We consider the K3 surfaces that arise as double covers of the elliptic modular surface of level 5, R5,5. Such surfaces have a natural elliptic fibration induced by the fibration on R5,5. Moreover, they admit several other elliptic fibrations. We describe such fibrations in terms of linear systems of curves on R5,5. 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 I5 and for which it is branched over two smooth fibers, giving a complete list of elliptic fibrations for these two scenarios.
Elliptic fibrations on covers of the elliptic modular surface of level 5 / F. Balestrieri, J. Desjardins, A. Garbagnati, C. Maistret, C. Salgado, I. Vogt (ASSOCIATION FOR WOMEN IN MATHEMATICS SERIES). - In: Women in Numbers Europe II : Contributions to Number Theory and Arithmetic Geometry / [a cura di] I.I. Bouw, E. Ozman, J. Johnson-Leung, R. Newton. - [s.l] : Springer, 2018. - ISBN 9783319749976. - pp. 159-197 [10.1007/978-3-319-74998-3_9]
Elliptic fibrations on covers of the elliptic modular surface of level 5
A. Garbagnati;
2018
Abstract
We consider the K3 surfaces that arise as double covers of the elliptic modular surface of level 5, R5,5. Such surfaces have a natural elliptic fibration induced by the fibration on R5,5. Moreover, they admit several other elliptic fibrations. We describe such fibrations in terms of linear systems of curves on R5,5. 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 I5 and for which it is branched over two smooth fibers, giving a complete list of elliptic fibrations for these two scenarios.File | Dimensione | Formato | |
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Balestrieri2018_Chapter_EllipticFibrationsOnCoversOfTh.pdf
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