On the uniqueness of elliptic K3 surfaces with maximal singular fibre

[Sur l’unicité des surfaces elliptiques de type K3 ayant une fibre singulière maximale]

Schütt, Matthias ; Schweizer, Andreas

Annales de l'Institut Fourier, Tome 63 (2013), p. 689-713 / Harvested from Numdam

Résumé

Nous déterminons des équations explicites pour les surfaces elliptiques de type $K3$ qui ont une section et une fibre singulière maximale. Si la caractéristique du corps sous-jacent est différente de $2$, pour chacun des deux types de fibre maximale, $I_{19}$ et $I^*_{14}$, la surface est unique. En caractéristique $2$ les fibres maximales sont de type $I_{18}$ ou $I^*_{13}$, et il y a deux, respectivement une, familles $1$-dimensionales de telles surfaces.

Abstract

We explicitly determine the elliptic $K3$ surfaces with section and maximal singular fibre. If the characteristic of the ground field is different from $2$, for each of the two possible maximal fibre types, $I_{19}$ and $I^*_{14}$, the surface is unique. In characteristic $2$ the maximal fibre types are $I_{18}$ and $I^*_{13}$, and there exist two (resp. one) one-parameter families of such surfaces.

Détail

Publié le : 2013-01-01

MR 3112845 | Zbl 1273.14078

DOI : https://doi.org/10.5802/aif.2773
Classification:  14J27,  14J28,  11G05
Mots clés: surface elliptique, surface de type $K3$, fibre singulière maximale, ramification sauvage

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