Irreducibility, BrillNoether loci, and Vojta's inequality
Authors:
Thomas J. Tucker and with an Appendix by Olivier Debarre
Journal:
Trans. Amer. Math. Soc. 354 (2002), 30113029
MSC (2000):
Primary 11G30, 11J68
Published electronically:
April 3, 2002
MathSciNet review:
1897388
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Abstract: This paper deals with generalizations of Hilbert's irreducibility theorem. The classical Hilbert irreducibility theorem states that for any cover of the projective line defined over a number field , there exist infinitely many rational points on the projective line such that the fiber of over is irreducible over . In this paper, we consider similar statements about algebraic points of higher degree on curves of any genus. We prove that Hilbert's irreducibility theorem admits a natural generalization to rational points on an elliptic curve and thus, via a theorem of Abramovich and Harris, to points of degree 3 or less on any curve. We also present examples that show that this generalization does not hold for points of degree 4 or more. These examples come from an earlier geometric construction of Debarre and Fahlaoui; some additional necessary facts about this construction can be found in the appendix provided by Debarre. We exhibit a connection between these irreducibility questions and the sharpness of Vojta's inequality for algebraic points on curves. In particular, we show that Vojta's inequality is not sharp for the algebraic points arising in our examples.
 [AH]
J.F.
Mestre, Familles de courbes hyperelliptiques à
multiplications réelles, Arithmetic algebraic geometry (Texel,
1989) Progr. Math., vol. 89, Birkhäuser Boston, Boston, MA,
1991, pp. 193–208 (French). MR 1085260
(92e:14022)
 [DF]
Debarre, O., Fahlaoui, R., Abelian varieties in , Comp. Math. 88 (1993), 235249.
 [L]
Robert
Lazarsfeld, BrillNoetherPetri without degenerations, J.
Differential Geom. 23 (1986), no. 3, 299–307.
MR 852158
(88b:14019)
 [R]
Igor
Reider, Vector bundles of rank 2 and linear systems on algebraic
surfaces, Ann. of Math. (2) 127 (1988), no. 2,
309–316. MR
932299 (89e:14038), http://dx.doi.org/10.2307/2007055
References to Appendix
 [AH]
 Abramovich, D., Harris, J., Abelian varieties and curves in , Comp. Math. 78 (1991), 227238. MR 92e:14022
 [DF]
 Debarre, O., Fahlaoui, R., Abelian varieties in , Comp. Math. 88 (1993), 235249.
 [L]
 Lazarsfeld, R., BrillNoetherPetri without Degenerations, J. Diff. Geom. 23 (1986), 299307. MR 88b:14019
 [R]
 Reider, I., Vector bundles of rank and linear systems on algebraic surfaces, Ann. of Math. 127 (1988), 309316. MR 89e:14038
References  [AH]
 D. Abramovich and J. Harris, Abelian varieties and curves in , Comp. Math. 78 (1991:2), 227238. MR 92c:14022
 [ACGH]
 E. Arbarello, M. Cornalba, P.A. Griffiths, and J. Harris, Geometry of algebraic curves I, SpringerVerlag, New York, 1985. MR 86h:14019
 [Ar]
 M. Artin, Lipman's proof of resolution of singularities for surfaces, in Arithmetic geometry (edited by G. Cornell and J. Silverman), SpringerVerlag, New York, 1986, pp. 267288. MR 89b:14029
 [Bo]
 E. Bombieri, Effective Diophantine Approximation on , Ann. Scuola Norm. Pisa Cl. Sci (4) 20 (1993), 6189. MR 94m:11086
 [Cr]
