Journal of the Mathematical Society of Japan

Van Geemen-Sarti involutions and elliptic fibrations on $K3$ surfaces double cover of $\mathbb{P}^2$


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In this paper we classify the elliptic fibrations on $K3$ surfaces which are the double cover of a blow up of $\mathbb{P}^2$ branched along rational curves and we give equations for many of these elliptic fibrations. Thus we obtain a classification of the van Geemen-Sarti involutions (which are symplectic involutions induced by a translation by a 2-torsion section on an elliptic fibration) on such a surface. Each van Geemen-Sarti involution induces a 2-isogeny between two $K3$ surfaces, which is described in this paper.

Article information

J. Math. Soc. Japan, Volume 66, Number 2 (2014), 479-522.

First available in Project Euclid: 23 April 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14J28: $K3$ surfaces and Enriques surfaces
Secondary: 14J27: Elliptic surfaces 14J50: Automorphisms of surfaces and higher-dimensional varieties

$K3$ surfaces automorphisms of $K3$ surfaces elliptic fibrations symplectic involutions van Geemen-Sarti involutions isogenies


COMPARIN, Paola; GARBAGNATI, Alice. Van Geemen-Sarti involutions and elliptic fibrations on $K3$ surfaces double cover of $\mathbb{P}^2$. J. Math. Soc. Japan 66 (2014), no. 2, 479--522. doi:10.2969/jmsj/06620479.

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