0igj):=[coefficient of a term 0`gj of equation GI].. Then f,(X, Y, Z)==O has a multiple intersection with f4(X, Y, Z)=O if. Coeff.(f4(-1-g3,g,g2),g`)=O for i=O,1 and (O/aX)F==O. In this case, (O/OX)F=:O, (O/OY)F==O, (a/OZ)F=O.. And the converse is similar. Q. E. D.. (c) Let. Sub.(X=1¥!+aiY+ ･･･,Y =Y'+biX+ ･･･, Z= Z'+ciX+ ･･･, GI); = [substitution X= X' +aiY+ ･･･, Y=:Y'+biX+ ･･･, Z== Z'+ciX.

(4) 50 T. TAKAHAsHi, K. WATANABE and T. HiGucHi + ･･･, to equation GI] for ai, ･･･,bi, ･･･,cb ･･･ are constants.. Since we consider at [1:O:O], set Z=1, Y==Y'---1 to obtain. CoX`+CiN3+CiX3Y+CioY`--4CioY3+6CioY2m4CioY+Cio+CiiY3. -3C,,Y-C,,+C,,Y2-2C,,Y+C,,+C,,Y--C,,+C,,+C,X3+C,X2Y2. -26,X2Y+C,X2+C,X2Y--C,X2+C,X2+C,XY3-3C,XY2. +3C,XY-C,X+C,XY2--2C,XY+C,X+C,XY-C,X+C,X+WXY. -WX+WY3-3WY2+3WY==O. This equation has an A-type singularity, since this equation has a term VVX. And we put Go:=this equation. Then !et Coeff.(Go, X`Yjl7Vk):==as above (2). If Coeff.(Go, Y2)= Coeff.(Go, XJ7V)=Coeff.(Go, YVTi)==O, then Go has an A2 singularity at [1, O, O, O]. Therefore we submit the following analytic transformations.. Gl:= Sub.(X== X' +3Y +3Y3+ Y`+ Y5, G,).. G2:==Sub.(VV=W'+Coeff.(Gl, XY)Y, Gl). G3:==Sub.(W==VV'+Coeff.(G2, XY2)Y2, G2).. s G4:=Sub.(W=W'+Coeff.(G3,XY3)Y3,G3). G5:=Sub.(l7Pi =W!+Coeff.(G4, XY`)Y`, G4).. G6:=Sub.(VV=W'+Coeff.(G5, XY5)Y5, G5).. G7:=Sub.(W=W'+Coeff.(G6, XY6)Y6, G6). Then we have an Ak-i singularity at [O, 1, -1, O] for k-tuple intersection of f3(X, Y, Z)==O with f4(X, Y, Z)=O (by (b)) by recognition principle (i.e. if Coeff.(G7, XY`)==O'fbr 1;Si<le Coeff.(G7, XYk)#O, we have an Alepi singu-. larity.). Q. E.D. (d) From (b),' at the first time we consider the case of le=O. Then the. condition of .k=:O is equivalent to CoiO.. The equation determines a Newton boundary, and all the terms of F=O other than X`, Y3, Z3, XYZ exist outside of its Newton boundary. Topological type of Tp,,,. singularity is determinned by its Newton boundary. Hence when le=O we have a T3,3,4 singularity at P.. Next we consider the case of le==1. Then Co=O and CitO and C2tO, and all the terms of F==O other than X5, Y3, Z3, XYZ exist outside of the Newton boundary (5, O, O), (O, 3, O), (O, O, 3), (1, 1, 1) bY the following analytic transforma-. tions:. Z==Z'-C,X2, Y==Y'-C,X2. And the cases of 2=< le S11 are also similar by the following analytic transformations 'list:.

(5) Classification of Quartic Surfaces with a Triple Point, I. 51. Analytic transformations list with respect to 2Sk.<,.,11 of T3,3,4+fe. singularity in case NC '' Z=Z'-C,X2,. Y=IY-i-3C?X3+2C,C,X3+3C,XZ'-C,X'Z',. fe= 2, 3, 4, 5, 6. Zi=Z"+C,C,X3-C,XYi. Eliminate inside terms of the each Newton. k=7, 8, 9, 10, 11. boundary by a term XYZ. We eliminate one parameter step by step. And from f3(X, Y, Z)=O, by a coordinates W=17V'-C4X-C7Y-CsZ) we can eliminate 3 parameters. And moreover 1 parameter is remaining for moduli in this case. Hence Max･(T3,3,r)=T3,3,is. This is also corresponding to the multiplicity of f,(X, Y, Z)==O with f,(X, Y, Z) =O. Dual graph of T3,3,. (r:$4) is as follows.. ttttl. o o. Z O .,. Xs,. T3,3,4. 1gk. T 3,3,4+k Fig. 3. If the multiplicities for two branches are at least 2, then from the parameter,. Co=Ci=C2=O.Hencewehaveanon-isolatedsingularityatP. Q.E.D. (e) If we have a singularity at [1, O, O, O], then the coefiEicients of X`, X3Y,. X3Z are O. And now since f3(X, Y, Z)=XYZ+Y3+Z3, we have a non-isolated. singularity at [1, O, O, O]. Q. E. D. ExAMpLE The following equations have T3,3,. singularity at [O, O, O, 1]. ExAMpLE 1. T3,3,ii. F=X3Y+X2Z2-XY2Z+Y3Z+PV(XYZ+Y3+Z3). ExAMpLE2. T3,3,i2. F=X3Y+X2Z2+Y4+YZ3+PV(XYZ+Y3+Z3). ExAMpLE 3. T3,3,i3. F=X3Y+X2Z2+XY3+2Y4+Y2Z2+YZ'3+VV(XYZ+Y3+Z3).. s.

