Abstract
The modern study of elliptic fibrations started in the early 1960s with seminal works by Kodaira and by Néron. Elliptic fibrations play a central role in the classification of algebraic surfaces, in many aspects of arithmetic geometry, theoretical physics, and string geometry. In these notes, we introduce the reader to basic geometric properties of elliptic fibrations over the complex numbers. We start with an introduction to the geometry of elliptic curves defined over the complex numbers. We then discuss Weierstrass models, Kodaira’s classification of singular fibers of elliptic surfaces, Tate’s algorithm, and Miranda’s regularization of elliptic threefolds.
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Notes
- 1.
Given three sets (\(A_1\), \(A_2\), and S) and two maps \(\varphi _1:A_1\rightarrow B\) and \(\varphi _2:A_2\rightarrow B\), we define the fibral product \(A_1\times _S A_2\) as the subset of \(A_1\times A_2\) composed of couples \((a_1,a_2)\) such that \(\varphi _1 (a_1)=\varphi _2(a_2)\).
- 2.
For example, if p is the generic point of a subvariety of B.
- 3.
As usual we take the convention in which the ring itself is not a prime ideal.
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Acknowledgements
I would like to thank the organizers and the participants of the Villa de Leyva Summer School of 2015. I am deeply grateful to my collaborators Ravi Jagadeesan, Patrick Jefferson, Monica Kang, Sabrina Pasterski, Julian Salazar, Shu-Heng Shao, and Shing-Tung Yau for helpful discussions. I am also thankful to Paolo Aluffi and Matilde Marcolli who have introduced me to the school. This work is partly supported by the National Science Foundation (NSF) grants DMS-1406925 and DMS-1701635 “Elliptic Fibrations and String Theory.”
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Esole, M. (2017). Introduction to Elliptic Fibrations. In: Cardona, A., Morales, P., Ocampo, H., Paycha, S., Reyes Lega, A. (eds) Quantization, Geometry and Noncommutative Structures in Mathematics and Physics. Mathematical Physics Studies. Springer, Cham. https://doi.org/10.1007/978-3-319-65427-0_7
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