By Hiroaki Hijikata

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51(1979) 189-203. R. Elkik: Solutions d'équations á coefficients dans un anneau henselian, Ann. Sei. École Normal Sup. 4« ser. 6(1973) 553-601. Β. Iversen: Generic local structure of the morphisms in commutative alge bra, Lec. Notes m Math. 310, Sprmger Verlag, BerUn 1973. H. Kurke, T. Pfister, G. Popescu and M. Roczen: Die Approximationseigenshaft lokaler Ringe, Lec. Notes in Math. 634, Springer Verlag, Berlm 1978. M. Nagata: On the theory of henseUan rings I, II, Nagoya Math. J. 5(1953) 45-57; 7(1954) 1-19.

3, there is a five-dimensional local rmg A such that A has a duahzing complex, is not a homomorphic image of a Gorenstein ring and is (S2). We note that Ass(A) = Assh(A) (cf. 4]). 2. We put Spec(A) | depth Ap = 2 < dimAp}. T{A) = {pe Then r(A) is not empty. Let α be an ideal such that V(a) = nonCM(A). As A is (S2), height α > 3. There is an A-regular sequence x^y in. a. Then we have T(A) c Ass(A/(x, y)) and T(A) is a finite set. We put s{A) = max{dim Ap | ρ e T(A)}, To(A) = {pe T{A) I dimAp = s{A)} and T^iA)=T{A)\To{A).

The search for equations of the sextic plane curves described above was begun in 1985 by the first author during his stay at Kyoto University. He is * Supported by DFG Research Grant Ba 423/2-1 Received December 13, 1986. 46 W . BARTH and R . M O O R E indebted to Professor M. Nagata and to JSPS for having made possible this visit. He is also indebted to D. 1 b) below. CONVENTION: The base field always is C. §1. S o m e P o l y n o m i a l s . The polynomial is homogeneous of bidegree 3, 3 in the two variables λ : μ and s :t ePi.

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