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PROJECTIVE CLASSIFICATION OF JETS OF SURFACES IN P4 6 JORGELUIZDEOLINDOSILVAANDYUTAROKABATA 1 0 2 n a Abstract. We are interested in the local extrinsic geometry of smooth sur- J facesin4-space,andclassifyjetsofMongeformsbyprojectivetransformations 3 accordingtoA3-types oftheircentralprojections. 2 1. Introduction ] G We are concerned with the geometry of the contact of smooth surfaces in the D projectivespaceRP4 withtheir tangentlines. Thecontactismeasuredbytypes of . map germs of central projections of the surfaces. In the present paper we classify h jets of generic surfaces by projective transformations which preserve the geometry t a of the contacts of surfaces with their tangent lines. m For surfaces in RP3, Platonova [20, 21] completed the classification of generic [ surfacesby projective transformations. The classificationof generic two parameter families of surfaces is done in [8] (see also [1, 10, 11, 12]). The study of surfaces 1 v in RP4 was proposedin [2] with relation to the classification of singularities which 5 appear in central projections of generic surfaces in R4 by D. Mond [15, 16, 17]. 5 Howeverthere have been no results about the classificationof surfacesso far. This 2 paper gives an answer to the proposalin [2] with the complete classification list of 6 jets of generic surfaces in RP4 by projective transformations. 0 . On the other hand, the study of geometric aspects of surfaces in 4-space is a 1 relatively new subject and has a lot of analogy to the study of surfaces in 3-space 0 which have been investigated in [5, 6, 13, 14, 18]. 6 1 Let M be a smooth surface embedded in R4 ⊂ RP4 containing the origin of : R4 where R4 is identified with the open subset {[x;y;z;w;1]} ⊂ RP4. We write v (z,w) = f(x,y) = (f (x,y),f (x,y)) as the Monge form of M at the origin where i 1 2 X f (0,0)= df (0,0)=0 for i =1,2. Two jets of surfaces at some points are said to i i r be projective equivalent if there is a projective transformationon RP4 sending one a to the other. Our result is the following. Theorem 1. There is an open everywhere dense subset O of the space of compact smooth surfaces M in RP4 such that the germ at each point on M in O is projectively equivalent to a germ with the k-jet of the Monge form of one of the cases in Table 1. Observe that the last column of Table 1 means the types of central projections of corresponding surfaces from view points on asymptotic lines. In section 2, we briefly explainthe stratificationofthe space ofjets of Mongeforms induced by the stratification of jets of germs of central projections, and review the stratification of the 3-jet space of Monge forms induced from the A3-stratification of central projection germs which was originally done in Ph. D thesis of Mond [15]. In Section3weobtainsimplenormalformsofjetsofMongeformsthatrepresenteach 1 2 JORGELUIZDEOLINDOSILVAANDYUTAROKABATA Type Normal form Condition cod Proj. Π (x2−y2+y2(φ +φ ),xy+ψ ) 0 − E 1 2 4 Π (x2+y3+yφ ,y2+αx3+xψ ) α6=0 0 S S 3 3 Π (x2+y3+yφ ,y2+xψ ) − 1 S,B B 3 3 Π (x2+yφ ,y2+xψ ) − 2 B 2B 3 3 Π (x2+βxy2+y3+yφ ,xy+xψ ) − 1 H H 3 3 Π (x2+xy2+λy4,xy+γy3+ψ ) γ,Λ6=0 2 P P 4 Π+ (x2+y2+k x2y+yφ ,ψ +ψ ) b −b 6=0 2 S,B I 1 3 3 4 30 12 Π− (xy+k x3+φ ,ψ +ψ ) b ,b 6=0,a =0 2 S,B,H I 2 4 3 4 30 03 22 Table 1. Strata of codimension ≤ 2 in the space of 4-jets of Monge forms corresponding to A3-types of central projections on asymptotic lines. Here φ = a xiyj,ψ = b xiyj, s i+j=s ij s i+j=s ij α,β,γ,λ,k ,k ,a ,b ∈R aPre moduli parametersPand Λ=6γ2+ 1 2 ij ij 4λ−15γ +5. Surface germs of Π -type do not have asymptotic E lines. stratum given in the previous section by projective transformations, and give the proof of Theorem 1. Remark 1.1. Our normal forms in Table 1 contain a lot of moduli parameters including coefficients of higher order terms of degree grater than 4. They must be interpreted as some projective differential invariants. For example, when we look at the A-types of the central projection of the Π -type surfaces germs, it is observed S that the central projection from a view point on the asymptotic line is A-equivalent toP (c):(x,xy2+cy4,xy+y3)wherec is amoduliparameter, and λ =c. Thefirst 3 γ author [7] found also that γ and λ are expressed by combinations of some cross- ratioinvariants andtheydeterminesthetopological typeofBDE(binarydifferential equations) of asymptotic curves. Acknowledgements: We would like to thank Takashi Nishimura and Farid Tari for organizing the JSPS-CAPES no.002/14 bilateral project in 2014-2016. The authors are supported by the project for their stays in ICMC-USP and Hokkaido University,respectively. The firstauthorthanks alsothe FAPESPno.2012/00066- 9 to support part of this work. We are also very grateful to Farid Tari and Toru Ohmoto for their supervisions. 2. The stratification of the 3-jet space of Monge forms Mond, in Chapter III of his Ph.D thesis [15], stratified the jet space of Monge forms according to A-types of germs of central projections. In this section we explain the way to stratify the jet space of Monge forms according to types of central projections, and review Mond’s stratification for the 3-jet space of Monge forms. Take a surface M in R4 ⊂ RP4 with the Monge form (z,w) = f(x,y) = (f (x,y),f (x,y)). Let V denote the space of polynomials in x,y of degree greater 1 2 ℓ than 1 and less than or equal to ℓ. Our aim here is to obtain a stratificationof the ℓ-jet space of Monge forms V ×V which is induced from the Aℓ-stratification of ℓ ℓ Jℓ(2,3) obtained in [15, 16] as follows. PROJECTIVE CLASSIFICATION OF JETS OF SURFACES IN P4 3 Type Normal form cod hyperbolic (x2,y2) 0 elliptic (x2−y2,xy) 0 parabolic (x2,xy) 1 inflection (x2+y2,0) or (xy,0) 2 degenerate inflection (x2,0) 3 degenerate inflection (0,0) 4 Table 2. TheclassificationofJ2(2,2)(whichisequaltothe 2-jet spaceofMongeformsf =(f ,f ))byGL(2)×GL(2)-actionsgiven 1 2 by Gibson in [9]. Consider a point p∈RP4, which is sometimes called a view point, not lying on M and define π : RP4−{p}→RP3 as the canonical projection which associates p x∈RP4−{p} to the line generatedby x−p. The central projection of the surface M from p is given by the composite map ϕ :=π ◦ι:M →RP3 p,M p (see also [8]). We denote the central projection of the surface germ expressed in Monge form f from a view point p by ϕ . We stratify V ×V by the difference of Aℓ-types of p,f ℓ ℓ jℓϕ for a view point p∈RP4−M and the ℓ-jet of a Monge form jℓf ∈V ×V . p,f ℓ ℓ Aℓ means the equivalence of jets of map germs, i.e., two jets jℓg,jℓh∈Jℓ(2,3) are equivalent if and only if there exist jets of diffeomorphism germs σ,τ of the source andthetargetattheoriginssuchthatjℓh=jℓ(τ◦g◦σ−1). Remarkthatevenforthe same ℓ-jet of the surface the central projection gives different Aℓ-types depending on the view point. For example, jℓϕ is always regular type (i.e. equivalent to p,f (x,y,0)) if and only if p is outside the tangentplane ofthe surfacegerm; and gives a singularity if and only if p is on the tangent plane to the surface. We say that a line on the tangent plane which goes through the origin in R4 is anasymptotic line of the surface given in Monge form f ifϕ isequivalenttoone p,f of singularities worse than a crosscap (S -type) for all view points p on the line. 0 In Section 3 we show that the classification of 2-jets of Monge forms by projective transformations coincides with the classification of J2(2,2) by GL(2) × GL(2)- actions in Table 2, and each orbit is characterized by the number of asymptotic lines [5, 16]. For elliptic type with the form j2f = (x2 + y2,xy), there are no asymptoticlines;forhyperbolictype withtheformj2f =(x2,y2), xandy-axisare twouniqueasymptoticlines;forparabolictypewiththeformj2f =(x2,xy),y-axis is a unique asymptotic line; for inflection type with the form j2f =(x2+y2,0) or (xy,0), all lines on the tangent plane which go through the origin are asymptotic lines. Thuswestudytypesofcentralprojectionsfromviewpointsontheasymptotic lines, and divide strata in Table 2 into finer ones. Defineasmoothmap,theMonge-TaylormapΘ:M →V ×V ,whichassociates ℓ ℓ toeachpointinM the ℓ-jetofMongeformf =(f ,f )atthe point. The following 1 2 is a natural extension of Bruce’s Theorem in [3]. 4 JORGELUIZDEOLINDOSILVAANDYUTAROKABATA Name Normal form S (x,y2,xy) 0 S (x,y2,y3+x2y) or (x,y2,y3) B (x,y2,x2y) or (x,y2,0) H (x,xy,y3) P (x,xy+y3,xy2) R (x,xy,xy2) T (x,xy+y3,0) U (x,xy,0) Table 3. A3-orbits of germs R2,0→R3,0 [15, 16]. Theorem 2. Let Z ⊂ V ×V be an GL(2)×GL(2)-invariant submanifold. For ℓ ℓ generic surface M in R4, the Monge-Taylor map Θ:M →V ×V is transverse to ℓ ℓ Z. Proof : The proof follows the same arguments in the proof of Theorem 1 in [3] (see also [6]). ✷ Since the strata are induced from Aℓ-types of central projections, they are GL(2)×GL(2)-invariant. ByTheorem2we consideronly stratawith codimension at most 2. In this section, we study