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

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

Sample text

Suppose X is not minimal and choose an e x c e p t i o n a l divisor t E. 9). Then Y has degree d'=d+k 2 and n ' = n + k ( k + l ) / 2 hence d'=2n'- -2+s' with st=s-k. If s'<0,Y must be ruled, hence so is X, contradic- 5~ ring pg(X)>l. NOW o b v i o u s l y s'

2. The above corollary is of folklore type. It was included as an illustration of the usefulness of Proposition 4 on one hand, and because it will be used in the proof of Theorem 8 below on the other hand. The analytic part of it was proved in [35j in the framework of Kodaira-Spencer's deformation theory (in the case where everything is smooth, but - except the K-3 surfaces - also including the two-dimensional case). 3. The proof of the corollary works also in the case when X is one of the O following: a quadric or a cubic in p 3 or a complete intersection of type (2,2) in p4.

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). 21. R. Hartshorne, Algebraic Geometry, Springer Verlag (1977). 22. R. Hartshorne, Curves with high self-intersection on algebraic surfaces, Publ.

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