(6) 52. T. TAKAHAsHi, K. WATANABE and T. HiGucHi ExAMpLE4. T3,3,i4. F==X3Y+X2Y2+X2Z2+XYZ2+2Y4-Y3Z+YZ3+W(XYZ+Y3+Z3). ExAMpLE5. T3,3,is. F==1(N3Y+X2Z2+2Y4+YZ3)+17V(XYZ+Y3+Z3). ExAMpLE 6. T3,3,ii. F==X3Y--4X2Y2+.i¥2Z2+6XY3+2XY2Z-4XYZ2+5Y3Z+6Y2Z2. +3YZ3+W(XYZ+Y3+Z3). ExAMpLE 7･ T3,3,u. F== X3Y -2X2Y2+ X2Z2-2XY2Z -2XYZ2+2Y4+3Y3Z. -YZ3+W(XYZ+Y3+Z3). ExAMpLE 8. T3,3,ii. F==X3Y+X2Z2-2XY3+2Y4-Y3Z-2Y2Z2+YZ3+VV(XYZ+Y3+Z3). ExAMpLE 9. T3,3,n. F==X3Y-2X2Y2+X2Z2+XY3-2XYZ2+Y3Z+Y2Z2+YZ3. +W(XYZ+Y3+Z3). ExAMpLE 10. T3,3,i2. F---X3Y-3X2Y2+X2Z2+3XY3-3XYZ2+Y4+3Y3Z+3Y2Z2+YZ3. +W(XYZ+Y3+Z3).･ ExAMpLE 11. T,,3,i2. F= X3Y+X2Y2+X2Z2-XY3+XYZ2+Y4-Y3Z-Y2Z2+YZ3 +W(XYZ+Y3+Z3). ExAMpLE 12. T3,3,i3. F=:X3Y-2X2Y2+X2Z2+XY3-2XYZ2+2Y4+2Y3Z+Y2Z2+YZ3. +W(XYZ+Y3+Z3). We consider these examples.. Example number. f,(0, g) (0==1). Partition away from [1, O, O, O]. Example 1. g7(g4+1). 14. Example 2. g8(g3+1). 13. Example 3. g9(g2+ 1). 12. Example 4. giO(q+1). 1. Example 5. 911. o. e.

(7) Classification of Quartic Surfaces with a Triple Point, I. Example 6. i. g7(g-1)4. Example 7. l. g7(g-1)3(g+1). Example 8. l. 1. Example 9. 1. Example 10 ,. Example 11. 1. Example 12. i. 53. 4 1. 3. 1. g7(g-l)2(g+1)2. 2. 2. i i. g7(g-1)2(g2+1). ;. 12. 2. g8(q-1)3. 3. g8(g-1)(g+1)2. L2. g9(g-1)2. 2. As a consequence of lemma 1, the singularities occurring are determined by the partition of 12, 11, 10, 9, 8, ･･･,2 given by coincideness of the intersection. points of f,(X, Y, Z)=f,(X, Y, Z) =O.. * When ColO. (This is the list of singularities other than P, where P is T3,3,4 singularity.). Partition. List 1iO.2. 1i2. Type of sing･ I 17. 2. 3. A,. A,A,. 17.5 l 1 l. 16. 6. 15. 2. 2. 3. 2A,. A,. 2A,A,. l. 14. 2, 2. 4. l. 3A,. 15. 7. A,A,. A,A,. A,. 14. 2. 3. 3. 14.2.6 l. 14. 4. 4. 14. 8. 13. 2. 2. 2. 3. A,. 3A,A,. 2A,A,. 13. 3. 3. 3. 3. 13. 3. 6. 1 13.4.s. }. 14. 3. 5. A,A,. A,2A,. A,A,. I tj. 13.2.2.5/ I' 13.2.3.4. A,A,A, 13. 9. 1 A,A, i. L. 4A,. A,A,. 12. 2. 2. 2. 2, 2. 12. 2. 2. 2. 4. 12. 2. 2. 3. 3. 5A,. 3A,A,. 2A,2A2. A,A,. r AiA3. I is.3.4. 2A,A,. 13. 2. 7. 16. 2. 4. 15. 2. 5. 4A,. 2A, 1I t!. 18. 2. 2. i 2A,. 16. 2. 2. 2. A,. 16. 3. 3. ; i. A,. A,. 18. 4. 14. 2, 2. 2. 2. i 19.3. 12.2.2.6 1 2A,A,. A, 12. 2. 3. 5. l, A,A,A, l. i.

(8) T. TAKAHAsHi, K. WATANABE and T. HiGucHi. 54. 12. 2. 4. 4. A,2A,. 12. 2. 8. 12. 3. 5. 12. 4. 6. A,A,. A,A,. A,A,. 12. 5. 5. 12. 10. A,. 2A,. 1. 2. 2. 2, 2, 3. 1. 2. 2. 2. 5. 1. 2. 2. 3. 4. 1, 2. 2. 7. 1. 2. 3. 6. 4A,A,. 3A,A,. 2A,A,A3. 2A,A,. A,A,A,. 1. 2. 4. 5. 1. 2. 9. 1. 3. 3. 5. 1. 3. 4. 4. 1. 3, 8. 1. 4. 7. A,A,A,. A,A,. 2A,A,. A,2A,. A,A,. A,A,. 1. 5. 6. 1. 11. 2. 2. 2. 2. 2, 2. 2. 2. 2. 2. 4. 2. 2, 2, 3.3. A,A,. Aio. 6A,. 4A,A,. 3A,2A2. 2. 2. 2. 6. 2. 2, 3. 5. 2. 2. 4. 4. 2. 2. 8. 2. 3. 3. 4. 2. 3. 7. i. 2A,A,. 2A,A,A4. 2A,2A,. 2A,A,. A,2A,A,. A,A,A,. l. 2. 4. 6. 2. 5. 5. 2. 10. 3. 3. 3, 3. 3. 3. 6. 3, 4. 5. A,A,A,. /1,2A,. A,A,. 4A,. 2A,A,. A,A,A,. 3. 9. 4. 4. 4. 4. 8. 5. 7. 6. 6. A,A,. 3A,. A,A,. A,A,. 2A,. '. ･･. 12. All. * When Co=C2=O and Ci40. . (This is the list of singularities other than P, where P is T3,3,s singularity.) :･. Partition. 1io. Type of sing. 15. 5. A,. 14. 2. 2. 2. 3A, ,. List. 18. 2. 17. 3. 16. 2. 2. 16, 4. 15. 2. 3. A,. A,. 2A,. A,. A,A,. 14. 2. 4. A,A,. 13. 2, 5. 13. 3. 4. 13. 7. A,A,. A,A,. A,. 12. 2. 6. 12. 3. 5,. 12. 4. 4. A,A, -･. A,A, '. 2A,. 14. 3. 3. 2A, 12. 2. 2. 2. 2. 4A,. 14. 6. 13. 2. 2. 3. A,. 2A,A,. 12. 2. 2. 4. 12. 2. 3. 3. 2A,A,. A,2A,. .12. 8. '1. 2. 2. 2. 3. A,. 3A,A,. 1.2.2:5,i. 2A,A,.