the stratification of 3-jets of Monge forms which is induced by the A3-orbits in [15, 16, 17] or their unions given in Table 3. Write f (x,y)= a xiyi, f (x,y)= b xiyi. 1 ij 2 ij i+Xj≥2 i+Xj≥2 Thenϕ canberegardedasamapgermR2,0→R3. Indeed,forp=[a;b;c;d;1]∈ p,f R4−M, we choose a6=0, then ϕ is given by p,f y−b f (x,y)−c f (x,y)−d 1 2 ϕ (x,y)= , , . p,f (cid:18)x−a x−a x−a (cid:19) On the other hand, if p is taken at infinity and written as p = [a;b;c;d;0], ϕ is p,f given by ϕ (x,y)=(y−ux,f (x,y)−vx,f (x,y)−wx) p,f 1 2 with (u,v,w)=(b, c, d) (see also [8]). a a a The following sums up the Propositions III. 2:2, 2:8, 2:14, 2:16 and 2:17 in [15]. Proposition 2.1. (Mond[15]) For a surface germ in Monge form (z,w)=f(x,y) we have the following. (i) Suppose that j2f =(x2,y2). Then: j3ϕ ∼S ⇔ a 6=0 (resp. b 6=0) p,f 03 30 j3ϕ ∼B ⇔ a =0 (resp. b =0) p,f 03 30 for p on the asymptotic line x=0 (resp. y =0). PROJECTIVE CLASSIFICATION OF JETS OF SURFACES IN P4 5 (ii) Suppose that j2f =(x2,xy). Then: j3ϕ ∼H ⇔ a 6=0 p,f 03 j3ϕ ∼P ⇔ a =0 and a ,b 6=0 p,f 03 12 03 j3ϕ ∼R ⇔ a =b =0 and a 6=0 p,f 03 03 12 j3ϕ ∼T ⇔ a =a =0 and b 6=0 p,f 03 12 03 j3ϕ ∼U ⇔ a =a =b =0 p,f 03 12 03 for p on the unique asymptotic line x=0. (iii) Suppose that j2f =(x2+y2,0). Then: j3ϕ ∼S orB p,f for any p on the xy-plane. (iv) Suppose that j2f =(xy,0). Then: j3ϕ ∼S,B orH ⇔ b 6=0 p,f 30 j3ϕ ∼S,B orP ⇔ b =0 and a ,b 6=0 p,f 30 30 21 j3ϕ ∼S,B orR ⇔ a =b =0 and b 6=0 p,f 30 30 21 j3ϕ ∼S,B orT ⇔ a =b =0 and b 6=0 p,f 30 21 30 j3ϕ ∼S,B orU ⇔ a =b =b =0 p,f 30 30 21 for any p on the xy-plane. Proof : Statement (i). It is easy to check that the x and y-axes are asymptotic lines at the origin for M in Monge-form with j2f = (x2,y2). Suppose p is at the y-axis and written as p=(0,a,0,0), then, by coordinate changes, we get j3ϕp,f ∼A3 (x,a(a21a−1)x2y+a03y3,y2). Ifa 6=0, j3ϕ (0) isofS-type,otherwiseit is ofB-type. If p isatinfinity onthe 03 p,f y-axis, we obtain j3ϕp,f ∼A3 (x,xy,a21x2y+a03y3). Again, if a 6=0, j3ϕ (0) is of S-type, otherwise it is of B-type. By exchanging 03 p,f x and y (also a and b ), the case of p on the x-axis is follows similarly. ij ij Statement (ii). Remark that y-axis is a unique asymptotic line at the origin when j2f(0) = (x2,xy), hence we put p = (0,a,0,0)∈ R4 ⊂ RP4 (a 6= 0) and we get j3ϕp,f ∼A3 (x,a12axy2+a03y3,xy+b03y3). Statement (ii) for p∈R4 naturally follows, and the case p at infinity is similar. Statement (iii). Let view point p=(a,b,0,0)∈R4 and a6=0, then we get j3ϕp,f ∼A3 (x,y2,ξ1x2y+ξ2y3) where ξ and ξ are homogeneous polynomials of degree 3 with variables a and b 1 2 whose coefficients consist of a and b . Statement (iii) for p = (a,b,0,0) ∈ R4 ij ij where a 6= 0 naturally follows, and it is easily seen that the projection gives just S -type for view points p = (0,b,0,0) ∈ R4 where b 6= 0. The case p at infinity is 1 similar. Statement (iv). For view point p = (a,b,0,0) ∈ R4 where a,b 6= 0, it is easily seenthatj3ϕ ∼S orB inthe similarwaytothe