Degree of the fibres of an elliptic fibration
Annales de l'Institut Fourier, Tome 33 (1983) no. 1, p. 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},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {33},
     number = {1},
     year = {1983},
     pages = {269-276},
     doi = {10.5802/aif.911},
     zbl = {0478.14001},
     mrnumber = {84j:14017},
     language = {en},
     url = {http://www.numdam.org/item/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. https://www.numdam.org/item/AIF_1983__33_1_269_0/

[1] A. Beauville, Surfaces algébriques complexes, Astérisque, 54 (1978). | MR 58 #5686 | Zbl 0394.14014

[2] E. Bombieri, Canonical models of surfaces of general type, Publ. Math. IHES, 42 (1972), 171-220. | Numdam | MR 47 #6710 | Zbl 0259.14005

[3] F. Enriques, Le superficie algebriche, Zanichelli, 1949. | MR 11,202b | Zbl 0036.37102

[4] P. Griffiths and J. Harris, Principles of algebraic geometry, John Wiley & Sons, New York, 1978. | MR 80b:14001 | Zbl 0408.14001

[5] K. Kodaira, On compact complex analytic surfaces II, Ann. of Math., 77 (1963). | MR 29 #2822 | Zbl 0118.15802

[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 48 #8502 | Zbl 0276.32018