(9) Classification of Quartic Surfaces with a Triple Point, I. 55. 1. 2. 3. 4. 1. 2. 7. 1. 3. 3. 3. 1. 3.6. 1. 4. 5. 1. 9. A,A,A,. A,A,. 3A,. A,A,. A,A,. A,. 2. 2. 2. 2. 2. 2. 2. 2. 4. 2. 2. 3.3. 2. 2.6. 2. 3. 5. 2. 4. 4. 5A,. 3A,A,. 2A,2A,. 2A,A,. A,A,A,. A,2A,. '. .. 2. 8. 3. 3. 4. 3. 7. 4. 6. 5. 5. 10. A,A,. 2A,A,. A,A,. A,A,. 2A,. A,. We can similarly make the lists of remaining cases (from le =2 to le = 10). And when fe=11, its quartic surface has a T3,3,is singularity at EO, O, o, 1] and no singularity except [O, O, O, 1] (see. example 5).. Case CC We may choose coordinates so that f,=X3-Y2Z.. Thus P is a Q-type. singularity (see Laufer [5]). x=o [O,O,1]. f4"O. [O,l,O] Z=O. /. LEMMA 2. Let. y =o. ' 4. Fig. ''. F==(X3-Y2'Z)W+f,(X, Y, Z), j. G =(I¥3-Y2Z)VV+g,(X, Y, Z).. , (a) F=O and G==O are projectively equivalent quartic surfaces, the equivalent fixing P, if and only if f4(02g, 03, g3) and g4(02g, 03, g3) are equivalent for. 12 degree equations.. (b) (1) For each singularity Q#P, the line PQ lies on V so that we have a common root of f3(X, Y, Z)=O and ft(.X, Y, Z)=O. This has multiplicity>1.. (2) Each root determines a line L through P on V. If the root has multiplicity >1, there is precisely one other singular point on L.. (c) A k-tuple root of f3(X, Y, Z)=O with f4(X, Y, Z)=O away from [O, O, 1] in P2 corresponds tO an Ak-i singularl'ty. (d) If f4(X, Y, Z) iEO (i.e., Ci4#O), we have a Qie singularity at P.. If Ci4=O and CglO, then we have a Qii singularity at P. If Ci4=Cg=O and Ci3tO, then we have a Qi2 singularity at P. (e) Point [O, O, 1, O] is not a singular point..

(10) 56 T. TAKAHAsHi, K. WATANABE and T. HiGucHi i. PRooF. (a) This holds as (a) of the above lemma 1. Q.E,D. (b) (1) By a change of coordinates we may suppose that Q is [O, O, 1, O] so that in f4(X, Y, Z)=O the Y`, XY3, Y3Z terms are zero. Thus f4(02g, 03, g3). has q2 as a factor, i.e., the root has multiplicity >=2, Q.E.D. (2) Suppose that (0, g)=(1, O) is a multiple root of f4(0, q) =O, then clearly Q==[O, 1,O,･ O] is a,singular point of V. Moreover there are no further points. alongthelinePQsince . ' OF Of, == - Y2ur +. oz. oz. vanishes on this line only atPand Q. Q,E.D. We will write. f4(02g, 03, g3)==C!oOi2+C60iig+C30'Og2+CiO9g3+CnO9g3+CoO8g`. +C,08g4+C,e7g5+C,,06g6+C,e6g6+C,05g7 +CsO`q8+Ci303g9+CgO2giO+Ci4gi2. (c) Suppose that the multiple root is at [1, O] for [0, g]. Then we obtain the following list.. 2-tuple root. if and only if Cio= C6 =O and C3#O. 3-tuple root. if and only if C io=C6= C3==O and CiSO. 4-tuple root. if and only if C io =C6==C3==Ci =O and C74O. 5-tuple root. if and only if Cio ==C6==C3=Ci= C7==O and. 6-tuple root. if and only if C io-=C6==C3 =Ci=C7=C4==O and C,,40. 7-tuple root. if and only if C lo=C6=C3=Cl==C,-=C4=Cz2==O and. 8-tuple root. if and only if Cio== C6==C3=Ci==C,=C4=Ci2=Cs==O. 9-tuple root. if and only if Cio= C,==C3==Ci :C,==C4 =Ci2== Cs==Cs==O. C,tO. C,#O. and C,#O. and. Ci3 l O. Cs=Cs=Ci3==O. 10-tuple root. if and only if Cio=C6=C3==Cl==C7=C4=Cl2==. 12-tuple root. if and only if C,o = C6 = C3 = Cl == C, = C4= Cl2 = Cs= C, =C 13 :' =- Cg. and Cg40. =O and Ci440. (There is no 11-tuple root at [O, 1, O, O], for there is no term 0gii in f4(0, g).). And setting Y=1, submit the following analytic transformation, First step ･････････Z =Z'+x3, Second step ･･････Z' = -Z",. Third step ･･････W==W'+2Ci2X3+3Ci3X6+4Ci4X9+C4X2+2CsX5+C7X+3CgX7 +2C,X4,.

