On the uniqueness of elliptic K3 surfaces with maximal singular fibre

Schütt, Matthias; Schweizer, Andreas

Annales de l'Institut Fourier, Tome 63 (2013) no. 2, pp. 689-713.

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.

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.

Classification : 14J27, 14J28, 11G05
Keywords: elliptic surface, $K3$ surface, maximal singular fibre, wild ramification.
Mot clés : surface elliptique, surface de type $K3$, fibre singulière maximale, ramification sauvage

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