By Hiroaki Hijikata

ISBN-10: 0123480310

ISBN-13: 9780123480316

**Read or Download Algebraic Geometry and Commutative Algebra. In Honor of Masayoshi Nagata, Volume 1 PDF**

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

**Example text**

0. Theorem 1 follows from Néron's p-desingularization [2] in the case η = 0. We will review the results we need here. -algebras. More precisely the following can be shown: T h e o r e m 2: ([2] §4): Let τ : R R' be a regular local morphism of discrete valuation rings and suppose that τ factors: R Β ^ R', where Β is a finite type R-algebra and σ is injective. Then there is a factorization of σ: Received December 3 , 1986. 36 Μ . ARTIN and C. ROTTHAUS R R' satisfying the following conditions: (i) C is a finite type smooth R-algebra, (ii) φ is injective, {ni) C C B\p-'].

T h e double plane Y. Here, for a: G P3 general, we consider the double cover y P2 branched over the six tritangents L i , . . , Le to the curve 5a.. To be precise, this double cover has fifteen nodes over the intersections L¿ HLj. By y = Υχ we mean its minimal desingularization. y is a K3-surface containing fifteen disjoint smooth rational curves 7V,j, one over each intersection Li Π Lj, 1 < i < j < 6. Over Sx the covering splits as 5 ' U S" with two rational curves 5 ' , 5 " C Y. By the reduced cover S' U S" 5a.

2 . T h e sextic s p a c e c u r v e 5 . 3). So 5 C P3 is a smooth rational curve of degree six. ( 2 . , three points on a trisecant L{s,t) to S. 48 W . B A R T H and R . M O O R E Proof. Ρ 3 ( λ , μ ; 5 , 0 = 0 impHes P | = d^Pi = 0. A glance at the explicit form of these polynomials (Section 1) shows that the three points χ(λ,,μ») lie on the two planes with dual coordinates βΨ 23ψ : βΨ : st^ : t{2s^-\-t^) : 5sH : s{s^ - 2t^) : Zs^ -1\ One easily checks their independence. 2) For fixed s : t with A{s^t) φ 0 the tangents to the three points x{Xi,ßi) e L{s,t) are dependent.

### Algebraic Geometry and Commutative Algebra. In Honor of Masayoshi Nagata, Volume 1 by Hiroaki Hijikata

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