(11) Classification of Quartic Surfaces with a Triple Point, I. 57. Fourth step ･･････VV'=W"-(Ci2+3Ci3X3+6Ci4X6+CsX2+CsX+3CgX`)Z", Fifth step･････････1717M=:PV(3)+C,,ZM2+4C,,X3ZM2-C,,Z"3+C,XZ"2,. ARd we rewrite Z", W(3) to Z, M/. Then F=O become as follows.. F=M/Z+C,X2+C,X3+C,X4+C,X5+C,,X6+C,X7+C,X8 +C,,X9+C,XiO+C,,Xi2. Next we suppose [0, q]=[1, 1] is a multiple root of f4(02g, 03, g3)=O, and let Coeff.(GI, 0`gj):=as above Lemma 1. Then f3(X, Y, Z)==O has ak-tuple intersection with f4(X, Y, Z)==O at [1, 1, 1, O] if Coeff.(f4(g-1, 1, (g-1)3), gi)=O for. 1$iS le-1 and Coeff.(f4(g-1, 1, (g-1)3), gle) #O.. And we submit the following transformations for F=O. Setting X==1,. Y=Y'+1 and Z= Z'+1, Y'= Y"-(1/2)Zt, YM == Y (3) + (3/8) Zi2,. y(3)=y(4)+(2560/8192)Zt3, Y(4) = Y(5) +(35840/131072)Zi4,. equation symbol. Y(5).= Y(6) +(132120576/536870912)Zi5, ････････････････････････････････････-･･････････････gi. PV= VV'+(1/2). Geeff. (gi, Y(6' Z')Z', ･････････････････････････････････････････''････････'''g2. W'= W"+(1/2). Coeff.(g2, Y(6)Z'2)Z'2,･･-･･･････････････-･･････････････'･''''''''''''''''g3. W"=W(3)+(1/2). Coeff.(g3, Y(6'Z'3)Z'3, ･････-･･････････････････････････････････････････g4 W(3)==W(`)+(1/2).Coeff.(g4, Y(6)Z'`)Z'`, ･･･････････････････････････････････････''''''gs. W(`)=W(5)+(1/2).Coeff.(gs, Y(6)Z'5)Z'5, ･･･････････････････････････････････-･････････g6 rewrite Y(6), W(5) to Y, W, ･････････････････････････････････････････････････'''････'･･''''g7. By the above transformations. Coeff.(g,,WZ`)=O for ISi$5 Coeff.(g,,YZi)=O for lgi.<.5.. Then. t/. /il COfe,ff,' (lg-.7'iig.Ztk)=-Ol ll if and only if ,il' COeff'(f`(P-1, 1, (97,1,)3)i,9,:ti)<=leO. lCoeff.(g7, Zle)7LO I･ if and only if l/ Coeff,(f4(g-1, 1,(g-1)3),gk)tO I. Hence a fe-tuple root of f4(g-1, 1, (g-1)3)==O corresponds to an Ak-i singularity (, but at [1, 1, 1, O] there is no fe=12 since Coeff.(g7, Zii)==12Ci4 and. Coeff. (g,, Zi2)=Ci,). Q. E. D. (d) Now Ci4 or Cg or Ci34O, since F=O has an isolated singularity. If. Ci4 7EO, weights for (X, Y, Z) of F=O is (1/3, 3/8, 1/4), then we have a Q,, singularity at P..

(12) T. TAKAHAsHi, K, WATANABE and T. HiGucHi. 58. If Ci4=O and CgtO, weights for (X, Y, Z) of F=O is (1/3, 7/18, 2/9), then we have a Qii singularity at P. If Ci4=Cg==O and Ci3#O, weights for (X, Y, Z) of F=O is (1/3, 2/5, 1/5),. then we haveaQi2 singularity at P. Q.E.D.. (e) If F==O has a singularity at [O, O, 1, O], the Z`, YZ3, XZ3 terms are. zero. But F==O has a non-isolated singularity at P. Q.E.D. When Ci4==O and Cg40, f3(X, Y, Z)=O has multiplicity=2 with f,(X, Y, Z). =O. When Ci4==Cg==O and Ci3tO, f3(X, Y,Z)=O has multiplicity=3 with f,(X, Y, Z)=O. Blow-up F=O at [O, O, 1, O]. Then F=:O becomes as follows. f3(X,Y,Z)=o i(. t. ,N. /. y=o. 1. y. y2.x3 y=o. <Tbr6W:ii51 w-up. 4. g. E3 (X,Y,z)=o t. /. t"N. x=o. Ns.d-. ts-.zZ. ×=o. y2.×3. y. <-Tbiil6W:{Ii51 w-up. g. Fig. 5.. This corresponds to the above result.. As a consequence of lemma 2, the singularities occurring as above are determined by the partitions of 12, 10, 9 given by coincideness of the intersection. points of f,(X, Y, Z)=O with f,(X, Y, Z)=O.. * When Ci4tO. (This is the list of singularities other than P, where P is a Qio singularity.) List. Partition. 112. 1iO. 2. 19. 3. 18. 2. 2. 18. 4. 2A,. A,. 1. Type of sing.. A,. A,.

(13) 59. Classi丘cation of Quartic Surfaces with a Triple Point，1. 17．2．3. 17．5. ．41／12. ．44. 16．2，2．2 3／41. 15，2．2．3. 15．2．5. 15．3．4. 2／11．42. ．41．44. ．42／13. 14」2．3．3. 14．2．6. 14．3．5. ．412、42. 、41．45. ．42．4．4. 13．2．2．5. 13．2．3．4. 13．2．7. 2、41、44. 。41、42．43. ノ11、46. 13．9．. ．48. n2左41. 12．2．2．2．2．2！2．2．2．2．4 5／11. ユ2．2．8. 3／11．43. 12．3．5. 16．2．4 ．41／13. 15．7 。46. 14．4．4 2．43. 16．3．3 2．42. 14．2．2．2，2 4／11. 14．8. 16．6 ．45. 14．2．2．4 2！41、43. 13．2．2．2．3. ノ17. 3．41．42. 13．3．6. 13・4・5. ．42z45. 4・孟・. 12．2．2．3．3. 12．2．2．6. 12・3・3・5. 2／112、41. 2／11．45. 。41／42／14. 1・．4．6. 1・．5．5. i・．10. 13．3．3．3 3．42. 且，. ．41／17. 。42、44. 1．2．2．2．2．3. 1．2．2．2．5. 1．2．1 Q．3．4. 4／11．42. 3、41／14. 2．41、42／13. 2、41・46. 1．2．9. 1．3．3．5. 1．3．4．4. 1．3．8. 1．4．7. 。41、48. 2、42、44. 、422、43. ．42∠し. ・43∠［6. 1．11. 2．2．2．2．2．2. 2．2．2．2．4. 2．2．2．3．3. 2．2．2．6. ．410. 6／11. 4／11、43. 2．2．4．4. 2．2．8. 2．3．3．4. 2．3．7. 2．4．6. 2．5．5. 2．4ユ2／13. 2．4ユノ17. 。4エ2．42／13. ノ4ユノ12．46. ．41．43。45. ．4ユ2／44. 2．10. 3．3．3．3. 3．3．6. 3．4．5. 3．9. 4．414. ．41、49. 4／12. 2．42／15. ．42．43．44. 、42．48. 3／13. 12. 4．8. 5．・7. 6．6. 且・4・. ．44／16. 2、4．5. ．43／15. 舶、. 。412、43. 1川μ廓. 13燃. ノ41ゴ. オ、且必、. 3／11．45. 1．2．玉. S15. 為ン1、オ、. 1．・6．16. 2．2．3．5・ 2／41／42／44 ’.

