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
DOI: https://doi.org/10.1090/jag/680
Published electronically: August 26, 2016
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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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Douglas Ulmer
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
Email: ulmer@math.gatech.edu

DOI: https://doi.org/10.1090/jag/680
Received by editor(s): August 4, 2014
Published electronically: August 26, 2016
Article copyright: © Copyright 2016 University Press, Inc.

American Mathematical Society