Degree of the fibres of an elliptic fibration
Buium, Alexandru
Annales de l'Institut Fourier, Tome 33 (1983), p. 269-276 / Harvested from Numdam
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Soit X→B 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 (ℒ,F) où ℒ 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 X→B. 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 X→B 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 (ℒ,F) where ℒ 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 X→B. 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},
     volume = {33},
     year = {1983},
     pages = {269-276},
     doi = {10.5802/aif.911},
     mrnumber = {84j:14017},
     zbl = {0478.14001},
     language = {en},
     url = {http://dml.mathdoc.fr/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) pp. 269-276. doi : 10.5802/aif.911. http://gdmltest.u-ga.fr/item/AIF_1983__33_1_269_0/

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