Rational curves on elliptic surfaces

Douglas Ulmer

doi:10.1090/jag/680

Rational curves on elliptic surfaces

Douglas Ulmer

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

We prove that a very general elliptic surface E → P1 over the complex numbers with a section and with geometric genus pg ≥ 2 contains no rational curves other than the section and components of singular fibers. Equivalently, if E/C(t) is a very general elliptic curve of height d ≥ 3 and if L is a finite extension of C(t) with L ∼= C(u), then the Mordell-Weil group E(L) = 0.

Original language	English (US)
Pages (from-to)	357-377
Number of pages	21
Journal	Journal of Algebraic Geometry
Volume	26
Issue number	2
DOIs	https://doi.org/10.1090/jag/680
State	Published - 2017
Externally published	Yes

ASJC Scopus subject areas

Algebra and Number Theory
Geometry and Topology

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10.1090/jag/680

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title = "Rational curves on elliptic surfaces",

abstract = "We prove that a very general elliptic surface E → P1 over the complex numbers with a section and with geometric genus pg ≥ 2 contains no rational curves other than the section and components of singular fibers. Equivalently, if E/C(t) is a very general elliptic curve of height d ≥ 3 and if L is a finite extension of C(t) with L ∼= C(u), then the Mordell-Weil group E(L) = 0.",

author = "Douglas Ulmer",

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year = "2017",

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N2 - We prove that a very general elliptic surface E → P1 over the complex numbers with a section and with geometric genus pg ≥ 2 contains no rational curves other than the section and components of singular fibers. Equivalently, if E/C(t) is a very general elliptic curve of height d ≥ 3 and if L is a finite extension of C(t) with L ∼= C(u), then the Mordell-Weil group E(L) = 0.

AB - We prove that a very general elliptic surface E → P1 over the complex numbers with a section and with geometric genus pg ≥ 2 contains no rational curves other than the section and components of singular fibers. Equivalently, if E/C(t) is a very general elliptic curve of height d ≥ 3 and if L is a finite extension of C(t) with L ∼= C(u), then the Mordell-Weil group E(L) = 0.

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DO - 10.1090/jag/680

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JF - Journal of Algebraic Geometry

IS - 2

ER -

Rational curves on elliptic surfaces

Abstract

ASJC Scopus subject areas

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