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arXiv · 1705.03527

Elliptic fibrations on covers of the elliptic modular surface of level 5

Francesca Balestrieri·Julie Desjardins·Alice Garbagnati·Céline Maistret·Cecília Salgado·Isabel Vogt
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Abstract

We consider the K3 surfaces that arise as double covers of the elliptic modular surface of level 5, $R_{5,5}$. Such surfaces have a natural elliptic fibration induced by the fibration on $R_{5,5}$. Moreover, they admit several other elliptic fibrations. We describe such fibrations in terms of linear systems of curves on $R_{5,5}$. This has a major advantage over other methods of classification of elliptic fibrations, namely, a simple algorithm that has as input equations of linear systems of curves in the projective plane yields a Weierstrass equation for each elliptic fibration. We deal in detail with the cases for which the double cover is branched over the two reducible fibers of type $I_5$ and for which it is branched over two smooth fibers, giving a complete list of elliptic fibrations for these two scenarios.

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Francesca Balestrieri, Julie Desjardins, Alice Garbagnati, Céline Maistret, Cecília Salgado, Isabel Vogt. 2022-05-09. Elliptic fibrations on covers of the elliptic modular surface of level 5. https://doi.org/10.1007/978-3-319-74998-3_9

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