By Lucian BĂdescu (auth.), Lucian Bădescu, Dorin Popescu (eds.)
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Additional info for Algebraic Geometry Bucharest 1982: Proceedings of the International Conference held in Bucharest, Romania, August 2–7, 1982
L~,%l, .... qi ) is rigid. n+l times 1 1 Theorem io (Char(k)= o). i) Let Y be P ~ P . Then the cone C(Y,O(a,b)) i_~s rigid for ever~ a>~2 and b ~ 2 ~ ii) Let Y be F 1 and p:Y cone C(Y,Oy(b)@p~O(a)) unless a - h = 2. ~pl the canonical pro~ectio n of Y. Then the is ri6id for ever~ a > b ~ / 2 . For the proof use Theorems 3 and 4, Proposition 4, the rigidity of p l ~ p1 and F 1 and the same method as in the proof of Theorem 8. Theorem 1 !. Let Y be the Grassmann variet~ Gn, r, wlth" n~>/5 and l ~ r ~ n - 1 .
Of K. By K%f~Ky+K we get Ky~0. Since pa(Y)=Pa(X)=l is K3 and we may conclude by lemma Suppose HK=3. 9). Castelnuovo from . 13), we get that there are finitely many smooth ratioSince dimIKI~l and divisors 3 it turns out that there exists an integral sor K which varie- it is very easy to find the struc- independentely nal curves on X of a given degree. 4) X is Castelnuovo. in . However ture of X in our case, Let f:X--Y be the contraction is not smooth rational. tion K2=0. ii) we get pg=2 map f K : X ~ 1 gives an elliptic for the integral ones.
A complex o~en ne~hbourhoo~) T' of o in T such that the fibre x t = f-l(t) over every k-rational point (resp, over ever~ point) t ~ T ' is isomorphic to a closed subscheme of pn havin6 the prqperty (P). Proof. Claim: There is an @tale (resp. an open in the complex topology) neighhourhood T I ~ T (resp. T I ~ T ) of the point 06 T and an invertible sheaf L on U I = U × T T 1 inducing on X ° the sheaf 0 X (i). o In order to prove the claim we shall distinguish between the algebraic and 27 and the analytic one.
Algebraic Geometry Bucharest 1982: Proceedings of the International Conference held in Bucharest, Romania, August 2–7, 1982 by Lucian BĂdescu (auth.), Lucian Bădescu, Dorin Popescu (eds.)