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

Let $X\to B$ 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 $\left(ℒ,F\right)$ 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 $X\to B$. We prove that ${n}_{e}={n}_{s}$ if and only if ${n}_{e}\ge 2m$ and that ${n}_{a}={n}_{v}$ if and only if ${n}_{a}\ge 3m$.

Soit $X\to B$ 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 $\left(ℒ,F\right)$$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 $X\to B$. On démontre que ${n}_{e}={n}_{s}$ si et seulement si ${n}_{e}\le 2m$ et que ${n}_{a}={n}_{v}$ si et seulement si ${n}_{a}\ge 3m$.

@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},
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, Volume 33 (1983) no. 1, pp. 269-276. doi : 10.5802/aif.911. http://www.numdam.org/item/AIF_1983__33_1_269_0/

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