By Kollar J., Lazarsfeld R., Morrison D. (eds.)
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T. Fujita, On the hyperplane section principle of Lefschetz, J. Math. Soc. Japan, 32 (198o) 153-169. 17. T. Fujita~ On the structure of polarized manifolds with total deficiency one, I, J. Math. Soc. Japan ]2 (198o) 709-725. 18. H. Gizatullin, On affine surfaces that can be completed by a non-singular rational curv~, Izv. Akad. Nauk USSR 34 (197o) 787-81o. 19. A. Grothendieck - J. Dieudozm@, El@ments de G@om@trie Algebrique, Chap. II, III, IV, Publ. Math. IHES, Bures sur yvette. 33 20. A. Grothendieck, Cohomologie locale des faisceaux cohdrents et th@or~mes de Lefschetz locaux et globaux, North Holland, Amsterdam (1968).
E=d_l_M (n_r) this bound and for which d~r(n-r)+2 A description of Castelnuovo is called a varieties is given in . Let X be a nondegenerate s=d-2n+2. i) LEMM~. section on X. Then H is a 2n-2+s in pn-i hence by Castelnuovo's genus we get: If s<0 then Pa(H)~n+s-l. If 0Ss~n-3 then Pa(H)S ~n~2s. 2) LEMMA. divisor on X. 3) L E M M A . I f d~2n-3 of ruledness we get: then X is ruled. Let us also note that for 0~s~n-3 and only if equality holds in nuovo curve. 4) LEMMA. of X is s+l. surface hence we get: For 0~s~n-3 Harris" bound on the g e o m e t r i c genus If X is linearly normal in en with 0gs~n-3 and H is then H2(Ox(H))=0 and Pa(X)=l+(i/2) (HK-s).
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.