(14) 60 T. TAKAHAsHi, K. WATANABE and T. HiGucHi * When Ci4=O and Cg#O. (P is a Qii singularity.) Partition. 1io. Type of sing.. 15. 2. 3. 15. 5. A,A,. A,. 13. 2. 2. 3. 13. 2. 5. 2A,A,. 18. 2. 17. 3. A,. A,. 2A,. 14, 2.4. 14. 3. 3. 14. 2, 2. 2. 3A,. 16. 2. 2. [ 16.4 A,. {. 14. 6. A,A,. 2A,. A,. 13. 3. 4. 12. 2. 2. 2. 2. 12. 2. 2. 4. 12. 2. 3. 3. A,A,. A,A,. 4A,. 2A,A,. A,2A,. 12. 2. 6. 12, 3. 5. 12. 4. 4. 12. 8. 1, 2. 2. 2. 3. 1. 2. 2. 5. A,A,. A,A,. 2A,. A,. 3A,A,. 2A,A,. 1. 2. 3. 4. 1. 2. 7. 1. 3. 3. 3. 1. 3. 6. 1. 4, 5. 1. 9. A,A,A,. A,A,. 3A,. A,A,. A,A,. A,. 2. 2. 2. 2. 2. 2. 2. 3. 3. 2. 2, 6. 2. 3. 5. 2. 4. 4. 2. 8. 5A,. 2A,2A2. 2A,A,. A,A,A,. A,2A,. A,A,. 3. 3. 4. 3, 7. 4, 6. 5. 5. 10. A,A,. 2A,. 2A,A, i A,A,. t. A,. * When Ci4=Cg==O and Ci3tO. (P is a Qi2 singularity.) Partition. lg. Type of sing.. 14. 2. 3. 14. 5. A,A,. A,. 12.2,2.3 ). 2A,A,. 12. 2. 5. A,A,. 17. 2. 16. 3. 15. 2. 2. 15. 4. A,. A,. 2A,. A,. 13. 2. 2. 2. 1- 13.2.4. 3A,. }. 12. 3. 4. A,A,. ). 13. 3. 3. 13. 6. A,A,. 2A,. A,. 12. 7. 1. 2. 2. 2. 2. 1. 2. 2. 4. A,. 4A,. 2A,A,.

(15) Classification of Quartic Surfaces with a Triple Point, I. 61. 1. 2. 3. 3. 1. 2. 6. 1. 3. 5. 1. 4. 4. L8. 2. 2. 2. 3. A,2A,. A,A,. A,A,. 2A,. A,. 3A,A,. 2. 2. 5. 2. 3. 4. 2. 7. 3. 3. 3. 3. 6. 4. 5. 9. 2A,A,. A,A,A,. A,A,. 3A,. A,A,. A,A,. A,. REMARK. In case NC, F=O did not have an Aio singularity at [O, 1, -1, O]. This corresponds to a multiplicity of Nodal point =2, i.e., when F= O through [1, O, O, O], F=O has a multiplicity =2 at [1, O, O, O].. Similarly in case CC, since [O,O, 1, O] itself has a multiplicity ==2, F=O does not have an Aio singularity at [O, 1, O, O]. But in case CC, we obtained an Aio singularity at [1, 1, 1, O]. Then F=O did not have an Aii singularity at that point. This will correspond to the multiplicity of a point other than [1, O, O, O] and [1, 1, 1, O]. This is also similar to the case NC.. About T,,,,. singularity, there is a result of the paper of Karras [7].. rnui:Yl?ill:il8¥oilo f4(x,y,z)=o Jt f4(×,YrZ)=O .:e---mult'iplicityZ2 at [O,lrO,O]. N "xf3(x,y,z)=o ". - f3(X,Y,Z)=o. multiplicity 2. at[o,o,1,o] ' '･･. multiplicity 12. '' ･u/atiL,L.L,o],. .rrtultiplicity 10. at[l,1,1,O] ･. A f,Tx,y,zfe6X'Y'Z'"O ' NIS},,,f,`-.iX,IY.6Z'"O. .,, multiplicityl multzplzcity2 at?. at [OrO,1,O]. Fig. 6.. Case QC We may choose coordinates so that f3(X, Y,Z)==Z(XY+Z2). Thus P is a T3,,,.singularity. And [1,O,O,O] and [O,1,O,O] are intersection points of. Z=O with XY+Z2==O. M Z=O ln.2 [1,O,O] ,. xy+z2 =o [O,1,O]. Fig.7. '. LEMMA3. Let F=(Z(XY+Z2))17V+f,(X, y, z).. '.

