Codimension two integral points on some rationally connected threefolds are potentially dense
Authors:
David McKinnon and Mike Roth
Journal:
J. Algebraic Geom. 31 (2022), 345-386
DOI:
https://doi.org/10.1090/jag/782
Published electronically:
December 20, 2021
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Abstract |
References |
Additional Information
Abstract: Let $X$ be a smooth, projective, rationally connected variety, defined over a number field $k$, and let $Z\subset X$ be a closed subset of codimension at least two. In this paper, for certain choices of $X$, we prove that the set of $Z$-integral points is potentially Zariski dense, in the sense that there is a finite extension $K$ of $k$ such that the set of points $P\in X(K)$ that are $Z$-integral is Zariski dense in $X$. This gives a positive answer to a question of Hassett and Tschinkel from 2001.
References
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References
- Arnaud Beauville, Surfaces algébriques complexes, Société Mathématique de France, Paris, 1978 (French). Avec une sommaire en anglais; Astérisque, No. 54. MR 0485887
- F. A. Bogomolov and Yu. Tschinkel, On the density of rational points on elliptic fibrations, J. Reine Angew. Math. 511 (1999), 87–93. MR 1695791, DOI 10.1515/crll.1999.511.87
- Enrico Bombieri and Walter Gubler, Heights in Diophantine geometry, New Mathematical Monographs, vol. 4, Cambridge University Press, Cambridge, 2006. MR 2216774, DOI 10.1017/CBO9780511542879
- A. Grothendieck, Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. II, Inst. Hautes Études Sci. Publ. Math. 17 (1963), 5–91 (French). MR 163911
- A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III, Inst. Hautes Études Sci. Publ. Math. 28 (1966), 5–255. MR 217086
- G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), no. 3, 349–366 (German). MR 718935, DOI 10.1007/BF01388432
- Tom Graber, Joe Harris, and Jason Starr, Families of rationally connected varieties, J. Amer. Math. Soc. 16 (2003), no. 1, 57–67. MR 1937199, DOI 10.1090/S0894-0347-02-00402-2
- Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
- Brendan Hassett and Yuri Tschinkel, Density of integral points on algebraic varieties, Rational points on algebraic varieties, Progr. Math., vol. 199, Birkhäuser, Basel, 2001, pp. 169–197. MR 1875174, DOI 10.1007/978-3-0348-8368-9_7
- V. A. Iskovskikh and Yu. G. Prokhorov, Algebraic geometry. V., Fano varieties. A translation of Algebraic geometry. 5 (Russian), Ross. Akad. Nauk, Vseross. Inst. Nauchn. i Tekhn. Inform., Moscow. Translation edited by A. N. Parshin and I. R. Shafarevich. Encyclopaedia of Mathematical Sciences, 47. Springer-Verlag, Berlin, 1999.
- János Kollár, Yoichi Miyaoka, and Shigefumi Mori, Rationally connected varieties, J. Algebraic Geom. 1 (1992), no. 3, 429–448. MR 1158625
- János Kollár, 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, DOI 10.1007/978-3-662-03276-3
- A. Levin, On the geometric and arithmetic puncturing problems, Preprint, 2020.
- D. McKinnon and Y. Zhu, The arithmetic puncturing problem and integral points, Preprint, 2018.
- Loïc Merel, Bornes pour la torsion des courbes elliptiques sur les corps de nombres, Invent. Math. 124 (1996), no. 1-3, 437–449 (French). MR 1369424, DOI 10.1007/s002220050059
- David Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, No. 5 Published for the Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, 1970. MR 0282985
- V. Shelestunova, Infinite sets of $D$-integral points on projective algebraic varieties, Master’s Thesis, University of Waterloo, 2005.
- C. L. Siegel, Über einige Anwendungen Diophantischer Approximationen, Abh. Preuss. Akad. Wiss. Phys. Math. Kl. (1929), 41–69.
- Paul Vojta, Diophantine approximations and value distribution theory, Lecture Notes in Mathematics, vol. 1239, Springer-Verlag, Berlin, 1987. MR 883451, DOI 10.1007/BFb0072989
Additional Information
David McKinnon
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
MR Author ID:
667698
Email:
dmckinnon@uwaterloo.ca
Mike Roth
Affiliation:
Department of Mathematics and Statistics, Queens University, Kingston, Ontario, Canada
MR Author ID:
717200
ORCID:
0000-0002-1182-9602
Email:
mike.roth@queensu.ca
Received by editor(s):
February 11, 2020
Received by editor(s) in revised form:
September 28, 2020, January 24, 2021, and February 2, 2021
Published electronically:
December 20, 2021
Additional Notes:
The first and second authors were partially supported by NSERC research grants.
Article copyright:
© Copyright 2021
University Press, Inc.
