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Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Rational curves on elliptic surfaces

Author: Douglas Ulmer
Journal: J. Algebraic Geom. 26 (2017), 357-377
Published electronically: August 26, 2016
MathSciNet review: 3606999
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Abstract | References | Additional Information

Abstract: We prove that a very general elliptic surface $\mathcal {E}\to \mathbb {P}^1$ over the complex numbers with a section and with geometric genus $p_g\ge 2$ contains no rational curves other than the section and components of singular fibers. Equivalently, if $E/\mathbb {C}(t)$ is a very general elliptic curve of height $d\ge 3$ and if $L$ is a finite extension of $\mathbb {C}(t)$ with $L\cong \mathbb {C}(u)$, then the Mordell-Weil group $E(L)=0$.

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Additional Information

Douglas Ulmer
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
MR Author ID: 175900
ORCID: 0000-0003-1529-4390

Received by editor(s): August 4, 2014
Published electronically: August 26, 2016
Article copyright: © Copyright 2016 University Press, Inc.