By Hiroaki Hijikata

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Additional resources for Algebraic Geometry and Commutative Algebra. In Honor of Masayoshi Nagata, Volume 1

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The twelve roots of Δ ( 5 , ί ) are the vertices of the icosahedron 5 : < = 0,οο,ε'^7;,εν (Λ = 0 , . . , 4 ) with 77 = ε + ε ^ η' = ε^^-ε^. The roots of Δ ( 5 , ί ) and Δ ( λ , μ ) correspond to each other under the relation defined by P3 = 0 as shown in Figure 1. This follows easily using the identity Ρζ{ηΛ\3,1) = η{8-ηΐ)\8-η'ί) and the synunetries above. We further use P¡ = sH^'XiX' -2μ')-]- 3Η^μ(2Χ^ + μ^) + t{23^ + ¿ ' ) A V ' + 5(5^ - 2 ¿ 5 ) A 2 / ^dxP¡ = 3Ψ{^Χ' - μ') + 53ΨΧ^μ + 2t(23^ + Í ' ) A V ' + S{3' - 2 ί 5 ) λ / ^-d,P¡=23H''X{X^ -2μ') + 5¿V(2A^ + /i^) + hsHX^ß" + (35^ - Í ^ ) A V ' .

B) The Hti are polynomials in i/^^ whose monomials have degree at least 2. ,r. Let φ : S'[Uij] —^ -ß[[X]] be the morphism defined by φ\3' = φ and φ{Uij) = 0. 4)(a),(b): α C Keτφ. Let φ : Si = S'[Uij]/a R[[X]] be the morphism induced by φ. Since S' is smooth over R[X], the defining equations show that Si is smooth over R[X] in a neighborhood of the locus {Uij = 0}. 6) Consider the morphism 7 : R[X,Y] -> S'[Uij] defined by 7(F^) = Zw Φ[Υ{ΐ) = O impües that σ : Ä[X,y] factors through η:σ = φοη.

Therefore depth C»p+dun C¿q/pC¿q > 1-f 1 = 2. Hence there is an integer r > d such that m''F-^(Ci) = 0 for ¿ = 1 , . . , d by virture of [6]. Take an element y from m** such that height (J + yA) = d + 1, and put J = r ( r + 2/A). 31 A Conjecture of Sharp =3 m^E + Ä+ for ¿ = 1 , . . -n Let R = R{A,J), ... = mi? -f generated A-algebra). = Note that R has a duaUzing complex (as Ä is a finitely T h e o r e m 3 . 1 . , 0^, (cf. 17] and [15,§1]). We put p = φη A. First suppose p is not a maximal ideal.

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