## On cubic elliptic varieties

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- by Jürgen Hausen, Antonio Laface, Andrea Luigi Tironi and Luca Ugaglia PDF
- Trans. Amer. Math. Soc.
**368**(2016), 689-708 Request permission

## Abstract:

Let $\pi \colon X\to \mathbb {P}^{n-1}$ be an elliptic fibration obtained by resolving the indeterminacy of the projection of a cubic hypersurface $Y$ of $\mathbb {P}^{n+1}$ from a line $L$ not contained in $Y$. We prove that the Mordell-Weil group of $\pi$ is finite if and only if the Cox ring of $X$ is finitely generated. We also provide a presentation of the Cox ring of $X$ when it is finitely generated.## References

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## Additional Information

**Jürgen Hausen**- Affiliation: Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany
- MR Author ID: 361664
- Email: juergen.hausen@uni-tuebingen.de
**Antonio Laface**- Affiliation: Departamento de Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile
- MR Author ID: 634848
- Email: alaface@udec.cl
**Andrea Luigi Tironi**- Affiliation: Departamento de Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile
- MR Author ID: 677961
- Email: atironi@udec.cl
**Luca Ugaglia**- Affiliation: Dipartimento di Matematica e Informatica, Università degli studi di Palermo, Via Archirafi 34, 90123 Palermo, Italy
- Email: luca.ugaglia@unipa.it
- Received by editor(s): August 14, 2013
- Received by editor(s) in revised form: December 2, 2013
- Published electronically: May 27, 2015
- Additional Notes: The second author was partially supported by Proyecto FONDECYT Regular N. 1110096

The third author was partially supported by Proyecto DIUC 211.013.036-1.0

The fourth author was partially supported by Università di Palermo (2012-ATE-0446) - © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**368**(2016), 689-708 - MSC (2010): Primary 14C20, 14Q15; Secondary 14E05, 14N25
- DOI: https://doi.org/10.1090/tran/6353
- MathSciNet review: 3413880