Algebraic Geometry Bucharest 1982: Proceedings of the by Lucian BĂdescu (auth.), Lucian Bădescu, Dorin Popescu (eds.)

By Lucian BĂdescu (auth.), Lucian Bădescu, Dorin Popescu (eds.)

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Extra info for Algebraic Geometry Bucharest 1982: Proceedings of the International Conference held in Bucharest, Romania, August 2–7, 1982

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X is said to be smoothable in pn if there exists a deformation (U,T, n o) of X in P such that d i m ( T ) > o, T connected and X is smooth for every t ~ ~. t is rigid in pn if for every deformation (U,T,o) of X in pn there exists a Zariski open neighbourhood T' of o in T such that for every k-rational point 26 t ET', X t is isomorphic to X. g. ,dr), etc. We say that every small deformation of X in pn has also the property (P) if for every deformation ~U,T,o) of X in P n there is a Zariski open neighbeurhcod T' of o in T such that X t is also a subscheme of pn having the property (P) for every k-rational point tET' Let X be an arbitrary proper scheme over k.

For ever~ s~/l the cone C(Y,Oy(S)) is rigid. Use Theorem 7, Proposition 4, the rigidity of the Grassmann variety and the same method as in the proof of Theorem 8. Theorem 12 (Char(k) = o). Let y be an elliptic curve, L a line bundle on Y of de~ree >~lo and X the projective cone C(Y,L). Let f:U > T be__~a proper flat m0rphism of algebraic schemes over k (resp. spaces) ....... such that the fibre of f over a k-rational point ~resp. over a point) ....... o~T is isomorphic to X. Then there is a Zariski open (reap.

Strictly follow that But on a non-runegative on the H i l b e r t selfin- scheme. lines on X. This w o u l d contradicting dimlcKlkl. dist(X)_<[3s-2-(1/c) ]=3s-3. Finally hyperplane suppose section. 12) such that r(C)~(3s-2)c-l. has it is i s o l a t e d If now there exists we may choose of C is a straight curve IcKl#@. and we c o n c l u d e find C~Ic]

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