 J. E. Cremona, Algorithms for modular elliptic curves, Cambridge University Press, Cambridge, 1992  121B, 175B, 225A. MR 93m:11053
 [DF]
 O. Debarre and R. Fahlaoui, Abelian varieties in and points of bounded degree on algebraic curves, Comp. Math. 88 (1993:3), 235249. MR 94h:14028
 [Fa 1]
 G. Faltings, Diophantine approximation on abelian varieties, Ann. of Math. (2) 133 (1991:3), 549576. MR 93d:11066
 [Fa 2]
 G. Faltings, The general case of S. Lang's conjecture, in Christante, V. and Messing, W. (eds.), Barsotti symposium in algebraic geometry, Perspectives in Mathematics 15, Academic Press, San Diego, Calif., 1994 pp. 175182. MR 95m:11061
 [Frey]
 G. Frey, Curves with infinitely many points of fixed degree, Israel J. Math. 85 (1994), 7983. MR 94m:11072
 [Fu]
 W. Fulton, Intersection Theory, SpringerVerlag, Berlin, 1984. MR 85k:14004
 [Ha]
 R. Hartshorne, Algebraic geometry, SpringerVerlag, Graduate Texts in Mathematics, vol. 52, New York, 1977. MR 57:3116
 [Hi]
 D. Hilbert, Über die Irreduzibilität ganzer rationaler Funktionen mit ganzähligen Koeffzienten, J. Reine Angew. Math. 110 (1892), 104120.
 [Ka]
 Y. Kawamata, On Bloch's conjecture, Invent. Math. 57 (1980), 97100. MR 81j:32030
 [L 1]
 S. Lang, Fundamentals of diophantine geometry, SpringerVerlag, New York, 1983. MR 85j:11005
 [L 2]
 S. Lang, Introduction to Arakelov theory, SpringerVerlag, New York, 1988. MR 89m:11059
 [L 3]
 S. Lang, Algebra (3rd ed.), AddisonWesley, Reading, MA, 1993. MR 86j:00003 (2nd ed.)
 [Mer]
 L. Merel, Bornes pour la torsion des courbes elliptiques sur les corps de nombres, (French) [Bounds for the torsion of elliptic curves over number fields], Invent. Math. 124 (1996), 437449. MR 96i:11057
 [Mum]
 D. Mumford, Abelian varieties, Oxford University Press, London, 1970. MR 44:219
 [Sch]
 W.M. Schmidt, Diophantine approximation, Lecture Notes in Math. 785, SpringerVerlag, New York, 1980. MR 81j:11038
 [Si]
 J. Silverman, Rational points on symmetric products of a curve, Amer. J. Math. 113 (1991), 471508. MR 92m:11060
 [ST 1]
 X. Song and T. J. Tucker, Dirichlet's Theorem, Vojta's inequality, and Vojta's conjecture, Comp. Math. 116 (1999:2), 219238. MR 2000d:11085
 [ST 2]
 X. Song and T. J. Tucker, Arithmetic discriminants and morphisms of curves, Trans. AMS, 353 (2001), 19211936. CMP 2001:08
 [V 1]
 P. Vojta, Arithmetic discriminants and quadratic points on curves, in Arithmetic algebraic geometry (Texel, 1989), Progr. Math. 89, Birkhäuser Boston, Boston, MA, 1991, pp. 359376. MR 92j:11059
 [V 2]
 P. Vojta, A generalization of theorems of Faltings and ThueSiegelRothWirsing, J. Amer. Math. Soc. 5 (1992:4), 763804. MR 94a:11093
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Additional Information
Thomas J. Tucker
Affiliation:
Department of Mathematics, University of Georgia, Athens, Georgia 30602
Email:
ttucker@math.uga.edu
with an Appendix by Olivier Debarre
Affiliation:
IRMAMathématique, Université Louis Pasteur, 67084 Strasbourg Cedex, France
Email:
debarre@math.ustrasbg.fr
DOI:
http://dx.doi.org/10.1090/S0002994702029045
PII:
S 00029947(02)029045
Received by editor(s):
May 20, 2001
Published electronically:
April 3, 2002
Article copyright:
© Copyright 2002 American Mathematical Society