above. Putp=(a,0,0,0)∈R4 p,f where a6=0 and we get a b j3ϕp,f ∼A3 (x,xy− 30y3,b21xy2− 30y3). a a Statement (iv) for p∈R4 naturally follows, and the case p at infinity is similar. ✷ 6 JORGELUIZDEOLINDOSILVAANDYUTAROKABATA Based onProposition2.1, we stratify the 3-jet space of Monge forms into strata with codimension at most 2 as in Table 4. In the next section we give simple normalforms of 4-jets of Monge forms which represent each stratum in Table 4 by projective transformations. Name Type of 2-jet Condition cod Proj. Π (x2−y2,xy) − 0 − E Π (x2,y2) a ·b 6=0 0 S S 03 30 Π b =0, a 6=0 1 S,B B 30 03 Π a =b =0 2 B 2B 03 30 Π (x2,xy) a 6=0 1 H H 03 Π a =0, a ·b 6=0 2 P P 03 12 03 Π+ (x2+y2,0) − 2 S,B I Π− (xy,0) b 6=0 2 S,B,H I 30 Table4. Strataofcodimension≤2inthespaceof3-jetsofMonge forms corresponding to A3-types of central projections from view points on asymptotic lines. Surface germs of Π -type do not have E asymptotic lines. Remark 2.1. By taking higher order terms of the simple normal forms in the Table 1, we can consider a finer stratification of the space of Monge forms which corresponds to A-types of central projections as Mond did in [15]. For instance, we take the Monge form of the Π -type in Table 1 and write f = (x2 + y3 + S a xiyj,y2+αx3 + b xiyj) with α,a ,b ∈ R, α 6= 0 and a = i+j≥4 ij i+j≥4 ij ij ij 40 Pb04 = 0. Then the conditioPn a31,b13 6= 0 determines the proper stratum of the Π -stratum where the A-types of the central projection can be determined as the S regular, crosscap, S or S -type depending on the position of the view points (See 1 2 [15]). 3. The classification of Monge forms by projective transformations and proof of Theorem 1 In this section we consider a classification of jets of Monge-forms of generic surfaces by projective transformations based on the stratification in Table 4. The projective linear group PGL(5) is defined as the quotient space GL(5)/ ∼, where A∼A′ if∃λ∈RsuchthatA=λA′. ToconsidertheactiononV ×V (ℓ-jet-spaces ℓ ℓ of Monge-forms), we define the following subgroup G(5):={Ψ∈PGL(5)|Ψ(0)=0, Ψ(W)=W} of PGL(5), where 0=[0;0;0;0;1]is the originand W is the xy-plane in R4. Thus G(5) form a 16-dimensional subgroup of PGL(5) and acts on V ×V . ℓ ℓ Let f =(f ,f ) and g =(g ,g ) be Monge forms of surface-germs at the origin. 1 2 1 2 We say that the k-jets of these Monge forms are projectively equivalent and write jkf ∼ jkg if there exists Ψ ∈ G(5) which transforms one to the other. Remark that Aℓ-types of central projections of jets of smooth surfaces are invariant under projective transformations of surfaces, that is, jℓϕ ∼ jℓϕ from view p,f Aℓ Φ(p),Φ(f) point p and Φ∈G(5). PROJECTIVE CLASSIFICATION OF JETS OF SURFACES IN P4 7 In this paper we check the equivalence of jets of Monge forms in the following way. With the coordinate (x,y,z,w) of R4, a projective transformation Ψ ∈ G(5) is regarded locally as a diffeomorphism germ R4,0→R4,0 given by q (x,y,z,w) q (x,y,z,w) q (x,y,z,w) q (x,y,z,w) 1 2 3 4 Ψ(x,y,z,w)= , , , , (cid:18) p(x,y,z,w) p(x,y,z,w) p(x,y,z,w) p(x,y,z,w) (cid:19) where q =q x+q y+q z+q w, for i=1,2, q =q z+q w, for j =3,4 and i i1 i2 i3 i4 j j3 j4 p=1+p x+p y+p z+p w. Define 1 2 3 4 F (x,y,z,w)= q3 −f (q1,q2) 1 p 1 p p F (x,y,z,w)= q4 −f (q1,q2). 