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

By Hiroaki Hijikata

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

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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. In fact, together with L{s,t) they span the plane W{s,t) with dual coordinates sH^ : : t{2s^-^t^) : s{s^-2t^).

A. Poona University Pune 411007 India Bhaskaracharya Pratishthana Pune 411004 India Algebraic Geometry and Commutative Algebra in Honor of Masayoshi Ν AG ATA pp. 27-34 (1987) A Conjecture of Sharp — The Case of Local Rings with dim nonCM < 1 or dim < 5 Yoichi A o Y A M A and Shiro G o t o * §1. I n t r o d u c t i o n . We continue to discuss a conjecture of Sharp on the existence of dualizing complexes from [3]. 11]). 1. Let A be a local nng. If A has a dualizing complex and dimnonCM(A) < 1, then A is a homomorphic image of a Gorenstein nng.

As A(s¿, ti) φ O, each trisecant L{si, ti) meets S in three distinct points. 2) the plane W(si,ti) touches 5 in these three points, hence L{si,ti) will not meet 5 in a fourth point. 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..

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