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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Approximating rational points on toric varieties


Authors: David McKinnon and Matthew Satriano
Journal: Trans. Amer. Math. Soc. 374 (2021), 3557-3577
MSC (2020): Primary 14G05; Secondary 11G50, 11J97
DOI: https://doi.org/10.1090/tran/8318
Published electronically: February 8, 2021
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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.


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Additional Information

David McKinnon
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
Email: dmckinnon@uwaterloo.ca

Matthew Satriano
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
Email: msatrian@uwaterloo.ca

DOI: https://doi.org/10.1090/tran/8318
Received by editor(s): May 19, 2020
Received by editor(s) in revised form: September 15, 2020
Published electronically: February 8, 2021
Additional Notes: The authors were partially supported by Discovery Grants from the Natural Sciences and Engineering Research Council
Article copyright: © Copyright 2021 American Mathematical Society