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On Self-Intersection Number of a Section on a Ruled Surface

Published online by Cambridge University Press:  22 January 2016

Masayoshi Nagata
Masayoshi Nagata*
Affiliation:
Department of Mathematics, Kyoto University
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Let E be a non-singular projective curve of genus g ≥ 0, P the projective line and let F be the surface E×P. Then it is well known that a ruled surface F* which is birational to F is biregular to a surface which is obtained by successive elementary transformations from F (for the notion of an elementary transformation, see [3]).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 37 , March 1970 , pp. 191 - 196
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1970

References

[1] Atiyah, M.F., Complex fibre bundles and ruled surfaces, Proc. London Math. Soc. (3) 5 (1955), 407–434.Google Scholar
[2] Maruyama, M., On classification of ruled surfaces, forthcoming in Lect. Math. Kyoto Univ. Google Scholar
[3] Nagata, M., On rational surfaces I, Mem. Coll. Sci. Univ. Kyoto, (A Math.) 32–3 (1960), 351–370.Google Scholar
[4] Nakai-M., Y. Nagata, , Algebraic geometry, Kyoritsu, Tokyo 1957 (in Japanese).Google Scholar
[5] Zariski, O., Theory and applications of holomorphic functions on algebraic varieties over arbitrary ground fields, Mem. A.M.S. 5 (1951).Google Scholar