(16) T. TAKAHAsHi, K. WATANABE and T. HiGucHi. 62. (f,(X, Y, Z)==C,X4+C,X3Y+C,X3Z+C,X2Y2+C,X2YZ+C,X2Z2. +C,XY3+C,XY2Z+C,XYZ2+C,XZ3+C,,Y4+C,,Y3Z +C,,Y2Z2+C,,YZ3+C,,Z4.) (a) Singularities of F==O other than that at P correspond to multiple intersections of f3(X, Y, Z)=O with f4(X, Y, Z) :O away from [1, O, O, O] and [O, 1, O, O].. (b) A le-tuple intersection of f3(X, Y, Z)=O with f4(X, Y, Z)=O away from [1, O, O, O] and [O, 1, O, O] corresponds to an Ak-i singularity. (c) If f,(1, O, O) ,E O and f4(O, 1, O)40, then we have a T3,4,4singularity at P. If f4(1, O, O)= O or f4(O, 1, O)=O, 'i.e., [1, O, O] is a leo-tuple intersection of. Z=O with f4(X, Y,Z)=O and a 7'o-tuple intersection of .[¥Y+Z2=O with f,(X, Y, Z)=O and [O, 1, O] is a lei-tuple intersection of Z=O with f4(X, Y, Z). =O and a A-tuple intersection of XY+Z2==O with f4(X, Y, Z)=O, then we have a T3,4+,,4+r singularity at P for q=Max.(leo, io), r=Max.(lei, 7'i) where ko+ki= O, 1, 2, 3, 4, io+ii=O, 1, 2, 3, 4, 5, 6, 7, 8 and ko, 7'o are not both at least 2, and. i are not both at least 2. (when ko==ki==io=1'i==O, that is above statement.) If feo, 1'o are both at least 2 or ki, 7'i are both at least 2, then F=O has a. lei, 7'. non-isolated singularity at P.. PRooF. (a) If Z =O has a multiple intersection with f4(X, Y, Z)=O, say at [1, 1, O, O], then f4(1, 1, O)=O and A(1, Y'+1, O) has a Y'2 as a multiple factor, and f4(1, 1, Z) has a Z2 as a multiple factor. Therefore let Sub.(･･･):=as above, Coeff.(･･･):==as above, its condition is as follows.. Sub. (X==1, Y==1, Z==O, f,)==O and Coeff.(f4(1, Y'+1, O), Y'O)==Coeff.(f,(1, Y'+1, O), Y') =O. and Coeff.(f4(1, 1, Z), ZO)=Coeff.(f,(1, 1, Z), Z)==O.. Then Sub.(X=1, Y=1, Z==O, VV==O, F)=O, Sub. (X=1, Y=1, Z =O, 17V= O, (O/OX)F)==O, Sub.(X==1, Y--1, Z==O, VIi= O, (O/OY)F)=O,. Sub.(X=1, Y=1, Z=O, I?V=O, (O/OZ)F)==O. Hence we have a singularity at [1, 1, O, O]. The converse is similar.. Sirnilarly if XY+Z2=O has a multiple intersection' with f4(X, Y, Z)==O say at [1, -1, O, O], i.e., when 0=1, g--1 of f4(02, -g2, 0g), then we haveasingu-. larity at [1, -1, 1, O]. The converse is similar. Q.E.D, (b) With the multiple point [1, 1, O, O] as in (a) we set. X=1, VJV=:Wr-C,X-C,Y-C,Z, Y'=Y'+1. Then F=O is as follows..

(17) Classification of Quartic Surfaces with a Triple Point, I 63. F= (Co+Ci+C3+C6+Cio)+(Ci+2C3+3C6+4Cio)Y +(C2+C,+C,+C,,)Z+(C,+3C,+6C,,)Y2. +(C,+4C,,)Y3+C,,Y4+WZ +------ For k==2, 3, 4 choose weights (1/le, 1/2, 1/2) for (Y, Z, W) and apply the recognition principle to obtain the result.. With the multiple point [1, -1, 1, O] as in (a) we set X==1, and moreover we submit the following ' transformations.. Y=Yi-1, Z =Z'+1,. equation symbol. Yi= YM-2Zt- Zi2,･･･････････-････････････････････････････････gi W==W'-Coeff. (gi, Y"Z')Z', ･･････････････････････････････････g2. W'= I7V"-Coeff. (g2, Y"Z'2)Z'2, ･････････････････････-･･'･'･'･''g3 I7V"=W(3)-Coeff.(g,, Y"Z'3)Z'3, ･･･････････････-･･････････････g4 M/(3) == J7V(`) -Coeff. (g,, Y"Z'4)Z'`,･････････'''''''''''''''''''''gs. Then for le==2, 3, 4, 5, 6, 7,8 choose weights (1/2, 1/2, 1/le) for (Y, I7V, Z) and apply the recognition principle to obtain the result.. (c) If f4(1, O, O)40 and f4(O, 1, O)IO, then ColO and CiotO, F==O determine a Newton boundary (4, O, O), (O, 4, O), (O, O, 3), (1, 1, 1). And eliminate the terms. other than X`, Y`, Z3, XYZ by analytic' transformations as above lemma 1.. x. (4tOrO) Newtonboundary (4tOtO)r(Or4tO)r(OtO,3)r(ltl,1). (1,1,1). z. <'OtO,3). Y o,4,O). Fig. 8.. Then F=O become as follows.. F=X`+Y`+Z3+LXYZ+g(X,Y,Z), (settingVV=1) where all the terms of g(X, Y,Z)=O are outside of the Newton boundary (4, O, O), (O, 4, O), (O, O, 3), (1, 1, 1).. Hence we have a T3,4,4 singularity at P. If f4(1, O, O)=O or f4(O, 1, O)==O, then as above lemma 1 we eliminate the terms other than Xa, YP, Z3, XYZ by a certain analytic transformations. Then. weobtaintheresult. -'' ･.