2 p 2 p p Then F (x,y,g ,g )=F (x,y,g ,g )=o(k) 1 1 2 2 1 2 where o is Landau’s symbol implies jkf ∼jkg. Hence, tocheckthe equivalence,wehaveto solvealgebraicequationsF =F = 1 2 o(k)intermsofq sandp sforagivenMongeformf =(f ,f )andsomesimplified i i 1 2 normal form g = (g ,g ). Recall that we already have a stratification of the 3-jet 1 2 space of Monge forms induced from the A3-stratification as in Table 4, hence our task is to find a simple normal form of each stratum by projective equivalence. We begin with simplifying 2-jets of Monge forms and then deal with higher jets. However we stop this process with the 4-jets, since the dimension of G(5) which act on the jet space of Monge forms is just 16 and it does not give so good normal forms for higher jets. 3.1. 2-jet. We first deal with the classification of 2-jets of Monge-forms. In the 2-jetspace,the conditionF =F =o(2)foranyj2f,j2g ∈V ×V givesequations 1 2 2 2 ofjustq ,q ,q ,q withi=1,2andj =3,4,andtheclassificationbyprojective i1 i2 j3 j4 transformationsis reducedto the classificationofV ×V ⊂J2(2,2)by the natural 2 2 action of G = GL(2,R)×GL(2,R). The G-orbits are classified in [9] described as Table 2. We classify now the higher jets of germs with a 2-jet as in Table 2. 3.2. Elliptic case. Suppose that j2f =(x2−y2,xy) and write j3(f ,f )=(x2−y2+ a xiyj,xy+ b xiyj) 1 2 ij ij i+Xj=3 i+Xj=3 where a ,b ∈R. The following equivalence ij ij j3(f ,f )∼(x2−y2+y2φ ,xy), 1 2 1 is given by projective transformationΦ with q =x+b z+(−a +b −b )w, q =y−b z+(−b +b +a )w, 1 03 31 12 30 2 30 21 03 30 q =z, q =w, p=1+(a +2b )x+(2b −a )y. 3 4 30 03 12 21 Here φ means homogeneous polynomials of degree k. Consider k j4(f ,f )=(x2−y2+y2φ + c xiyj,xy+ d xiyj) 1 2 1 ij ij i+Xj=4 i+Xj=4 where c ,d ∈R, then ij ij j4(f ,f )∼(x2−y2+y2(φ +φ ),xy+φ ), 1 2 1 2 4 by Φ with q =x, q =y, q =z, q =w, p=1+c z+c w. 1 2 3 4 40 31 8 JORGELUIZDEOLINDOSILVAANDYUTAROKABATA 3.3. Hyperbolic case. Suppose that j2f =(x2,y2) and write j3(f ,f )=(x2+ a xiyj,y2+ b xiyj) 1 2 ij ij i+Xj=3 i+Xj=3 where a ,b ∈R. The following equivalence ij ij j3(f ,f )∼(x2+a y3,y2+b x3) 1 2 03 30 is given by projective transformationΦ with q =x+ 1(−a +b )z− 1a w, q =y− 1b z+ 1(a −b )w, 1 2 30 12 2 12 2 2 21 2 21 03 q =z, q =w, p=1+b x+a y. 3 4 12 21 We can eliminate more two coefficients in 4-jet. Put j4(f ,f )=(x2+a y3+ c xiyj,y2+b x3+ d xiyj) 1 2 03 ij 30 ij i+Xj=4 i+Xj=4 where c ,d ∈R, then ij ij j4(f ,f )∼(x2+a y3+yφ ,y2+b x3+xψ ), 1 2 03 3 30 3 by Φ with q =x, q =y, q =z, q =w, p=1−c z−d w. Here φ and ψ 1 2 3 4 40 04 3 3 means homogeneous polynomials of degree 3. Then (x2+y3+yφ ,y2+αx3+xψ ), α∈R∗ if a ,b 6=0; 3 3 03 30 j4(f1,f2)∼ (x2+y3+yφ3,y2+xψ3) if a03 6=0 and b30 =0;  (x2+yφ ,y2+xψ ) if a =b =0. 