Abstract
In this paper, we prove two results regarding the arithmetics of separably \(\mathbb {A}^1\)-connected varieties of rank one. First we prove that over a large field, there is an \(\mathbb {A}^1\)-curve through any rational point of the boundary, if the boundary divisor is smooth and separably rationally connected. Secondly, we generalize a theorem of Hassett–Tschinkel for the Zariski density of integral points over function fields of curves.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abramovich, D., Chen, Q.: Stable logarithmic maps to Deligne–Faltings pairs II. Asian J. Math. 18(3), 465–488 (2014)
Abramovich, D., Chen, Q., Gillam, D., Marcus, S.: The evaluation space of logarithmic stable maps (2010). arXiv:1012.5416v1 (preprint)
Abrmovich, D., Chen, Q., Marcus, S., Wise, J.: Boundedness of the space of stable log maps (2014). arXiv:1408.0869
Chen, Q.: Stable logarithmic maps to Deligne–Faltings pairs I. Ann. Math. (2). 180(2), 455–521 (2014)
Chen, Q., Zhu, Y.: \(\mathbb{A}^{1}\)-curves on log smooth varieties (2014). arXiv:1407.5476
Chen, Q., Zhu, Y.: Very free curves on Fano complete intersections. Algebr. Geom. 1(5), 558–572 (2014)
Graber, T., Harris, J., Starr, J.: Families of rationally connected varieties. J. Am. Math. Soc. 16(1), 57–67 (2003). (electronic)
Gross, M., Siebert, B.: Logarithmic Gromov–Witten invariants. J. Am. Math. Soc. 26(2), 451–510 (2013)
Hassett, B., Tschinkel, Y.: Weak approximation over function fields. Invent. Math. 163(1), 171–190 (2006)
Hassett, B., Tschinkel, Y.: Log Fano varieties over function fields of curves. Invent. Math. 173(1), 7–21 (2008)
Kato, K.: Logarithmic Structures of Fontaine–Illusie,Algebraic Analysis, Geometry, and Number Theory (Baltimore, MD, 1988), pp. 191–224. Johns Hopkins Univ. Press, Baltimore (1989)
Kim, B.: Logarithmic stable maps, new developments in algebraic geometry, integrable systems and mirror symmetry (RIMS, Kyoto, 2008). In: Adv. Stud. Pure Math., vol. 59, pp. 167–200. Math. Soc. Japan, Tokyo (2010)
Kollár, J., Miyaoka, Y., Mori, S.: Rationally connected varieties. J. Algebraic Geom. 1(3), 429–448 (1992)
Kollár, J.: Rational curves on algebraic varieties. In: 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, Berlin (1996)
Kollár, J.: Rationally connected varieties over local fields. Ann. Math. (2) 150(1), 357–367 (1999)
Olsson, M.C.: Logarithmic geometry and algebraic stacks. Ann. Sci. École Norm. Sup. (4) 36(5), 747–791 (2003)
Olsson, M.C.: Universal log structures on semi-stable varieties. Tohoku Math. J. (2) 55(3), 397–438 (2003)
Olsson, M.C.: The logarithmic cotangent complex. Math. Ann. 333(4), 859–931 (2005)
Rosen, M.: Number theory in function fields. In: Graduate Texts in Mathematics, vol. 210. Springer, New York (2002)
Wise, J.: Moduli of morphisms of logarithmic schemes (2014). arXiv:1408.0037
Acknowledgments
The authors would like to thank the anonymous referee for his/her detailed comments and suggestions on the manuscript.
Author information
Authors and Affiliations
Corresponding author
Additional information
Q. Chen is partially supported by NSF Grant DMS-1403271.
Rights and permissions
About this article
Cite this article
Chen, Q., Zhu, Y. \(\mathbb {A}^1\)-connected varieties of rank one over nonclosed fields. Math. Ann. 364, 1505–1515 (2016). https://doi.org/10.1007/s00208-015-1257-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-015-1257-1