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
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 $E(L)=0$.
References
- Fedor Bogomolov, Brendan Hassett, and Yuri Tschinkel, Constructing rational curves on K3 surfaces, Duke Math. J. 157 (2011), no. 3, 535โ550. MR 2785829, DOI https://doi.org/10.1215/00127094-1272930
- Fedor Bogomolov and Yuri Tschinkel, Algebraic varieties over small fields, Diophantine geometry, CRM Series, vol. 4, Ed. Norm., Pisa, 2007, pp. 73โ91. MR 2349648
- David Cox and Ron Donagi, On the failure of variational Torelli for regular elliptic surfaces with a section, Math. Ann. 273 (1986), no. 4, 673โ683. MR 826466, DOI https://doi.org/10.1007/BF01472138
- James Carlson, Mark Green, Phillip Griffiths, and Joe Harris, Infinitesimal variations of Hodge structure. I, Compositio Math. 50 (1983), no. 2-3, 109โ205. MR 720288
- David A. Cox, The Noether-Lefschetz locus of regular elliptic surfaces with section and $p_g\ge 2$, Amer. J. Math. 112 (1990), no. 2, 289โ329. MR 1047301, DOI https://doi.org/10.2307/2374717
- Igor Dolgachev, Weighted projective varieties, Group actions and vector fields (Vancouver, B.C., 1981) Lecture Notes in Math., vol. 956, Springer, Berlin, 1982, pp. 34โ71. MR 704986, DOI https://doi.org/10.1007/BFb0101508
- Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
- Jรกnos Kollรกr, Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 32, Springer-Verlag, Berlin, 1996. MR 1440180
- Serge Lang, Hyperbolic and Diophantine analysis, Bull. Amer. Math. Soc. (N.S.) 14 (1986), no. 2, 159โ205. MR 828820, DOI https://doi.org/10.1090/S0273-0979-1986-15426-1
- Rick Miranda, The moduli of Weierstrass fibrations over ${\bf P}^{1}$, Math. Ann. 255 (1981), no. 3, 379โ394. MR 615858, DOI https://doi.org/10.1007/BF01450711
- F. Pakovich, Algebraic curves $P(x)-Q(y)=0$ and functional equations, Complex Var. Elliptic Equ. 56 (2011), no. 1-4, 199โ213. MR 2774592, DOI https://doi.org/10.1080/17476930903394838
- Tetsuji Shioda, An explicit algorithm for computing the Picard number of certain algebraic surfaces, Amer. J. Math. 108 (1986), no. 2, 415โ432. MR 833362, DOI https://doi.org/10.2307/2374678
- Joseph H. Silverman, A bound for the Mordell-Weil rank of an elliptic surface after a cyclic base extension, J. Algebraic Geom. 9 (2000), no. 2, 301โ308. MR 1735774
- Peter F. Stiller, The Picard numbers of elliptic surfaces with many symmetries, Pacific J. Math. 128 (1987), no. 1, 157โ189. MR 883383
- Douglas Ulmer, $L$-functions with large analytic rank and abelian varieties with large algebraic rank over function fields, Invent. Math. 167 (2007), no. 2, 379โ408. MR 2270458, DOI https://doi.org/10.1007/s00222-006-0018-x
- Douglas Ulmer, Elliptic curves over function fields, Arithmetic of $L$-functions, IAS/Park City Math. Ser., vol. 18, Amer. Math. Soc., Providence, RI, 2011, pp. 211โ280. MR 2882692, DOI https://doi.org/10.1090/pcms/018/09
- Douglas Ulmer, Explicit points on the Legendre curve, J. Number Theory 136 (2014), 165โ194. MR 3145329, DOI https://doi.org/10.1016/j.jnt.2013.09.010
- Claire Voisin, On a conjecture of Clemens on rational curves on hypersurfaces, J. Differential Geom. 44 (1996), no. 1, 200โ213. MR 1420353
References
- Fedor Bogomolov, Brendan Hassett, and Yuri Tschinkel, Constructing rational curves on K3 surfaces, Duke Math. J. 157 (2011), no. 3, 535โ550. MR 2785829 (2012d:14061), DOI https://doi.org/10.1215/00127094-1272930
