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

By Hiroaki Hijikata

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Al­ gebra, 81 (1983), 29-57. G. Faltings, Uber die Annulatoren lokaler Kohomologiegruppen, Arch. (Basel), 30 (1978), 473-476. G. Faltings, Über Macaulayfizierung, Math. , 238 (1978), 175-192. S. Goto and K. Yamagishi, The theory of unconditioned strong d-sequences and modules of finite local cohomology. Preprint. J. E. Hall, Fundamental dualizing complexes for commutative noetherian rings, Quart. J. Math. (2), 3 0 (1979), 21-32. R. Hartshorne, Residues and duality, Lect. Notes in Math. 20, Springer Verlag, 1966.

2) appUed to 5¿, ti instead of λ^, /z¿ shows i = j . So for all points χ(λ, μ) e S but finitely many, the three trisecants L{si,ti) through χ are distinct. On the projected curve 5χ they define three distinct singularities p¿. At each p¿ two branches of 5a. 2) they meet not transversally. This impHes S{pi) > 2; and in view of Σ δ{ρ) = 6, these points p¿ are the only singularities on 5a.. The lines L(5¿,íj) are the only trisecants through x. By the dimension argument above, this also holds for the points χ G 5 where Δ ( λ , μ ) = 0.

If h^{C)-^ > 4, the sections 3 e H^{C)'^ vanishmg (doubly) at 67 would form a vector space of dimension > 3. Counting intersections of the curves {s = 0} this is possible only if all these curves are reducible. However, the pullback of the general Une through the node of Sx corresponding to 67 is easily seen to be ureducible, a contradiction. 56 W . BARTH and R . 2) Forx, y G P 3 \ ( i ^ u r u i ^ ) assume = Ay {as polarized Then the branch curves Sx, Sy C Ρ2 are projectively equivalent. surfaces).

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