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**

These scrolls are defined by elhptic curves with level-5 structure. Using as parameter λ : μ G P i , F. Klein's "Ikosaeder Transzendente" parametrizing this curve, its image in P3 = P H ° { ^ ) , upon the right choice of coordinates, is just the curve S from Section 2. The equation Pz{\,ß\s,t) = 0 describes for given λ : μ the three parameters Si : ti G P i corresponding to the three 2-torsion quotients of the elhptic curve belonging to λ : μ. The essential new observation is that the abehan surface A c P4 corre sponding to a: G PH^{J^) is reconstructed as a fourfold cover of the plane P2 starting with a double plane X ramified over the projected curve 3χ.

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-']. Theorem 2 impUes the following well-known statement which we will use for the proof of Theorem 1: T h e o r e m 3 : Let R be an excellent discrete valuation nng, R its comple tion, X = ( X i , . . , X n ) variables, and let τ : R[X] R[[X]] be the canonical embedding. Moreover suppose that there is given a commutative diagram: R{X\ (•) Β where Β is a R[X]-algebra of finite type and σ is injective.

2). , not on D , Γ , orW. 3) Assume χ ^ DOWOT. 6). Proof If a double tangent of Sx is not the projection of a double tangent to 5 , it determines a plane W{s,t) containing two tangents of 5 , cf. 4). So the double tangent is a tritangent. §4. T h e double plane X. Here we describe the surface X = Χχ, the double cover of P2 branched over Sx for χ G P3 general. To be precise: This double cover has ten nodes over the ten double points of 5 , ; by Χ we mean its minimal desingularization. X is a K3-surface containing ten smooth rational curves iV,, one over each double pomt Pi e Sx.

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

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