3 3 03 30 3.4. Parabolic case. Suppose that j2f =(x2,xy) and write j3(f ,f )=(x2+ a xiyj,xy+ b xiyj) 1 2 ij ij i+Xj=3 i+Xj=3 where a ,b ∈R. It is easy to show that ij ij j3(f ,f )∼(x2+a xy2+a y3,xy+¯b xy2+b y3) 1 2 12 03 12 03 where¯b =b − 1a . If a 6=0, then 12 12 2 21 03 j3(f ,f )∼(x2+(a +3b )xy2+a y3,xy) 1 2 12 03 03 with the equivalence given by Ψ with q =x− (−¯b12a03+3a12b03+3b203)w, 1 a03 q = b03x+y+ b203(a12b03−a03¯b12)z− b03(2b203+¯b12a03)w, q =z, 2 a03 a303 a203 3 q = b03z+w, p=1+ b203(a12+b03)x− (−2¯b12a03+4a12b03+3b203)y. 4 a03 a203 a03 Then the 4-jet can be written in the form j4(f ,f )∼(x2+βxy2+y3+yφ ,xy+xψ ), 1 2 3 3 where β = (a12+3b03), φ and ψ mean homogeneous polynomials of degree 3. a303/2 3 3 If a =0 but a 6=0 we obtain 03 12 j3(f ,f )∼(x2+xy2,xy+γy3) 1 2 with the projective transformation Ψ given by q =a x+a b w, q =y, q =a2 z, 1 12 12 12 2 3 12 q =a w, p=1+2b y, 4 12 12 PROJECTIVE CLASSIFICATION OF JETS OF SURFACES IN P4 9 where γ = b03. If we put a12 j4(f ,f )=(x2+xy2+ a xiyj,xy+βy3+ b xiyj), 1 2 ij ij i+Xj=4 i+Xj=4 then γ 6=0 leads to j4(f ,f )∼(x2+xy2+λy4,xy+γy3+φ ) 1 2 4 by a projective transformation Ψ with q =x+ 1(−q2 +p )z+(3γq −3q )w, 1 2 21 1 21 21 q =y+q x+ 1(−2γq3 +q3 +p q )z+ 1(−q2 +p )w, 2 21 2 21 21 1 21 2 21 1 q =z, q =q z+w, p=1+p x−(−6γq −4q )y+p z+p w, 3 4 21 1 21 21 3 4 where p = 1 ξ , p = 1 ξ , p = 1 ξ , q = −a13, ξ are combinations of the 1 Λ2 1 3 Λ4 2 4 Λ3 3 21 Λ i coefficientsof4-jetandΛ=6γ2+4λ−15γ+56=0. IfΛ=0thetermsofj4(f1,f2) of order 4 can not be removed. The φ is a homogeneous polynomials of degree 4. 4 3.5. Inflection case. Suppose that j2f =(x2+y2,0) and write j3(f ,f )=(x2+y2+ a xiyj, b xiyj). 1 2 ij ij i+Xj=3 i+Xj=3 Let b −b 6=0. It follows that 30 12 j3(f ,f )∼(x2+y2+k x2y,φ ) 1 2 1 3 by Ψ with q =x, q =y, q =z+(a −a )w, q =(b −b )w, 1 2 3 30 12 4 30 12 p=1− (a30b12−a12b30)x− (a30b03−a12b03−a03b30+a03b12)y. (b30−b12) (b30−b12) Here, k is scalar constant. Now, we take 1 j4(f ,f )=(x2+y2+k x2y+ c xiyj,φ + d xiyj), 1 2 1 ij 3 ij i+Xj=4 i+Xj=4 where c ,d ∈R, then it follows that ij ij j4(f ,f )∼(x2+y2+k x2y+yψ ,φ +φ ) 1 2 1 3 3 4 by Ψ with q = x, q = y, q = z, q = w,p = 1+c z. Here φ and ψ means 1 2 3 4 40 k k homogeneous polynomials of degree k. Next, suppose j3(f ,f )=(xy+ a xiyj, b xiyj). 1 2 ij ij i+Xj=3 i+Xj=3 If b 6=0, then 03 j3(f ,f )∼(xy+k x3,φ ) 1 2 2 3 by Ψ with q =x, q =y, q =z+(a +a )w, q =b w, 1 2 3 21 03 4 03 p=1− (a21b03−a03b21)x− (a21b03−a03b12)y. b03 b03 The k is a scalar constants. Finally, we consider 2 j4(f ,f )=(xy+k x3+ c xiyj,φ + d xiyj), 1 2 2 ij 3 ij i+Xj=4 i+Xj=4 10 JORGELUIZDEOLINDOSILVAANDYUTAROKABATA where c ,d ∈R. Thus, it follows that ij ij j4(f ,f )∼(xy+k x3+ξ¯,φ +φ ) 1 2 2 4 3 4 byΨwithq =x, q =y, q =z, q =w, p=1+c z.Theφ meanshomogeneous 1 2 3 4 22 k polynomials of degree k and ξ¯ is a homogeneous polynomials of degree 4 without 4 the term x2y2. 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