Approximating rational points on toric varieties

McKinnon, David; Satriano, Matthew

doi:10.1090/tran/8318

Approximating rational points on toric varieties
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by David McKinnon and Matthew Satriano PDF

Trans. Amer. Math. Soc. 374 (2021), 3557-3577 Request permission

Abstract:

Given a smooth projective variety $X$ over a number field $k$ and $P\in X(k)$, the first author conjectured that in a precise sense, any sequence that approximates $P$ sufficiently well must lie on a rational curve. We prove this conjecture for smooth split toric surfaces conditional on Vojta’s conjecture. More generally, we show that if $X$ is a $\mathbb {Q}$-factorial terminal split toric variety of arbitrary dimension, then $P$ is better approximated by points on a rational curve than by any Zariski dense sequence.

References

Buczyńska, W., Fake weighted projective spaces, English translation of Toryczne przestrzenie rzutowe, Magister thesis, https://arxiv.org/pdf/arXiv:0805.1211.pdf, 2002.
Castañeda Santos, D., Rational approximations on smooth rational surfaces, PhD thesis, University of Waterloo, 2019.
David A. Cox, John B. Little, and Henry K. Schenck, Toric varieties, Graduate Studies in Mathematics, vol. 124, American Mathematical Society, Providence, RI, 2011. MR 2810322, DOI 10.1090/gsm/124
F. J. Dyson, The approximation to algebraic numbers by rationals, Acta Math. 79 (1947), 225–240. MR 23854, DOI 10.1007/BF02404697
Osamu Fujino, Equivariant completions of toric contraction morphisms, Tohoku Math. J. (2) 58 (2006), no. 3, 303–321. With an appendix by Fujino and Hiroshi Sato. MR 2273272
William Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. MR 1234037, DOI 10.1515/9781400882526
Z. Huang, Rational approximations on toric varieties, https://arxiv.org/pdf/1809.02001.pdf, 2018.
Kasprzyk, A., Classifying terminal weighted projective space, https://arxiv.org/pdf/1304.3029.pdf, 2013.
János Kollár and Shigefumi Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti; Translated from the 1998 Japanese original. MR 1658959, DOI 10.1017/CBO9780511662560
Liouville, Nouvelle démonstration d’un théorème sur les irrationnelles algébriques, Comptes rendus hebdomadaires des séances de l’Académie des sciences, Tome XVIII, séance de 20 mai 1844, 910–911.
Kenji Matsuki, Introduction to the Mori program, Universitext, Springer-Verlag, New York, 2002. MR 1875410, DOI 10.1007/978-1-4757-5602-9
David McKinnon, A conjecture on rational approximations to rational points, J. Algebraic Geom. 16 (2007), no. 2, 257–303. MR 2274515, DOI 10.1090/S1056-3911-06-00458-9
David McKinnon and Mike Roth, An analogue of Liouville’s theorem and an application to cubic surfaces, Eur. J. Math. 2 (2016), no. 4, 929–959. MR 3572552, DOI 10.1007/s40879-016-0113-5
David McKinnon and Mike Roth, Seshadri constants, diophantine approximation, and Roth’s theorem for arbitrary varieties, Invent. Math. 200 (2015), no. 2, 513–583. MR 3338009, DOI 10.1007/s00222-014-0540-1
K. F. Roth, Rational approximations to algebraic numbers, Mathematika 2 (1955), 1–20; corrigendum, 168. MR 72182, DOI 10.1112/S0025579300000644
Carl Siegel, Approximation algebraischer Zahlen, Math. Z. 10 (1921), no. 3-4, 173–213 (German). MR 1544471, DOI 10.1007/BF01211608
Axel Thue, Über Annäherungswerte algebraischer Zahlen, J. Reine Angew. Math. 135 (1909), 284–305 (German). MR 1580770, DOI 10.1515/crll.1909.135.284
Paul Vojta, Diophantine approximations and value distribution theory, Lecture Notes in Mathematics, vol. 1239, Springer-Verlag, Berlin, 1987. MR 883451, DOI 10.1007/BFb0072989