Download Algebraic Geometry and Commutative Algebra. In Honor of by Hiroaki Hijikata PDF

By Hiroaki Hijikata

ISBN-10: 0123480310

ISBN-13: 9780123480316

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

Example text

Xn-l,Yvj,Aj] and define p : To --. j. Then p induces a morphism p : T o/{ Gj, Fij) --. R[[X1 , • •• ,Xn - 1 ]]. 40 Μ . ARTIN and C . , X„_i]-algebra 5i of finite type: t Si Let Ylj and A'- denote the images of Yuj and Aj in Si. ,Wn], φ:8'—. by iP\Si = φι; ^ ( X , ) = Xn\ Φ{\ν,) and define morphisms: R[[X]] = w. W^ -f E ^ J where A ' = + ^ i ^ í ^ í ^ i - Then σ factors over the smooth il[Xj-aJgebra S' hy σ = rp o μ. 10) C l a i m : There is an element u e S' such that is invertible in R[[X]] and ρ ( Χ , Γ ' ) divides fi{X,Y') in for all t = 1 , .

They span a plane which should not contain any tcingent of S', If A(s,t) φ 0 , the trisecants L* and M* meet 5 " transversally in together five points, so there can be no sixth point in which 5 " touches the plane. §3. T h e p r o j e c t e d c u r v e s 5 χ . For X G P3, by 5a. C P2 we denote the projection of S from x. As a curve in P2 it is determined up to automorphisms of P2 only. Whenever χ ^ 5 , we have deg 5 χ = 6. We denote by D C P3 the (closure of the) surface swept out by all the tnsecants to 5 .

If the reduced cover would be trivial over one α;», it would therefore be trivial over all Ui, hence trivial over S^. 1). 2) means that both curves 5 ' , S" C Y are smooth over each node Pi e Sx. So both of them are smooth rational curves on Y. Picking one of them at random and calling it NQ, we have sixteen disjoint smooth rational curves NQ, NÍJ on Y. Again by NikuHn's theorem [N] the surface y is a Kummer surface. Let A y be the double cover, EQ, EÍJ C A the exceptional curves over iVo, Nij and σ: A A their blow-down.

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


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