- Fedor Bogomolov and Yuri Tschinkel, Algebraic varieties over small fields, Diophantine geometry, CRM Series, vol. 4, Ed. Norm., Pisa, 2007, pp. 73โ91. MR 2349648 (2009c:14042)
- David Cox and Ron Donagi, On the failure of variational Torelli for regular elliptic surfaces with a section, Math. Ann. 273 (1986), no. 4, 673โ683. MR 826466 (87h:14028), DOI https://doi.org/10.1007/BF01472138
- James Carlson, Mark Green, Phillip Griffiths, and Joe Harris, Infinitesimal variations of Hodge structure. I, Compositio Math. 50 (1983), no. 2-3, 109โ205. MR 720288 (86e:32026a)
- David A. Cox, The Noether-Lefschetz locus of regular elliptic surfaces with section and $p_g\ge 2$, Amer. J. Math. 112 (1990), no. 2, 289โ329. MR 1047301, DOI https://doi.org/10.2307/2374717
- Igor Dolgachev, Weighted projective varieties, Group actions and vector fields (Vancouver, B.C., 1981) Lecture Notes in Math., vol. 956, Springer, Berlin, 1982, pp. 34โ71. MR 704986 (85g:14060), DOI https://doi.org/10.1007/BFb0101508
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157 (57 \#3116)
- Jรกnos Kollรกr, Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 32, Springer-Verlag, Berlin, 1996. MR 1440180 (98c:14001)
- Serge Lang, Hyperbolic and Diophantine analysis, Bull. Amer. Math. Soc. (N.S.) 14 (1986), no. 2, 159โ205. MR 828820, DOI https://doi.org/10.1090/S0273-0979-1986-15426-1
- Rick Miranda, The moduli of Weierstrass fibrations over $\textbf {P}^{1}$, Math. Ann. 255 (1981), no. 3, 379โ394. MR 0615858, DOI https://doi.org/10.1007/BF01450711
- F. Pakovich, Algebraic curves $P(x)-Q(y)=0$ and functional equations, Complex Var. Elliptic Equ. 56 (2011), no. 1-4, 199โ213. MR 2774592, DOI https://doi.org/10.1080/17476930903394838
- Tetsuji Shioda, An explicit algorithm for computing the Picard number of certain algebraic surfaces, Amer. J. Math. 108 (1986), no. 2, 415โ432. MR 0833362, DOI https://doi.org/10.2307/2374678
- Joseph H. Silverman, A bound for the Mordell-Weil rank of an elliptic surface after a cyclic base extension, J. Algebraic Geom. 9 (2000), no. 2, 301โ308. MR 1735774 (2001a:11107)
- Peter F. Stiller, The Picard numbers of elliptic surfaces with many symmetries, Pacific J. Math. 128 (1987), no. 1, 157โ189. MR 0883383
- Douglas Ulmer, $L$-functions with large analytic rank and abelian varieties with large algebraic rank over function fields, Invent. Math. 167 (2007), no. 2, 379โ408. MR 2270458 (2007k:11101), DOI https://doi.org/10.1007/s00222-006-0018-x
- Douglas Ulmer, Elliptic curves over function fields, Arithmetic of $L$-functions, IAS/Park City Math. Ser., vol. 18, Amer. Math. Soc., Providence, RI, 2011, pp. 211โ280. MR 2882692
- Douglas Ulmer, Explicit points on the Legendre curve, J. Number Theory 136 (2014), 165โ194. MR 3145329, DOI https://doi.org/10.1016/j.jnt.2013.09.010
- Claire Voisin, On a conjecture of Clemens on rational curves on hypersurfaces, J. Differential Geom. 44 (1996), no. 1, 200โ213. MR 1420353
Additional Information
Douglas Ulmer
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
MR Author ID:
175900
ORCID:
0000-0003-1529-4390
Email:
ulmer@math.gatech.edu
Received by editor(s):
August 4, 2014
Published electronically:
August 26, 2016
Article copyright:
© Copyright 2016
University Press, Inc.