(18) 64 T. TAKAHAsHi, K. WATANABE and T. If feo, 7'o are both at least 2, then Co==Ci=C2==O, singularity at P. Similarly if lei, A are both at Ieast. HIGUcHI. we have. a non-isolated. 2, then C,==Cio =Cii= O,. Q. E. D.. we have a non-isolated singularity at P.. Dual graph of T3,q,r. l- '" -" '---:i"0'sli'"----l. I il lI i ili li 1･ l. l i ttt tl1. l--v i'oi--..it l' l tt. 2O=q :O-r Fig. 9. On the point [1, 1, O, O] there exists precisely a le-tuple root. Now 1-tuple root is clear. And from (b), for le-tuple intersection its condition is as follows.. C,+C,+C,,+C,+C, =O ･･････････････････････････････(1). C,+4C,,+C,+3C,=O･･･････････････････t･.･.........(2) 6C,,tC,±3C,==O････････････････････････････････････(3) 4C,,+C,==O ････････････････････････････････････････(4). From (1), Ci=-Ce-Cio-C3-C6'''''''''''''''･'･''''(1'). From (1') and (2), C3=Co-3Ciom2C6 '''''･････････････････････････････(2') From (2') and (3), C,==-Co-3C,,････････････････････(3'). From (3') and (4), Cio=Co･ Hence there is fe ==1, 2, 3, 4.. For a le-tuple root of XY+Z2=O with f4(X, Y, Z)==O, this is similar.. And by parameters or linear system (resolution by blow-up), we obtain Max,(3+q+r)==21. Hence Max･(T3,q,r)=T3,7,n･. ExAMpLE 1. T3,4,i2 The following equation has a T3,4,i2 singularity at P.. F=X3Y+X2Z2+Y4+VV(XYZ+Z3)..

(19) Classification of Quartic Surfaces with a Triple Point, I 65 ExAMpLE 2. T3,7,ii The following equation has a. T3,7,ii singularity at P.. F=R(.X3Y+X2Z2+Y3Z)+W(XYZ+Z3). Applying the above lemma to various configuration of a quartic and QP one obtains the following list.. * When feo=io=fei==7'i =O･ (This is the list of singularities other than P. P is a T3,4,4 singularity･) '. Intersection symbol. 14. 16, 2. 14. 15, 3. 14. 14, 2, 2. A,. A,. 2A,. 14. 12, 2, 2, 2. 14. 12, 2, 4. 14. 12, 3, 3. A,. 3A,. A,A,. 2A,. 14. 1, 2, 5. 14. 1, 3, 4. 14. 18. Singularities. 14. 14, 4. 14. 13, 2, 3. A,. A,A,. 14. 12, 6. 14. 1, 2, 2, 3. A,. 2A,A,. 14. 2, 2, 4. 14. 2, 3, 3. 2A,A,. 14. 13, 5'. A,A,. A,A,. 14. 2, 6. 14. 3, 5. A,2A,. A,A,. 12, 2. 18. 12, 2. 16, 2. 12, 2. 15, 3. A,. 2A,. . A,A,. 12, 2. 13, 5. A,A,. 4A, 14. 8. A,A,. 2A,. Ati. 12, 2. 14, 2, 2. 12, 2, 14. 4. 12, 2. 13, 2, 3. 1. I. 2A,A,. A,A,. 12, 2. 12, 3, 3. 12, 2. 12, 6. 2A,A,. A,2A,. A,A,. 12, 2. 1, 7. 12, 2. 2, 2, 2, 2. 12, 2. 2, 2J 4. 12, 2. 2, 3, 3. 5A,. 3A,A,. 2A,2A2. 12, 2. 8. 1, 3. 18. 1, 3. 16, 2. 12, 2. 1･, 3, 4. 2A,A,. A,A,A,. 12, 2. 2, 6. 12, 2. 3, 5. 12, 2. 4, 4. 2A,A,. A,A,A,. A,2A,. 1, 3. 15, 3. 1,'3. 14, 2, 2. 1, 3. 14, 4. 1, 3. 13, 2, 3. A,A,. i A,2 A2" .. ' 2A,A,. A, 14. 4, 4. 12, 2. 1, 23 5. 2A,. 14. 2, 2, 2, 2. 3A,. 12, 2. 12, 2, 2, 2 I 12, 2. 12, 2, 4. ,4A,. 14, 1, 7. A,A,. AiA,. A, 1, 3. 13, 5'. A,A. ,. 12, 2. 1, 2, 2, 3. 3A,A,. A,A, 1, 3. 12, 2, 2, 2･-. L 3A,Ab. rv,,.

(20) T. TAKAHAsHi, K. WATANABE and T. HiGucHi. 6i6. 1, 3. 12, 2, 4. A,A,A,. 1, 3. 12, 3, 3. s. 1, 3.' l2, 6. 3A,. A,A,. 1, 3. 1, 2, 2, 3. l 2A,2A,. 1, 3. 1, 2, 5. 1, 3. 1, 3, 4. A,A,A,. 2A,A,. 1, 3. 1, 7. 1, 3. 2, 2, 2, 2. 1, 3. 2, 2, 4. 1, 3. 2, 3, 3. 1, 3. 2, 6. A,A,. 4A,A,. 2A,A,A3. A,3A,. A,A,A,. 2A,A,. 1, 3. 4, 4. 1, 3. 8. 2, 2. 16, 2. 2, 2. 15, 3. 2, 2. 14, 2, 2. A,2A,. A,A,. 2, 2. 14, 4. 2, 2. 13, 2, 3. 2A,A, l 2, 2. 12, 6. 3A,A, 2, 2, 1, 2, 2, 3. 2A,A, ). 4A,A,. l 2,2.ls 2A,. 2, 2. 13, 5. I 2A,A,. 3A,. 2A,A,. 2, 2. 12, 2, 2, 2 2, 2. 12, 2, 4. 5A,. 2, 2. 1, 2,5 i 2, 2. 1, 3,4. l 3A,A, i2A,A,A,. 1, 3. 3, 5･. 4A,. 2, 2. 12, 3, 3. 3A,A,. 2A,2A2. 2, 2. 1,7. 2, 2. 2, 2, 2, 2. 2A,A,. 6A,. 2, 2. 2, 3, 3. 2, 2. 2, 6. 2, 2. 3, 5. 2, 2. 4, 4. 2, 2. 8. 3A,2A,. 3A,A,. 2A,A,A4. 2A,2A,. 2A,A,. 4. 18. 4. 16, 2. 4. 15, 3. 4. 14, 2, 2. 4. 14, 4. 4. 13, 2, 3. A,. A,A,. A,A,. 2A,A,. 2A3. A,A,A,. 4. 12, 2, 4. 4. 13, 3, 3. 4. 12, 6. 4. 1, 2, 2, 3. 3A,A,. A,2A,. 2A,A･,. A,A,. 2A,A,A3. 4. 1, 2, 5. 4. 1, 3, 4. 4. 1, 7. 4. 2, 2, 2, 2. 4. 2, 2, 4. 4. 2, 3, 3. A,A,A,. A,2A,. l A,A,. 4A,A,. 2A,2A3. A,2A,A,. 4. 2, 6. 4. 3, 5. 4. 4, 4. 4. 8. A,A,A,. A,A,A,. 3A,. A,A,. 2, 2. 2, 2, 4. 4A,A, ]. 4. 13, 5. A,A,. i 4. 12, 2, 2,2. The notatiQn 12,2.12,2,2,2, for example, means that f4(.X,,Y, Z)==O has two simple and one double intersections with Z==O and'/two simple. and three double. intersections with XY+Z2=O..

