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On cubic elliptic varieties


Authors: Jürgen Hausen, Antonio Laface, Andrea Luigi Tironi and Luca Ugaglia
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
Published electronically: May 27, 2015
MathSciNet review: 3413880
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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.


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Additional Information

Jürgen Hausen
Affiliation: Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany
Email: juergen.hausen@uni-tuebingen.de

Antonio Laface
Affiliation: Departamento de Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile
Email: alaface@udec.cl

Andrea Luigi Tironi
Affiliation: Departamento de Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile
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

DOI: https://doi.org/10.1090/tran/6353
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)
Article copyright: © Copyright 2015 American Mathematical Society