Irreducibility, Brill-Noether loci, and Vojta's inequality

Authors:
Thomas J. Tucker and with an Appendix by Olivier Debarre

Journal:
Trans. Amer. Math. Soc. **354** (2002), 3011-3029

MSC (2000):
Primary 11G30, 11J68

DOI:
https://doi.org/10.1090/S0002-9947-02-02904-5

Published electronically:
April 3, 2002

MathSciNet review:
1897388

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Abstract | References | Similar Articles | Additional Information

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]**Abramovich, D., Harris, J.,*Abelian varieties and curves in*, Comp. Math.**78**(1991), 227-238. MR**92e:14022****[DF]**Debarre, O., Fahlaoui, R.,*Abelian varieties in*, Comp. Math.**88**(1993), 235-249.**[L]**Lazarsfeld, R.,*Brill-Noether-Petri without Degenerations*, J. Diff. Geom.**23**(1986), 299-307. MR**88b:14019****[R]**Reider, I.,*Vector bundles of rank and linear systems on algebraic surfaces*, Ann. of Math.**127**(1988), 309-316. MR**89e:14038**

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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:
IRMA-Mathématique, Université Louis Pasteur, 67084 Strasbourg Cedex, France

Email:
debarre@math.u-strasbg.fr

DOI:
https://doi.org/10.1090/S0002-9947-02-02904-5

Received by editor(s):
May 20, 2001

Published electronically:
April 3, 2002

Article copyright:
© Copyright 2002
American Mathematical Society