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

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

102 First Claim: If F is

102 First Claim: If F is semiweighted homogeneous for equivariant bifurcation equivalence, then the bifurcation character for F is the same as that for its initial part F 0. It is immediate that the bifurcation character remains unchanged under equivariant imperfect bifurcation equivalence (since this equivalence both preserves the sets where ~. >0, = 0, and < 0; as well as commutes with the group action). This is still true for topological imperfect bifurcation equivalence. Hence, we can first apply the topological determinacy theorem [DI] as it applies to this equivalence [D2] to conclude that the bifurcation character is the same for both the germ F which is semi-weighted homogeneous for equivariant bifurcation equivalence and its initial part F 0. Thus, in the semiweighted homogeneous case we may as well assume that F itself is weighted homogeneous. Then, to find a formula for the bifurcation character using an analogue of the Jacobian algebra, we are going to modify the proof of the corresponding result for the permutation character given in [I)4]. This proof, in turn, was a simple modification of the proof given in [D3] for the nonequivariant case. Hence we return to the notation of [I33]. We denote the complexification of the curve singularity X = F-l(0) by C. We also let n: (C, 0 ) ~ (C, 0) denote the normalization of C given in IO3]. The rings (~C,0 and 19~, 0 are denoted by A and A and A has a maximal ideal m A. There was defined a meromorphic one form (z = "/-1.dt/f = (dtl/t 1 ..... dtr/tr), where t i denotes the local coordinate for the component \$i of C" Z J_ \$i, and ,/-1 and f are special germs in A such that co --- dt/fis a real generator for coA, the module of dualizing differentials, and 7 -1 has the special property that y-l.m A c A but ~1 ~ A. The form ct used in [D3] counted all half branches positively. To account for the sign of ~. on each branch, it is natural to consider instead ct = ~.~/-1.dt/f. In theorem 3 of [I)4] we proved an equivariant version of the result of Montaldi-van Straten [MvS]. To apply this result, we must compute the residue pairing on Rot+ where Rct+ = tOA/( ta A n A.tx) Rtz- = A.tz/( co A n A.o0 This time since k e m A , by the special property of 7 -1, ct e to A so Rot- = 0. Thus it is only necessary to compute Rtz+. For this we shall explicitly determine a. Via the natural isomorphism of meromorphic forms n*: f~6-(,) "- f~(,), Hence, n*(7-1.dt/f) k.y-l.dt/f = = (dtl/t 1 ...... dtr/t r) = (1/wt(~.)).n*(d~,/X). (1/wt(~.)).clX- (Jac(F)/wt(~.)).dX/Jac(F). However, as explained in §6, co = dX/Jac(F) is defined on each branch of F-l(0) and is a

103 generator for m A, the module of dualizing differentials. Since to A is a free A-module on this generator, we conclude P, tz + = t0A/A-ot ~, A/(Jac(F)).{co} = q~).{co}. Moreover, in [W] it is shown that in A, J(F) = {x • A: x. 7 • A}. ff h • ~L.J(F) then h = X.h' with h' • J(F), so that h.dt/f = (h'.y).2L.y'l.dt/f • A.ct. Hence, ~,.J(F) c B(F). Second, the pairing on Ra+ is given by (h.co, h'.o)) g--+ Res ((h/(Jac(F)/wt(2L))).h'.o)) = Res (h.h'. (wt(~).d,X/Jac(F)2)). Thus, via the identification q(F) :, qKF).to, it is a composition of the multiplication pairing on q(F) with res' (h, h') ~ ( hh' ) ~ res'(hh') where res'(h) = Res(h. (wt(~.).d~/Jac(F)2)) Since wt(~,).d~/Jac~) 2 is G-invariant and Res is G-invariant (e.g. see the proof of theorem 3 in [D4]), res' is G-invariant. Also, we recall that by the discussion in [MvS], Jac(F2).o~ = d(Jac(F)), hence res'(Jac(F2) ) = Res(Jac(F2). (wt(~,).d~/Jac~)2)) = Res(wt(~.).d(Jac(F))/Jac(F)) > 0. Via the G-isormorphism qKF) "~ q~-').{o)), we have identified the residue pairing on ROt + with the multiplication pairing on B(F) composed with res' and sigG(B(F)) = sigG(Rot+). Lastly, we must identify the middle weight in B(F). However, the residue is only nonzero on the weight zero part. Hence, res'(h) ~ 0 implies However, wt(Jac(F)) --- s. Thus, wt(h) - wt(~,) + 2wt(Jac(F)) = 0. wt(h) = 2s - wt(~.). We conclude that 2s - wt(2L) is the top weight and so s b = s - 1/2-wt(~L) is the middle weight, n Proof of 3) and 1) of Theorem 2 From the preceding, we see that if we use the form 0t = d~, then the preceding proof applies and we still conclude that in the nonweighte.d homogeneous case, Xb is given by sigG(qkCF)) = sigG(Q(F2) ). Lastly, we see that an alternate way to compute Xp is to use the form (x = ~,.d~. This

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