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SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

216 Z 2 (rn~)Z29 denote

216 Z 2 (rn~)Z29 denote the ideal in Eq generated by the Xij and the quadratic monomials in the Yi and w i, and let 3 denote the ideal generated by (mq) z29 and the Yi- We claim that the two normal forms given in the theorem are M-determined, where M is the subspace of Z 2 Epq given by:- Z 2 M = ElX t .{ A7,z1A3,z2A3 } + ZI.{ A4,A5,A6,zI,X1A,x1A2,z2,z2A,z2A2 } Z 2 + (m2q) z2 .{ A,A2A 3 } + Eq .{1}. We prove the claim for the f'ast normal form - the calculations for the second are very + similar. The proof consists of showing that M is a (~N intrinsic subspace of. TN(F) where N is the unipotent group defined below. The claim then follows from Theorem 3.4. Decompose W*~V as ~IR2~IR2~IR 3 where the 4 components are the linear subspaces with coordinates {L}, {Xl,X2}, {Wl,W 2} and {Yl,Y2,Y3} respectively. Then N is defined to be the group of matrices of the form:- "I, 00 0 I 0 0 00 I. O00l with respect to this decomposition. Here I is the identity matrix, • is an arbitrary matrix and the 0s above the diagonal are forced by the 72 symmetry. With this choice of N the tangent space TN(F ) is given by:- Z2 z2 z2 Z2 2 z2 TN(F) = ElXl "{ mtxl'g0"F' ¢q.F, tx2.F } + Eq .{ mq .13ij.F, 3.yk.F, (mq) .Gz.F, 1} where:- OtO=~3 ,m ~1 =xl~----, ~2=xz ~ , ~=x.~t_~_ (i d= 1,2), yk= ~ (k= 1,2), and 8~ = 3 (£ = 1,2,3). ~Yz

217 In terms of the invariants the fn'st normal form can be written:- 3 F = A 4 + Z yjA.i + {(tXl+Wl)Xl + (Ot2+w2)x2)A2 + X2 A + Xl. j=l Easy calculations give:- 3 °t0"F -- 8A4 + Z 2jyjAJ + 5{(otl+Wl)X1 + (ot2+w2)x2}A2 + 3XEA + X1 j=l Otl.F = ct2.F = 1311.F = Xl{ 1 + (tXt+wl)A2 } [521-F = X2{ 1 + (Otl+Wl)A2 } YI.F = Xl A2 Y2.F = X2 A2 3 8xIA3 + X 2jyj Z1Aj-t + 5{(Ctl+Wl)Xll + ((x2+w2)X12)A2 + 3X12 A + Xtl j=l 3 8x2A3 + X 2jYj X2Aj-1 + 5{(°~l+Wl)X12 + (ct2+w2)XE2}A2 + 3X22A ÷ X12 j=l Using these expressions it is easily seen that:- ~12.F = x1A{ 1 + (ct2+w2)A } ~22.F = x2A{ 1 + (ot2+w2)A } Gz.F = A ~ ~ = 1,2,3. Z 2 3.{ A4,A5,A6,xI,X1A,ztA2,z2,x2A,z2A2 } + (m~) Z2 .{ A,A2A 3 } + Eq .{1} is contained in TN(F). It therefore remains to show that:- Z 2 Epq .{ A7,zIA3,z2A3 } is contained in TN(F). To do this it is sufficient, by Nakayama's Lemma, to show that:- Z 2 Z 2 Z 2 z 2 Z 2 ElX l .{A7,xIA3,x2 A3} E Epq .{mpq.Ct0.F , ot1.F , ot2.F} + mpq.{mpq.Ct0.F , txl.F , ot2.F}. Z 2 Z 2 Since modulo mlx t.{mlx t.ct0.F , ott.F , tx2.F) we have:- A3.o~0.F = 8A 7 + xIA 3 , al.F = 8X1A3, o~2.F = 8x2A3 , the result follows. The proof that the Fwst normal form is M-determined is completed by noting that M is

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