(21) Classification of Quartic Surfaces with a Triple Point, I. 67. EXAMPLE. z=o. L2,2.12r2t2,2. f4(XtY,Z)=o. xy+z2=o in E2' [1,O,O]. Z-tuple. 2-tuple '. 2-tuple l--tuple 2--tuple. 1-tuple. 2-tuple Io,L,o]. l-tuple. Fig. 10. We can make the lists of remaining cases.. Case QT. So we take f3(X, Y, Z)=Z(XZ'+Y2). Thus P is a S-type singularity.. LEMMA 4. Let F==W(Z(XZ+IY2))+A(X, Y, Z). (a) Singularities of P=O other than at P correspond to inultiple intersec-. tions of Z=O with f,(X, Y, Z)==O and XZ+Y2=O with f,(X, Y, Z)=O away. from [1, O, O]. '. (b) A le-tuple intersection of Z =O with f4(X, Y, Z)=O or XZ+Y2=O with. f4(X, Y, Z,)= 0 away from [1, O, O] corresponds :to an Ale-i singularity.. (c) Let J be the multiplicity of f4(X, Y, Z)=O with;Z=O at [1,-O, O], and let L be the 'multiplicity ef- A(X, Y, Z)=:O with XZ+Y2=O at [1, O, O]. Then the singular point P is determined by the following condition list.. ' Condition list s, K. L. o. o. Sil. 1. 1. Si2. 2. 2. Si4. 3. 2. t- ' s13+.2. Type. of ,Singularity. '.

(22) T. TAKAHAsHI, K. WATANABE and T. HiGucHi. 68. 4. 2. S(3+3. 2. 3. S13+2. 2. 4. S13+3. 2. 5. S13+4. 2. 6. S13+s. 2. 7. S13+6. 2. 8. S13+7. If L L are both at least 3, then we have a non-isolated singularity at P.. PRooF. (a) and (b) hold as above lemma 3, (c) When 1= L=O (i.e., Co 7E O), choose weights in terms X`, XZ2, Y2Z and apply the recognition principle. Then we have a Sii singularity at P. Similarly. when J==L=1, we have a Si2 singularity at P, when J=L==2, we have a Si4 singularity at P by using the recognition principle.. And in remaining cases F--O become the following equations by analytic transformations.. (L= 2) (W == 1) F==Z(XZ-2Y2)=2X2Y2+X5+X`-JY'+g(X, Ys Z),. (K==2). F=Z(XZ-2Y2)+X2Y2+(Y2+X3)X"+i+g(X, Y, Z), where all terms of g(X, Y, Z) are outside of the Newton boundary of F==O. Hence we have a SI3+2.+3b-g singularity (when L=2) or Si3+n singularity. (when K=2) at P. ,. ･If L L are both at least 3, then Co==Ci=C2=O. Hence we have a non-. isolatedsingularityatP. , ･ Q.E.D. Duai graph of s'J3+n 1 ,-. ,. O="2. '' '. '. 1 ･'･ .-3 , ' ･.----<n-2)-- '･. t.t t1 ..., ,, ..,,-i,. ' '. '. --4 Dual graPh Of SJ3+n ･-. ---in-. C n-2 )-----. -3. Fig. 11.. 4.

(23) Classification of Quartic Surfaces with a Triple Point, I. We. 69. can make the list as above.. (When 1=L=O) list. Intersection symbol. 14. 18. Singularities. 14. 16, 2. 14. 15, 3. 14. 14, 2, 2. A,. A,. 2A,. 14. 14, 4. 14. 13, 2, 3. 14. 13, 5. 14. 12, 2, 2, 2. 14. 12, 2, 4. A,. A,A,. A,. 3A,. A,A,. ----e---e--------------ttt-------------et--------}}--------- 'e ････････････････････To be continued.. .. The notation is as same as Case QC. We can make the lists of remaining cases.. EXAMPLE.. in p2. f4(X,Y,Z)=o. [x,o,o]. xz+y =o. .. J=2 L=8. z=o Fig. 12.. P is a Si3+7 singularity.. References [1] JEssop, C.M.: Quartic surfaces with singular points. Cambridge at the univ. press, (1916).. [2] ARNoLD, V.I.: Normal forms of functions near degenerate critical points, the Weyl groups Ak, Dk, Ek and Largrange singularities. Func. AnaL App. 6, (1972), 254-272.. [3] ARNoLD, V.I,: Normal forms of functions in the neighborhoods of degenerate critical points. Russian Math. Surveys, 29(2), (1974), 11-49.. [4] [5]. BRucE, J.W. and WALL, C.T.C.: On the classification of cubic surfaces. J. London Math. Soc. (2), 19 (1979), 245-256. LAuFER, H.B.: On minimally elliptic singularities. Amer. J. Math. 99, (1977), 1257-1295. ･.

(24) 7. 70. T. TAKAHAsAHr, K. WATANABE and T. HiGucHi. [6] UMEzu, Y.: On Normal Projective Surfaces with Trivial Dualizing Sheaf. Tokyo J. Math., 4 (1981), 343-354.. [7]. KARRAs, U.: Deformations of cusp singularities. Proc. Symp. in Pure Math. A.M.S. 30, (1977), 37-44.. [8] HiGucHi, T., YosHiNAGA, E. and WATANABE, K.: Introduction to complex analysis of several variables (in Japanese), Morikita Library in Mathematics 51 (1980). c. b. ..

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