Degree of the fibres of an elliptic fibration
Annales de l'Institut Fourier, Tome 33 (1983) no. 1, pp. 269-276.

Soit XB une fibration elliptique et soit F une fibre générale. Soit n e ,n s ,n a ,n v les minima des valeurs non-nulles des nombres d’intersection (,F) parcourt successivement les ensembles suivants : diviseurs effectifs sur X, faisceaux inversibles engendrés par sections globales, diviseurs amples et diviseurs très amples. Soit m le maximum des multiplicités des fibres de XB. On démontre que n e =n s si et seulement si n e 2m et que n a =n v si et seulement si n a 3m.

Let XB an elliptic fibration with general fibre F. Let n e ,n s ,n a ,n v be the minima of the non-zero intersection numbers (,F) where runs successively through the following sets: effective divisors on X, invertible sheaves spanned by global sections, ample divisors and very ample divisors. Let m be the maximum of the multiplicities of the fibres of XB. We prove that n e =n s if and only if n e 2m and that n a =n v if and only if n a 3m.

@article{AIF_1983__33_1_269_0,
     author = {Buium, Alexandru},
     title = {Degree of the fibres of an elliptic fibration},
     journal = {Annales de l'Institut Fourier},
     pages = {269--276},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {33},
     number = {1},
     year = {1983},
     doi = {10.5802/aif.911},
     mrnumber = {84j:14017},
     zbl = {0478.14001},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.911/}
}
TY  - JOUR
AU  - Buium, Alexandru
TI  - Degree of the fibres of an elliptic fibration
JO  - Annales de l'Institut Fourier
PY  - 1983
SP  - 269
EP  - 276
VL  - 33
IS  - 1
PB  - Institut Fourier
PP  - Grenoble
UR  - http://www.numdam.org/articles/10.5802/aif.911/
DO  - 10.5802/aif.911
LA  - en
ID  - AIF_1983__33_1_269_0
ER  - 
%0 Journal Article
%A Buium, Alexandru
%T Degree of the fibres of an elliptic fibration
%J Annales de l'Institut Fourier
%D 1983
%P 269-276
%V 33
%N 1
%I Institut Fourier
%C Grenoble
%U http://www.numdam.org/articles/10.5802/aif.911/
%R 10.5802/aif.911
%G en
%F AIF_1983__33_1_269_0
Buium, Alexandru. Degree of the fibres of an elliptic fibration. Annales de l'Institut Fourier, Tome 33 (1983) no. 1, pp. 269-276. doi : 10.5802/aif.911. http://www.numdam.org/articles/10.5802/aif.911/

[1] A. Beauville, Surfaces algébriques complexes, Astérisque, 54 (1978). | Numdam | MR | Zbl

[2] E. Bombieri, Canonical models of surfaces of general type, Publ. Math. IHES, 42 (1972), 171-220. | Numdam | MR | Zbl

[3] F. Enriques, Le superficie algebriche, Zanichelli, 1949. | MR | Zbl

[4] P. Griffiths and J. Harris, Principles of algebraic geometry, John Wiley & Sons, New York, 1978. | MR | Zbl

[5] K. Kodaira, On compact complex analytic surfaces II, Ann. of Math., 77 (1963). | MR | Zbl

[6] C.P. Ramanujam, Remarks on the Kodaira vanishing theorem, J. of the Indian Math. Soc., 36 (1972), 41-51 ; Supplement to the article “Remarks on the Kodaira vanishing theorem”, J. of the Indian Math. Soc., 38 (1974), 121-124. | MR | Zbl

Cité par Sources :